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+The Project Gutenberg EBook of Pioneers of Science, by Oliver Lodge
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Pioneers of Science
+
+Author: Oliver Lodge
+
+Release Date: April 26, 2009 [EBook #28613]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK PIONEERS OF SCIENCE ***
+
+
+
+
+Produced by Audrey Longhurst, Greg Bergquist and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by The Internet Archive/American Libraries.)
+
+
+
+
+
+
+Transcriber's Note
+
+The punctuation and spelling from the original text have been faithfully
+preserved. Only obvious typographical errors have been corrected.
+
+There are several mathematical formulas within the text. They are
+represented as follows:
+ Superscripts: x^3
+ Subscripts: x_3
+ Square Root: [square root] Greek Letters: [pi], [theta].
+
+Greek star names are represented as [alpha], [gamma], for example.
+
+
+
+
+PIONEERS OF SCIENCE
+
+[Illustration]
+
+[Illustration: NEWTON
+
+_From the picture by Kneller, 1689, now at Cambridge_]
+
+
+
+
+  PIONEERS OF SCIENCE
+
+  BY
+  OLIVER LODGE, F.R.S.
+
+
+  PROFESSOR OF PHYSICS IN VICTORIA UNIVERSITY COLLEGE, LIVERPOOL
+
+  _WITH PORTRAITS AND OTHER ILLUSTRATIONS_
+
+
+  London
+  MACMILLAN AND CO.
+  AND NEW YORK
+  1893
+
+  RICHARD CLAY AND SONS, LIMITED,
+  LONDON AND BUNGAY.
+
+
+
+
+PREFACE
+
+
+This book takes its origin in a course of lectures on the history and
+progress of Astronomy arranged for me in the year 1887 by three of my
+colleagues (A.C.B., J.M., G.H.R.), one of whom gave the course its name.
+
+The lectures having been found interesting, it was natural to write them
+out in full and publish.
+
+If I may claim for them any merit, I should say it consists in their
+simple statement and explanation of scientific facts and laws. The
+biographical details are compiled from all readily available sources,
+there is no novelty or originality about them; though it is hoped that
+there may be some vividness. I have simply tried to present a living
+figure of each Pioneer in turn, and to trace his influence on the
+progress of thought.
+
+I am indebted to many biographers and writers, among others to Mr.
+E.J.C. Morton, whose excellent set of lives published by the S.P.C.K.
+saved me much trouble in the early part of the course.
+
+As we approach recent times the subject grows more complex, and the men
+more nearly contemporaries; hence the biographical aspect diminishes and
+the scientific treatment becomes fuller, but in no case has it been
+allowed to become technical and generally unreadable.
+
+To the friends (C.C.C., F.W.H.M., E.F.R.) who with great kindness have
+revised the proofs, and have indicated places where the facts could be
+made more readily intelligible by a clearer statement, I express my
+genuine gratitude.
+
+  UNIVERSITY COLLEGE, LIVERPOOL,
+  _November, 1892_.
+
+
+
+
+CONTENTS
+
+
+  _PART I_
+
+  LECTURE I
+
+                                                   PAGE
+
+  COPERNICUS AND THE MOTION OF THE EARTH              2
+
+
+  LECTURE II
+
+  TYCHO BRAH� AND THE EARLIEST OBSERVATORY           32
+
+
+  LECTURE III
+
+  KEPLER AND THE LAWS OF PLANETARY MOTION            56
+
+
+  LECTURE IV
+
+  GALILEO AND THE INVENTION OF THE TELESCOPE         80
+
+
+  LECTURE V
+
+  GALILEO AND THE INQUISITION                       108
+
+
+  LECTURE VI
+
+  DESCARTES AND HIS THEORY OF VORTICES              136
+
+
+  LECTURE VII
+
+  SIR ISAAC NEWTON                                  159
+
+
+  LECTURE VIII
+
+  NEWTON AND THE LAW OF GRAVITATION                 180
+
+
+  LECTURE IX
+
+  NEWTON'S "PRINCIPIA"                              203
+
+
+  _PART II_
+
+  LECTURE X
+
+  ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT      232
+
+
+  LECTURE XI
+
+  LAGRANGE AND LAPLACE--THE STABILITY OF THE SOLAR SYSTEM,
+  AND THE NEBULAR HYPOTHESIS                        254
+
+
+  LECTURE XII
+
+  HERSCHEL AND THE MOTION OF THE FIXED STARS        273
+
+
+  LECTURE XIII
+
+  THE DISCOVERY OF THE ASTEROIDS                    294
+
+
+  LECTURE XIV
+
+  BESSEL--THE DISTANCES OF THE STARS, AND THE DISCOVERY OF
+  STELLAR PLANETS                                   304
+
+
+  LECTURE XV
+
+  THE DISCOVERY OF NEPTUNE                          317
+
+
+  LECTURE XVI
+
+  COMETS AND METEORS                                331
+
+
+  LECTURE XVII
+
+  THE TIDES                                         353
+
+
+  LECTURE XVIII
+
+  THE TIDES, AND PLANETARY EVOLUTION                379
+
+
+
+
+ILLUSTRATIONS
+
+
+  FIG.                                                                PAGE
+
+  1. ARCHIMEDES                                                          8
+
+  2. LEONARDO DA VINCI                                                  10
+
+  3. COPERNICUS                                                         12
+
+  4. HOMERIC COSMOGONY                                                  15
+
+  5. EGYPTIAN SYMBOL OF THE UNIVERSE                                    16
+
+  6. HINDOO EARTH                                                       17
+
+  7. ORDER OF ANCIENT PLANETS CORRESPONDING TO THE DAYS OF
+     THE WEEK                                                           19
+
+  8. PTOLEMAIC SYSTEM                                                   20
+
+  9. SPECIMENS OF APPARENT PATHS OF VENUS AND OF MARS
+     AMONG THE STARS                                                    21
+
+  10. APPARENT EPICYCLIC ORBITS OF JUPITER AND SATURN                   22
+
+  11. EGYPTIAN SYSTEM                                                   24
+
+  12. TRUE ORBITS OF EARTH AND JUPITER                                  25
+
+  13. ORBITS OF MERCURY AND EARTH                                       25
+
+  14. COPERNICAN SYSTEM AS FREQUENTLY REPRESENTED                       26
+
+  15. SLOW MOVEMENT OF THE NORTH POLE IN A CIRCLE AMONG
+      THE STARS                                                         29
+
+  16. TYCHONIC SYSTEM, SHOWING THE SUN WITH ALL THE PLANETS
+      REVOLVING ROUND THE EARTH                                         38
+
+  17. PORTRAIT OF TYCHO                                                 41
+
+  18. EARLY OUT-DOOR QUADRANT OF TYCHO                                  43
+
+  19. MAP OF DENMARK, SHOWING THE ISLAND OF HUEN                        45
+
+  20. URANIBURG                                                         46
+
+  21. ASTROLABE                                                         47
+
+  22. TYCHO'S LARGE SEXTANT                                             48
+
+  23. THE QUADRANT IN URANIBURG                                         49
+
+  24. TYCHO'S FORM OF TRANSIT CIRCLE                                    50
+
+  25. A MODERN TRANSIT CIRCLE                                           51
+
+  26. ORBITS OF SOME OF THE PLANETS DRAWN TO SCALE                      60
+
+  27. MANY-SIDED POLYGON OR APPROXIMATE CIRCLE ENVELOPED
+      BY STRAIGHT LINES                                                 61
+
+  28. KEPLER'S IDEA OF THE REGULAR SOLIDS                               62
+
+  29. DIAGRAM OF EQUANT                                                 67
+
+  30. EXCENTRIC CIRCLE SUPPOSED TO BE DIVIDED INTO EQUAL AREAS          68
+
+  31. MODE OF DRAWING AN ELLIPSE                                        70
+
+  32. KEPLER'S DIAGRAM PROVING EQUABLE DESCRIPTION OF AREAS
+      FOR AN ELLIPSE                                                    71
+
+  33. DIAGRAM OF A PLANET'S VELOCITY IN DIFFERENT PARTS OF ITS ORBIT    72
+
+  34. PORTRAIT OF KEPLER                                                76
+
+  35. CURVE DESCRIBED BY A PROJECTILE                                   82
+
+  36. TWO FORMS OF PULSILOGY                                            87
+
+  37. TOWER OF PISA                                                     91
+
+  38. VIEW OF THE HALF-MOON IN SMALL TELESCOPE                          97
+
+  39. PORTION OF THE LUNAR SURFACE MORE HIGHLY MAGNIFIED                98
+
+  40. ANOTHER PORTION OF THE LUNAR SURFACE                              99
+
+  41. LUNAR LANDSCAPE SHOWING EARTH                                    100
+
+  42. GALILEO'S METHOD OF ESTIMATING THE HEIGHT OF LUNAR MOUNTAIN      101
+
+  43. SOME CLUSTERS AND NEBUL�                                         102
+
+  44. STAGES OF THE DISCOVERY OF JUPITER'S SATELLITES                  103
+
+  45. ECLIPSES OF JUPITER'S SATELLITES                                 105
+
+  46. OLD DRAWINGS OF SATURN BY DIFFERENT OBSERVERS, WITH
+      THE IMPERFECT INSTRUMENTS OF THAT DAY                            111
+
+  47. PHASES OF VENUS                                                  112
+
+  48. SUNSPOTS AS SEEN WITH LOW POWER                                  113
+
+  49. A PORTION OF THE SUN'S DISK AS SEEN IN A POWERFUL MODERN
+      TELESCOPE                                                        114
+
+  50. SATURN AND HIS RINGS                                             115
+
+  51. MAP OF ITALY                                                     118
+
+  52. PORTRAIT OF GALILEO                                              126
+
+  53. PORTRAIT OF DESCARTES                                            148
+
+  54. DESCARTES'S EYE DIAGRAM                                          151
+
+  55. DESCARTES'S DIAGRAM OF VORTICES FROM HIS "PRINCIPIA"             152
+
+  56. MANOR-HOUSE OF WOOLSTHORPE                                       162
+
+  57. PROJECTILE DIAGRAM                                               170
+
+  58.   }                                                            { 171
+  59.   } DIAGRAMS ILLUSTRATIVE OF THOSE NEAR THE BEGINNING          { 174
+  60.   } OF NEWTON'S "PRINCIPIA"                                    { 175
+  61-2. }                                                            { 175
+
+  63. PRISMATIC DISPERSION                                             182
+
+  64. A SINGLE CONSTITUENT OF WHITE LIGHT IS CAPABLE OF NO
+      MORE DISPERSION                                                  183
+
+  65. PARALLEL BEAM PASSING THROUGH A LENS                             184
+
+  66. NEWTON'S TELESCOPE                                               186
+
+  67. THE SEXTANT, AS NOW MADE                                         187
+
+  68. NEWTON WHEN YOUNG                                                196
+
+  69. SIR ISAAC NEWTON                                                 200
+
+  70. ANOTHER "PRINCIPIA" DIAGRAM                                      207
+
+  71. WELL-KNOWN MODEL EXHIBITING THE OBLATE SPHEROIDAL
+      FORM AS A CONSEQUENCE OF SPINNING ABOUT A CENTRAL
+      AXIS                                                             219
+
+  72. JUPITER                                                          221
+
+  73. DIAGRAM OF EYE LOOKING AT A LIGHT REFLECTED IN A DISTANT
+      MIRROR THROUGH THE TEETH OF A REVOLVING WHEEL                    238
+
+  74. FIZEAU'S WHEEL, SHOWING THE APPEARANCE OF DISTANT
+      IMAGE SEEN THROUGH ITS TEETH                                     239
+
+  75. ECLIPSES OF ONE OF JUPITER'S SATELLITES                          241
+
+  76. A TRANSIT INSTRUMENT FOR THE BRITISH ASTRONOMICAL EXPEDITION,
+      1874                                                             243
+
+  77. DIAGRAM OF EQUATORIALLY MOUNTED TELESCOPE                        245
+
+  78. ABERRATION DIAGRAM                                               250
+
+  79. SHOWING THE THREE CONJUNCTION PLACES IN THE ORBITS OF
+      JUPITER AND SATURN                                               259
+
+  80. LORD ROSSE'S DRAWING OF THE SPIRAL NEBULA IN CANES
+      VENATICI                                                         269
+
+  81. SATURN                                                           271
+
+  82. PRINCIPLE OF NEWTONIAN REFLECTOR                                 278
+
+  83. HERSCHEL'S 40-FOOT TELESCOPE                                     283
+
+  84. WILLIAM HERSCHEL                                                 285
+
+  85. CAROLINE HERSCHEL                                                287
+
+  86. DOUBLE STARS                                                     288
+
+  87. OLD DRAWING OF THE CLUSTER IN HERCULES                           290
+
+  88. OLD DRAWING OF THE ANDROMEDA NEBULA                              291
+
+  89. THE GREAT NEBULA IN ORION                                        292
+
+  90. PLANETARY ORBITS TO SCALE                                        297
+
+  91. DIAGRAM ILLUSTRATING PARALLAX                                    307
+
+  92. THE K�NIGSBERG HELIOMETER                                        312
+
+  93. PERTURBATIONS OF URANUS                                          320
+
+  94. URANUS' AND NEPTUNE'S RELATIVE POSITIONS                         325
+
+  95. METEORITE                                                        333
+
+  96. METEOR STREAM CROSSING FIELD OF TELESCOPE                        334
+
+  97. DIAGRAM OF DIRECTION OF EARTH'S ORBITAL MOTION                   335
+
+  98. PARABOLIC AND ELLIPTIC ORBITS                                    340
+
+  99. ORBIT OF HALLEY'S COMET                                          341
+
+  100. VARIOUS APPEARANCES OF HALLEY'S COMET WHEN LAST SEEN            342
+
+  101. HEAD OF DONATI'S COMET OF 1858                                  343
+
+  102. COMET                                                           344
+
+  103. ENCKE'S COMET                                                   345
+
+  104. BIELA'S COMET AS LAST SEEN IN TWO PORTIONS                      346
+
+  105. RADIANT POINT PERSPECTIVE                                       348
+
+  106. PRESENT ORBIT OF NOVEMBER METEORS                               349
+
+  107. ORBIT OF NOVEMBER METEORS BEFORE AND AFTER ENCOUNTER
+       WITH URANUS                                                     351
+
+  108. THE MERSEY                                                      355
+
+  109. CO-TIDAL LINES, SHOWING THE WAY THE TIDAL WAVE
+       REACHES THE BRITISH ISLES FROM THE ATLANTIC                     359
+
+  110. WHIRLING EARTH MODEL                                            364
+
+  111. EARTH AND MOON MODEL                                            365
+
+  112. EARTH AND MOON (EARTH'S ROTATION NEGLECTED)                     366
+
+  113. MAPS SHOWING HOW COMPARATIVELY FREE FROM LAND OBSTRUCTION
+       THE OCEAN IN THE SOUTHERN HEMISPHERE IS                         369
+
+  114. SPRING AND NEAP TIDES                                           370
+
+  115. TIDAL CLOCK                                                     371
+
+  116. SIR WILLIAM THOMSON (LORD KELVIN)                               373
+
+  117. TIDE-GAUGE FOR RECORDING LOCAL TIDES                            375
+
+  118. HARMONIC ANALYZER                                               375
+
+  119. TIDE-PREDICTER                                                  376
+
+  120. WEEKLY SHEET OF CURVES                                          377
+
+
+
+
+PIONEERS OF SCIENCE
+
+
+
+
+PART I
+
+_FROM DUSK TO DAYLIGHT_
+
+
+
+
+DATES AND SUMMARY OF FACTS FOR LECTURE I
+
+
+_Physical Science of the Ancients._ Thales 640 B.C., Anaximander 610
+B.C., PYTHAGORAS 600 B.C., Anaxagoras 500 B.C., Eudoxus 400 B.C.,
+ARISTOTLE 384 B.C., Aristarchus 300 B.C., ARCHIMEDES 287 B.C.,
+Eratosthenes 276 B.C., HIPPARCHUS 160 B.C., Ptolemy 100 A.D.
+
+_Science of the Middle Ages._ Cultivated only among the Arabs; largely
+in the forms of astrology, alchemy, and algebra.
+
+_Return of Science to Europe._ Roger Bacon 1240, Leonardo da Vinci 1480,
+(Printing 1455), Columbus 1492, Copernicus 1543.
+
+_A sketch of Copernik's life and work._ Born 1473 at Thorn in Poland.
+Studied mathematics at Bologna. Became an ecclesiastic. Lived at
+Frauenburg near mouth of Vistula. Substituted for the apparent motion of
+the heavens the real motion of the earth. Published tables of planetary
+motions. Motion still supposed to be in epicycles. Worked out his ideas
+for 36 years, and finally dedicated his work to the Pope. Died just as
+his book was printed, aged 72, a century before the birth of Newton. A
+colossal statue by Thorwaldsen erected at Warsaw in 1830.
+
+
+
+
+PIONEERS OF SCIENCE
+
+
+
+
+LECTURE I
+
+COPERNICUS AND THE MOTION OF THE EARTH
+
+
+The ordinary run of men live among phenomena of which they know nothing
+and care less. They see bodies fall to the earth, they hear sounds, they
+kindle fires, they see the heavens roll above them, but of the causes
+and inner working of the whole they are ignorant, and with their
+ignorance they are content.
+
+"Understand the structure of a soap-bubble?" said a cultivated literary
+man whom I know; "I wouldn't cross the street to know it!"
+
+And if this is a prevalent attitude now, what must have been the
+attitude in ancient times, when mankind was emerging from savagery, and
+when history seems composed of harassments by wars abroad and
+revolutions at home? In the most violently disturbed times indeed, those
+with which ordinary history is mainly occupied, science is quite
+impossible. It needs as its condition, in order to flourish, a fairly
+quiet, untroubled state, or else a cloister or university removed from
+the din and bustle of the political and commercial world. In such places
+it has taken its rise, and in such peaceful places and quiet times true
+science will continue to be cultivated.
+
+The great bulk of mankind must always remain, I suppose, more or less
+careless of scientific research and scientific result, except in so far
+as it affects their modes of locomotion, their health and pleasure, or
+their purse.
+
+But among a people hurried and busy and preoccupied, some in the pursuit
+of riches, some in the pursuit of pleasure, and some, the majority, in
+the struggle for existence, there arise in every generation, here and
+there, one or two great souls--men who seem of another age and country,
+who look upon the bustle and feverish activity and are not infected by
+it, who watch others achieving prizes of riches and pleasure and are not
+disturbed, who look on the world and the universe they are born in with
+quite other eyes. To them it appears not as a bazaar to buy and to sell
+in; not as a ladder to scramble up (or down) helter-skelter without
+knowing whither or why; but as a fact--a great and mysterious fact--to
+be pondered over, studied, and perchance in some small measure
+understood. By the multitude these men were sneered at as eccentric or
+feared as supernatural. Their calm, clear, contemplative attitude seemed
+either insane or diabolic; and accordingly they have been pitied as
+enthusiasts or killed as blasphemers. One of these great souls may have
+been a prophet or preacher, and have called to his generation to bethink
+them of why and what they were, to struggle less and meditate more, to
+search for things of true value and not for dross. Another has been a
+poet or musician, and has uttered in words or in song thoughts dimly
+possible to many men, but by them unutterable and left inarticulate.
+Another has been influenced still more _directly_ by the universe around
+him, has felt at times overpowered by the mystery and solemnity of it
+all, and has been impelled by a force stronger than himself to study it,
+patiently, slowly, diligently; content if he could gather a few crumbs
+of the great harvest of knowledge, happy if he could grasp some great
+generalization or wide-embracing law, and so in some small measure enter
+into the mind and thought of the Designer of all this wondrous frame of
+things.
+
+These last have been the men of science, the great and heaven-born men
+of science; and they are few. In our own day, amid the throng of
+inventions, there are a multitude of small men using the name of science
+but working for their own ends, jostling and scrambling just as they
+would jostle and scramble in any other trade or profession. These may be
+workers, they may and do advance knowledge, but they are never pioneers.
+Not to them is it given to open out great tracts of unexplored
+territory, or to view the promised land as from a mountain-top. Of them
+we shall not speak; we will concern ourselves only with the greatest,
+the epoch-making men, to whose life and work we and all who come after
+them owe so much. Such a man was Thales. Such was Archimedes,
+Hipparchus, Copernicus. Such pre-eminently was Newton.
+
+Now I am not going to attempt a history of science. Such a work in ten
+lectures would be absurd. I intend to pick out a few salient names here
+and there, and to study these in some detail, rather than by attempting
+to deal with too many to lose individuality and distinctness.
+
+We know so little of the great names of antiquity, that they are for
+this purpose scarcely suitable. In some departments the science of the
+Greeks was remarkable, though it is completely overshadowed by their
+philosophy; yet it was largely based on what has proved to be a wrong
+method of procedure, viz the introspective and conjectural, rather than
+the inductive and experimental methods. They investigated Nature by
+studying their own minds, by considering the meanings of words, rather
+than by studying things and recording phenomena. This wrong (though by
+no means, on the face of it, absurd) method was not pursued exclusively,
+else would their science have been valueless, but the influence it had
+was such as materially to detract from the value of their speculations
+and discoveries. For when truth and falsehood are inextricably woven
+into a statement, the truth is as hopelessly hidden as if it had never
+been stated, for we have no criterion to distinguish the false from the
+true.
+
+[Illustration: FIG. 1.--Archimedes.]
+
+Besides this, however, many of their discoveries were ultimately lost to
+the world, some, as at Alexandria, by fire--the bigoted work of a
+Mohammedan conqueror--some by irruption of barbarians; and all were
+buried so long and so completely by the night of the dark ages, that
+they had to be rediscovered almost as absolutely and completely as
+though they had never been. Some of the names of antiquity we shall have
+occasion to refer to; so I have arranged some of them in chronological
+order on page 4, and as a representative one I may specially emphasize
+Archimedes, one of the greatest men of science there has ever been, and
+the father of physics.
+
+The only effective link between the old and the new science is afforded
+by the Arabs. The dark ages come as an utter gap in the scientific
+history of Europe, and for more than a thousand years there was not a
+scientific man of note except in Arabia; and with the Arabs knowledge
+was so mixed up with magic and enchantment that one cannot contemplate
+it with any degree of satisfaction, and little real progress was made.
+In some of the _Waverley Novels_ you can realize the state of matters in
+these times; and you know how the only approach to science is through
+some Arab sorcerer or astrologer, maintained usually by a monarch, and
+consulted upon all great occasions, as the oracles were of old.
+
+In the thirteenth century, however, a really great scientific man
+appeared, who may be said to herald the dawn of modern science in
+Europe. This man was Roger Bacon. He cannot be said to do more than
+herald it, however, for we must wait two hundred years for the next name
+of great magnitude; moreover he was isolated, and so far in advance of
+his time that he left no followers. His own work suffered from the
+prevailing ignorance, for he was persecuted and imprisoned, not for the
+commonplace and natural reason that he frightened the Church, but merely
+because he was eccentric in his habits and knew too much.
+
+The man I spoke of as coming two hundred years later is Leonardo da
+Vinci. True he is best known as an artist, but if you read his works you
+will come to the conclusion that he was the most scientific artist who
+ever lived. He teaches the laws of perspective (then new), of light and
+shade, of colour, of the equilibrium of bodies, and of a multitude of
+other matters where science touches on art--not always quite correctly
+according to modern ideas, but in beautiful and precise language. For
+clear and conscious power, for wide-embracing knowledge and skill,
+Leonardo is one of the most remarkable men that ever lived.
+
+About this time the tremendous invention of printing was achieved, and
+Columbus unwittingly discovered the New World. The middle of the next
+century must be taken as the real dawn of modern science; for the year
+1543 marks the publication of the life-work of Copernicus.
+
+[Illustration: FIG. 2.--Leonardo da Vinci.]
+
+Nicolas Copernik was his proper name. Copernicus is merely the Latinized
+form of it, according to the then prevailing fashion. He was born at
+Thorn, in Polish Prussia, in 1473. His father is believed to have been a
+German. He graduated at Cracow as doctor in arts and medicine, and was
+destined for the ecclesiastical profession. The details of his life are
+few; it seems to have been quiet and uneventful, and we know very little
+about it. He was instructed in astronomy at Cracow, and learnt
+mathematics at Bologna. Thence he went to Rome, where he was made
+Professor of Mathematics; and soon afterwards he went into orders. On
+his return home, he took charge of the principal church in his native
+place, and became a canon. At Frauenburg, near the mouth of the Vistula,
+he lived the remainder of his life. We find him reporting on coinage for
+the Government, but otherwise he does not appear as having entered into
+the life of the times.
+
+He was a quiet, scholarly monk of studious habits, and with a reputation
+which drew to him several earnest students, who received _viv� voce_
+instruction from him; so, in study and meditation, his life passed.
+
+He compiled tables of the planetary motions which were far more correct
+than any which had hitherto appeared, and which remained serviceable for
+long afterwards. The Ptolemaic system of the heavens, which had been the
+orthodox system all through the Christian era, he endeavoured to improve
+and simplify by the hypothesis that the sun was the centre of the system
+instead of the earth; and the first consequences of this change he
+worked out for many years, producing in the end a great book: his one
+life-work. This famous work, "De Revolutionibus Orbium Coelestium,"
+embodied all his painstaking calculations, applied his new system to
+each of the bodies in the solar system in succession, and treated
+besides of much other recondite matter. Towards the close of his life it
+was put into type. He can scarcely be said to have lived to see it
+appear, for he was stricken with paralysis before its completion; but a
+printed copy was brought to his bedside and put into his hands, so that
+he might just feel it before he died.
+
+[Illustration: FIG. 3.--Copernicus.]
+
+That Copernicus was a giant in intellect or power--such as had lived in
+the past, and were destined to live in the near future--I see no reason
+whatever to believe. He was just a quiet, earnest, patient, and
+God-fearing man, a deep student, an unbiassed thinker, although with no
+specially brilliant or striking gifts; yet to him it was given to effect
+such a revolution in the whole course of man's thoughts as is difficult
+to parallel.
+
+You know what the outcome of his work was. It proved--he did not merely
+speculate, he proved--that the earth is a planet like the others, and
+that it revolves round the sun.
+
+Yes, it can be summed up in a sentence, but what a revelation it
+contains. If you have never made an effort to grasp the full
+significance of this discovery you will not appreciate it. The doctrine
+is very familiar to us now, we have heard it, I suppose, since we were
+four years old, but can you realize it? I know it was a long time before
+I could. Think of the solid earth, with trees and houses, cities and
+countries, mountains and seas--think of the vast tracts of land in Asia,
+Africa, and America--and then picture the whole mass spinning like a
+top, and rushing along its annual course round the sun at the rate of
+nineteen miles every second.
+
+Were we not accustomed to it, the idea would be staggering. No wonder it
+was received with incredulity. But the difficulties of the conception
+are not only physical, they are still more felt from the speculative and
+theological points of view. With this last, indeed, the reconcilement
+cannot be considered complete even yet. Theologians do not, indeed, now
+_deny_ the fact of the earth's subordination in the scheme of the
+universe, but many of them ignore it and pass it by. So soon as the
+Church awoke to a perception of the tremendous and revolutionary import
+of the new doctrines, it was bound to resist them or be false to its
+traditions. For the whole tenor of men's thought must have been changed
+had they accepted it. If the earth were not the central and
+all-important body in the universe, if the sun and planets and stars
+were not attendant and subsidiary lights, but were other worlds larger
+and perhaps superior to ours, where was man's place in the universe?
+and where were the doctrines they had maintained as irrefragable? I by
+no means assert that the new doctrines were really utterly
+irreconcilable with the more essential parts of the old dogmas, if only
+theologians had had patience and genius enough to consider the matter
+calmly. I suppose that in that case they might have reached the amount
+of reconciliation at present attained, and not only have left scientific
+truth in peace to spread as it could, but might perhaps themselves have
+joined the band of earnest students and workers, as so many of the
+higher Catholic clergy do at the present day.
+
+But this was too much to expect. Such a revelation was not to be
+accepted in a day or in a century--the easiest plan was to treat it as a
+heresy, and try to crush it out.
+
+Not in Copernik's life, however, did they perceive the dangerous
+tendency of the doctrine--partly because it was buried in a ponderous
+and learned treatise not likely to be easily understood; partly,
+perhaps, because its propounder was himself an ecclesiastic; mainly
+because he was a patient and judicious man, not given to loud or
+intolerant assertion, but content to state his views in quiet
+conversation, and to let them gently spread for thirty years before he
+published them. And, when he did publish them, he used the happy device
+of dedicating his great book to the Pope, and a cardinal bore the
+expense of printing it. Thus did the Roman Church stand sponsor to a
+system of truth against which it was destined in the next century to
+hurl its anathemas, and to inflict on its conspicuous adherents torture,
+imprisonment, and death.
+
+To realize the change of thought, the utterly new view of the universe,
+which the Copernican theory introduced, we must go back to preceding
+ages, and try to recall the views which had been held as probable
+concerning the form of the earth and the motion of the heavenly bodies.
+
+[Illustration: FIG. 4.--Homeric Cosmogony.]
+
+The earliest recorded notion of the earth is the very natural one that
+it is a flat area floating in an illimitable ocean. The sun was a god
+who drove his chariot across the heavens once a day; and Anaxagoras was
+threatened with death and punished with banishment for teaching that the
+sun was only a ball of fire, and that it might perhaps be as big as the
+country of Greece. The obvious difficulty as to how the sun got back to
+the east again every morning was got over--not by the conjecture that he
+went back in the dark, nor by the idea that there was a fresh sun every
+day; though, indeed, it was once believed that the moon was created once
+a month, and periodically cut up into stars--but by the doctrine that in
+the northern part of the earth was a high range of mountains, and that
+the sun travelled round on the surface of the sea behind these.
+Sometimes, indeed, you find a representation of the sun being rowed
+round in a boat. Later on it was perceived to be necessary that the sun
+should be able to travel beneath the earth, and so the earth was
+supposed to be supported on pillars or on roots, or to be a dome-shaped
+body floating in air--much like Dean Swift's island of Laputa. The
+elephant and tortoise of the Hindu earth are, no doubt, emblematic or
+typical, not literal.
+
+[Illustration: FIG. 5.--Egyptian Symbol of the Universe.
+
+The earth a figure with leaves, the heaven a figure with stars, the
+principle of equilibrium and support, the boats of the rising and
+setting sun.]
+
+Aristotle, however, taught that the earth must be a sphere, and used all
+the orthodox arguments of the present children's geography-books about
+the way you see ships at sea, and about lunar eclipses.
+
+To imagine a possible antipodes must, however, have been a tremendous
+difficulty in the way of this conception of a sphere, and I scarcely
+suppose that any one can at that time have contemplated the possibility
+of such upside-down regions being inhabited. I find that intelligent
+children invariably feel the greatest difficulty in realizing the
+existence of inhabitants on the opposite side of the earth. Stupid
+children, like stupid persons in general, will of course believe
+anything they are told, and much good may the belief do them; but the
+kind of difficulties felt by intelligent and thoughtful children are
+most instructive, since it is quite certain that the early philosophers
+must have encountered and overcome those very same difficulties by their
+own genius.
+
+[Illustration: FIG. 6.--Hindoo Earth.]
+
+However, somehow or other the conception of a spherical earth was
+gradually grasped, and the heavenly bodies were perceived all to revolve
+round it: some moving regularly, as the stars, all fixed together into
+one spherical shell or firmament; some moving irregularly and apparently
+anomalously--these irregular bodies were therefore called planets [or
+wanderers]. Seven of them were known, viz. Moon, Mercury, Venus, Sun,
+Mars, Jupiter, Saturn, and there is little doubt that this number seven,
+so suggested, is the origin of the seven days of the week.
+
+     The above order of the ancient planets is that of their supposed
+     distance from the earth. Not always, however, are they thus quoted
+     by the ancients: sometimes the sun is supposed nearer than Mercury
+     or Venus. It has always been known that the moon was the nearest of
+     the heavenly bodies; and some rough notion of its distance was
+     current. Mars, Jupiter, and Saturn were placed in that order
+     because that is the order of their apparent motions, and it was
+     natural to suppose that the slowest moving bodies were the furthest
+     off.
+
+     The order of the days of the week shows what astrologers considered
+     to be the order of the planets; on their system of each successive
+     hour of the day being ruled over by the successive planets taken in
+     order. The diagram (fig. 7) shows that if the Sun rule the first
+     hour of a certain day (thereby giving its name to the day) Venus
+     will rule the second hour, Mercury the third, and so on; the Sun
+     will thus be found to rule the eighth, fifteenth, and twenty-second
+     hour of that day, Venus the twenty-third, and Mercury the
+     twenty-fourth hour; so the Moon will rule the first hour of the
+     next day, which will therefore be Monday. On the same principle
+     (numbering round the hours successively, with the arrows) the first
+     hour of the next day will be found to be ruled by Mars, or by the
+     Saxon deity corresponding thereto; the first hour of the day after,
+     by Mercury (_Mercredi_), and so on (following the straight lines of
+     the pattern).
+
+     The order of the planets round the circle counter-clockwise, _i.e._
+     the direction of their proper motions, is that quoted above in the
+     text.
+
+To explain the motion of the planets and reduce them to any sort of law
+was a work of tremendous difficulty. The greatest astronomer of ancient
+times was Hipparchus, and to him the system known as the Ptolemaic
+system is no doubt largely due. But it was delivered to the world mainly
+by Ptolemy, and goes by his name. This was a fine piece of work, and a
+great advance on anything that had gone before; for although it is of
+course saturated with error, still it is based on a large substratum of
+truth. Its superiority to all the previously mentioned systems is
+obvious. And it really did in its more developed form describe the
+observed motions of the planets.
+
+Each planet was, in the early stages of this system, as taught, say, by
+Eudoxus, supposed to be set in a crystal sphere, which revolved so as to
+carry the planet with it. The sphere had to be of crystal to account for
+the visibility of other planets and the stars through it. Outside the
+seven planetary spheres, arranged one inside the other, was a still
+larger one in which were set the stars. This was believed to turn all
+the others, and was called the _primum mobile_. The whole system was
+supposed to produce, in its revolution, for the few privileged to hear
+the music of the spheres, a sound as of some magnificent harmony.
+
+[Illustration: FIG. 7.--Order of ancient planets corresponding to the
+days of the week.]
+
+The enthusiastic disciples of Pythagoras believed that their master was
+privileged to hear this noble chant; and far be it from us to doubt
+that the rapt and absorbing pleasure of contemplating the harmony of
+nature, to a man so eminently great as Pythagoras, must be truly and
+adequately represented by some such poetic conception.
+
+[Illustration: FIG. 8.--Ptolemaic system.]
+
+The precise kind of motion supposed to be communicated from the _primum
+mobile_ to the other spheres so as to produce the observed motions of
+the planets was modified and improved by various philosophers until it
+developed into the epicyclic train of Hipparchus and of Ptolemy.
+
+It is very instructive to observe a planet (say Mars or Jupiter) night
+after night and plot down its place with reference to the fixed stars
+on a celestial globe or star-map. Or, instead of direct observation by
+alignment with known stars, it is easier to look out its right ascension
+and declination in _Whitaker's Almanac_, and plot those down. If this be
+done for a year or two, it will be found that the motion of the planet
+is by no means regular, but that though on the whole it advances it
+sometimes is stationary and sometimes goes back.[1]
+
+[Illustration: FIG. 9.--Specimens of Apparent paths of Venus and of Mars
+among the stars.]
+
+[Illustration: FIG. 10.--Apparent epicyclic orbits of Jupiter and
+Saturn; the Earth being supposed fixed at the centre, with the Sun
+revolving in a small circle. A loop is made by each planet every year.]
+
+These "stations" and "retrogressions" of the planets were well known to
+the ancients. It was not to be supposed for a moment that the crystal
+spheres were subject to any irregularity, neither was uniform circular
+motion to be readily abandoned; so it was surmised that the main sphere
+carried, not the planet itself, but the centre or axis of a subordinate
+sphere, and that the planet was carried by this. The minor sphere could
+be allowed to revolve at a different uniform pace from the main sphere,
+and so a curve of some complexity could be obtained.
+
+A curve described in space by a point of a circle or sphere, which
+itself is carried along at the same time, is some kind of cycloid; if
+the centre of the tracing circle travels along a straight line, we get
+the ordinary cycloid, the curve traced in air by a nail on a
+coach-wheel; but if the centre of the tracing circle be carried round
+another circle the curve described is called an epicycloid. By such
+curves the planetary stations and retrogressions could be explained. A
+large sphere would have to revolve once for a "year" of the particular
+planet, carrying with it a subsidiary sphere in which the planet was
+fixed; this latter sphere revolving once for a "year" of the earth. The
+actual looped curve thus described is depicted for Jupiter and Saturn in
+the annexed diagram (fig. 10.)
+
+     It was long ago perceived that real material spheres were
+     unnecessary; such spheres indeed, though possibly transparent to
+     light, would be impermeable to comets: any other epicyclic gearing
+     would serve, and as a mere description of the motion it is simpler
+     to think of a system of jointed bars, one long arm carrying a
+     shorter arm, the two revolving at different rates, and the end of
+     the short one carrying the planet. This does all that is needful
+     for the first approximation to a planet's motion. In so far as the
+     motion cannot be thus truly stated, the short arm may be supposed
+     to carry another, and that another, and so on, so that the
+     resultant motion of the planet is compounded of a large number of
+     circular motions of different periods; by this device any required
+     amount of complexity could be attained. We shall return to this at
+     greater length in Lecture III.
+
+     The main features of the motion, as shown in the diagram, required
+     only two arms for their expression; one arm revolving with the
+     average motion of the planet, and the other revolving with the
+     apparent motion of the sun, and always pointing in the same
+     direction as the single arm supposed to carry the sun. This last
+     fact is of course because the motion to be represented does not
+     really belong to the planet at all, but to the earth, and so all
+     the main epicyclic motions for the superior planets were the same.
+     As for the inferior planets (Mercury and Venus) they only appear
+     to oscillate like the bob of a pendulum about the sun, and so it is
+     very obvious that they must be really revolving round it. An
+     ancient Egyptian system perceived this truth; but the Ptolemaic
+     system imagined them to revolve round the earth like the rest, with
+     an artificial system of epicycles to prevent their ever getting far
+     away from the neighbourhood of the sun.
+
+     It is easy now to see how the Copernican system explains the main
+     features of planetary motion, the stations and retrogressions,
+     quite naturally and without any complexity.
+
+     [Illustration: FIG. 11.--Egyptian system.]
+
+     Let the outer circle represent the orbit of Jupiter, and the inner
+     circle the orbit of the earth, which is moving faster than Jupiter
+     (since Jupiter takes 4332 days to make one revolution); then
+     remember that the apparent position of Jupiter is referred to the
+     infinitely distant fixed stars and refer to fig. 12.
+
+     Let E_1, E_2, &c., be successive positions of the earth; J_1,
+     J_2, &c., corresponding positions of Jupiter. Produce the lines
+     E_1 J_1, E_2 J_2, &c., to an enormously greater circle
+     outside, and it will be seen that the termination of these lines,
+     representing apparent positions of Jupiter among the stars,
+     advances while the earth goes from E_1 to E_3; is almost
+     stationary from somewhere about E_3 to E_4; and recedes from
+     E_4 to E_5; so that evidently the recessions of Jupiter are
+     only apparent, and are due to the orbital motion of the earth. The
+     apparent complications in the path of Jupiter, shown in Fig. 10,
+     are seen to be caused simply by the motion of the earth, and to be
+     thus completely and easily explained.
+
+     [Illustration: FIG. 12.--True orbits of Earth and Jupiter.]
+
+     The same thing for an inferior planet, say Mercury, is even still
+     more easily seen (_vide_ figure 13).
+
+     The motion of Mercury is direct from M'' to M''', retrograde from
+     M''' to M'', and stationary at M'' and M'''. It appears to
+     oscillate, taking 72�5 days for its direct swing, and 43�5 for its
+     return swing.
+
+     [Illustration: FIG. 13.--Orbit of Mercury and Earth.]
+
+     On this system no artificiality is required to prevent Mercury's
+     ever getting far from the sun: the radius of its orbit limits its
+     real and apparent excursions. Even if the earth were stationary,
+     the motions of Mercury and Venus would not be _essentially_
+     modified, but the stations and retrogressions of the superior
+     planets, Mars, Jupiter, &c., would wholly cease.
+
+     The complexity of the old mode of regarding apparent motion may be
+     illustrated by the case of a traveller in a railway train unaware
+     of his own motion. It is as though trees, hedges, distant objects,
+     were all flying past him and contorting themselves as you may see
+     the furrows of a ploughed field do when travelling, while you
+     yourself seem stationary amidst it all. How great a simplicity
+     would be introduced by the hypothesis that, after all, these things
+     might be stationary and one's self moving.
+
+[Illustration: FIG. 14.--Copernican system as frequently represented.
+But the cometary orbit is a much later addition, and no attempt is made
+to show the relative distances of the planets.]
+
+Now you are not to suppose that the system of Copernicus swept away the
+entire doctrine of epicycles; that doctrine can hardly be said to be
+swept away even now. As a description of a planet's motion it is not
+incorrect, though it is geometrically cumbrous. If you describe the
+motion of a railway train by stating that every point on the rim of each
+wheel describes a cycloid with reference to the earth, and a circle with
+reference to the train, and that the motion of the train is compounded
+of these cycloidal and circular motions, you will not be saying what is
+false, only what is cumbrous.
+
+The Ptolemaic system demanded large epicycles, depending on the motion
+of the earth, these are what Copernicus overthrew; but to express the
+minuter details of the motion smaller epicycles remained, and grew more
+and more complex as observations increased in accuracy, until a greater
+man than either Copernicus or Ptolemy, viz. Kepler, replaced them all by
+a simple ellipse.
+
+One point I must not omit from this brief notice of the work of
+Copernicus. Hipparchus had, by most sagacious interpretation of certain
+observations of his, discovered a remarkable phenomenon called the
+precession of the equinoxes. It was a discovery of the first magnitude,
+and such as would raise to great fame the man who should have made it in
+any period of the world's history, even the present. It is scarcely
+expressible in popular language, and without some technical terms; but I
+can try.
+
+The plane of the earth's orbit produced into the sky gives the apparent
+path of the sun throughout a year. This path is known as the ecliptic,
+because eclipses only happen when the moon is in it. The sun keeps to it
+accurately, but the planets wander somewhat above and below it (fig. 9),
+and the moon wanders a good deal. It is manifest, however, in order that
+there may be an eclipse of any kind, that a straight line must be able
+to be drawn through earth and moon and sun (not necessarily through
+their centres of course), and this is impossible unless some parts of
+the three bodies are in one plane, viz. the ecliptic, or something very
+near it. The ecliptic is a great circle of the sphere, and is usually
+drawn on both celestial and terrestrial globes.
+
+The earth's equator also produced into the sky, where it may still be
+called the equator (sometimes it is awkwardly called "the equinoctial"),
+gives another great circle inclined to the ecliptic and cutting it at
+two opposite points, labelled respectively [Aries symbol] and [Libra
+symbol], and together called "the equinoxes." The reason for the name is
+that when the sun is in that part of the ecliptic it is temporarily also
+on the equator, and hence is symmetrically situated with respect to the
+earth's axis of rotation, and consequently day and night are equal all
+over the earth.
+
+Well, Hipparchus found, by plotting the position of the sun for a long
+time,[2] that these points of intersection, or equinoxes, were not
+stationary from century to century, but slowly moved among the stars,
+moving as it were to meet the sun, so that he gets back to one of these
+points again 20 minutes 23-1/4 seconds before it has really completed a
+revolution, _i.e._ before the true year is fairly over. This slow
+movement forward of the goal-post is called precession--the precession
+of the equinoxes. (One result of it is to shorten our years by about 20
+minutes each; for the shortened period has to be called a year, because
+it is on the position of the sun with respect to the earth's axis that
+our seasons depend.) Copernicus perceived that, assuming the motion of
+the earth, a clearer account of this motion could be given. The ordinary
+approximate statement concerning the earth's axis is that it remains
+parallel to itself, _i.e._ has a fixed direction as the earth moves
+round the sun. But if, instead of being thus fixed, it be supposed to
+have a slow movement of revolution, so that it traces out a cone in the
+course of about 26,000 years, then, since the equator of course goes
+with it, the motion of its intersection with the fixed ecliptic is so
+far accounted for. That is to say, the precession of the equinoxes is
+seen to be dependent on, and caused by, a slow conical movement of the
+earth's axis.
+
+The prolongation of each end of the earth's axis into the sky, or the
+celestial north and south poles, will thus slowly trace out an
+approximate circle among the stars; and the course of the north pole
+during historic time is exhibited in the annexed diagram.
+
+It is now situated near one of the stars of the Lesser Bear, which we
+therefore call the Pole star; but not always was it so, nor will it be
+so in the future. The position of the north pole 4000 years ago is shown
+in the figure; and a revolution will be completed in something like
+26,000 years.[3]
+
+[Illustration: FIG. 15.--Slow movement of the north pole in a circle
+among the stars. (Copied from Sir R. Ball.)]
+
+This perception of the conical motion of the earth's axis was a
+beautiful generalization of Copernik's, whereby a multitude of facts
+were grouped into a single phenomenon. Of course he did not explain the
+motion of the axis itself. He stated the fact that it so moved, and I do
+not suppose it ever struck him to seek for an explanation.
+
+An explanation was given later, and that a most complete one; but the
+idea even of seeking for it is a brilliant and striking one: the
+achievement of the explanation by a single individual in the way it
+actually was accomplished is one of the most astounding things in the
+history of science; and were it not that the same individual
+accomplished a dozen other things, equally and some still more
+extraordinary, we should rank that man as one of the greatest
+astronomers that ever lived.
+
+As it is, he is Sir Isaac Newton.
+
+We are to remember, then, as the life-work of Copernicus, that he placed
+the sun in its true place as the centre of the solar system, instead of
+the earth; that he greatly simplified the theory of planetary motion by
+this step, and also by the simpler epicyclic chain which now sufficed,
+and which he worked out mathematically; that he exhibited the precession
+of the equinoxes (discovered by Hipparchus) as due to a conical motion
+of the earth's axis; and that, by means of his simpler theory and more
+exact planetary tables, he reduced to some sort of order the confused
+chaos of the Ptolemaic system, whose accumulation of complexity and of
+outstanding errors threatened to render astronomy impossible by the mere
+burden of its detail.
+
+There are many imperfections in his system, it is true; but his great
+merit is that he dared to look at the facts of Nature with his own eyes,
+unhampered by the prejudice of centuries. A system venerable with age,
+and supported by great names, was universally believed, and had been
+believed for centuries. To doubt this system, and to seek after another
+and better one, at a time when all men's minds were governed by
+tradition and authority, and when to doubt was sin--this required a
+great mind and a high character. Such a mind and such a character had
+this monk of Frauenburg. And it is interesting to notice that the
+so-called religious scruples of smaller and less truly religious men did
+not affect Copernicus; it was no dread of consequences to one form of
+truth that led him to delay the publication of the other form of truth
+specially revealed to him. In his dedication he says:--
+
+"If there be some babblers who, though ignorant of all mathematics, take
+upon them to judge of these things, and dare to blame and cavil at my
+work, because of some passage of Scripture which they have wrested to
+their own purpose, I regard them not, and will not scruple to hold their
+judgment in contempt."
+
+I will conclude with the words of one of his biographers (Mr. E.J.C.
+Morton):--
+
+"Copernicus cannot be said to have flooded with light the dark places of
+nature--in the way that one stupendous mind subsequently did--but still,
+as we look back through the long vista of the history of science, the
+dim Titanic figure of the old monk seems to rear itself out of the dull
+flats around it, pierces with its head the mists that overshadow them,
+and catches the first gleam of the rising sun,
+
+  "'... like some iron peak, by the Creator
+    Fired with the red glow of the rushing morn.'"
+
+
+
+
+DATES AND SUMMARY OF FACTS FOR LECTURE II
+
+
+Copernicus lived from 1473 to 1543, and was contemporary with Paracelsus
+and Raphael.
+
+  Tycho Brah�     from 1546 to 1601.
+  Kepler          from 1571 to 1630.
+  Galileo         from 1564 to 1642.
+  Gilbert         from 1540 to 1603.
+  Francis Bacon   from 1561 to 1626.
+  Descartes       from 1596 to 1650.
+
+_A sketch of Tycho Brah�'s life and work._ Tycho was a Danish noble,
+born on his ancestral estate at Knudstorp, near Helsinborg, in 1546.
+Adopted by his uncle, and sent to the University of Copenhagen to study
+law. Attracted to astronomy by the occurrence of an eclipse on its
+predicted day, August 21st, 1560. Began to construct astronomical
+instruments, especially a quadrant and a sextant. Observed at Augsburg
+and Wittenberg. Studied alchemy, but was recalled to astronomy by the
+appearance of a new star. Overcame his aristocratic prejudices, and
+delivered a course of lectures at Copenhagen, at the request of the
+king. After this he married a peasant girl. Again travelled and observed
+in Germany. In 1576 was sent for to Denmark by Frederick II., and
+established in the island of Huen, with an endowment enabling him to
+devote his life to astronomy. Built Uraniburg, furnished it with
+splendid instruments, and became the founder of accurate instrumental
+astronomy. His theories were poor, but his observations were admirable.
+In 1592 Frederick died, and five years later, Tycho was impoverished and
+practically banished. After wandering till 1599, he was invited to
+Prague by the Emperor Rudolf, and there received John Kepler among other
+pupils. But the sentence of exile was too severe, and he died in 1601,
+aged 54 years.
+
+A man of strong character, untiring energy, and devotion to accuracy,
+his influence on astronomy has been immense.
+
+
+
+
+LECTURE II
+
+TYCHO BRAH� AND THE EARLIEST OBSERVATORY
+
+
+We have seen how Copernicus placed the earth in its true position in the
+solar system, making it merely one of a number of other worlds revolving
+about a central luminary. And observe that there are two phenomena to be
+thus accounted for and explained: first, the diurnal revolution of the
+heavens; second, the annual motion of the sun among the stars.
+
+The effect of the diurnal motion is conspicuous to every one, and
+explains the rising, southing, and setting of the whole visible
+firmament. The effect of the annual motion, _i.e._ of the apparent
+annual motion, of the sun among the stars, is less obvious, but it may
+be followed easily enough by observing the stars visible at any given
+time of evening at different seasons of the year. At midnight, for
+instance, the position of the sun is definite, viz. due north always,
+but the constellation which at that time is due south or is rising or
+setting varies with the time of year; an interval of one month producing
+just the same effect on the appearance of the constellations as an
+interval of two hours does (because the day contains twice as many hours
+as the year contains months), _e.g._ the sky looks the same at midnight
+on the 1st of October as it does at 10 p.m. on the 1st of November.
+
+All these simple consequences of the geocentric as opposed to the
+heliocentric point of view were pointed out by Copernicus, in addition
+to his greater work of constructing improved planetary tables on the
+basis of his theory. But it must be admitted that he himself felt the
+hypothesis of the motion of the earth to be a difficulty. Its acceptance
+is by no means such an easy and childish matter as we are apt now to
+regard it, and the hostility to it is not at all surprising. The human
+race, after having ridiculed and resisted the truth for a long time, is
+apt to end in accepting it so blindly and unimaginatively as to fail to
+recognize the real achievement of its first propounders, or the
+difficulties which they had to overcome. The majority of men at the
+present day have grown accustomed to hear the motion of the earth spoken
+of: their acceptance of it means nothing: the attitude of the paradoxer
+who denies it is more intelligent.
+
+It is not to be supposed that the idea of thus explaining some of the
+phenomena of the heavens, especially the daily motion of the entire
+firmament, by a diurnal rotation of the earth had not struck any one. It
+was often at this time referred to as the Pythagorean theory, and it had
+been taught, I believe, by Aristarchus. But it was new to the modern
+world, and it had the great weight of Aristotle against it.
+Consequently, for long after Copernicus, only a few leading spirits
+could be found to support it, and the long-established venerable
+Ptolemaic system continued to be taught in all Universities.
+
+The main objections to the motion of the earth were such as the
+following:--
+
+1. The motion is unfelt and difficult to imagine.
+
+     That it is unfelt is due to its uniformity, and can be explained
+     mechanically. That it is difficult to imagine is and remains true,
+     but a most important lesson we have to learn is that difficulty of
+     conception is no valid argument against reality.
+
+2. That the stars do not alter their relative positions according to
+the season of the year, but the constellations preserve always the same
+aspect precisely, even to careful measurement.
+
+     This is indeed a difficulty, and a great one. In June the earth is
+     184 million miles away from where it was in December: how can we
+     see precisely the same fixed stars? It is not possible, unless they
+     are at a practically infinite distance. That is the only answer
+     that can be given. It was the tentative answer given by Copernicus.
+     It is the correct answer. Not only from every position of the
+     earth, but from every planet of the solar system, the same
+     constellations are visible, and the stars have the same aspect. The
+     whole immensity of the solar system shrinks to practically a point
+     when confronted with the distance of the stars.
+
+     Not, however, so entirely a speck as to resist the terrific
+     accuracy of the present century, and their microscopic relative
+     displacement with the season of the year has now at length been
+     detected, and the distance of many thereby measured.
+
+3. That, if the earth revolved round the sun, Mercury and Venus ought to
+show phases like the moon.
+
+     So they ought. Any globe must show phases if it live nearer the sun
+     than we do and if we go round it, for we shall see varying amounts
+     of its illuminated half. The only answer that Copernicus could give
+     to this was that they might be difficult to see without extra
+     powers of sight, but he ventured to predict that the phases would
+     be seen if ever our powers of vision should be enhanced.
+
+4. That if the earth moved, or even revolved on its own axis, a stone or
+other dropped body ought to be left far behind.
+
+     This difficulty is not a real one, like the two last, and it is
+     based on an ignorance of the laws of mechanics, which had not at
+     that time been formulated. We know now that a ball dropped from a
+     high tower, so far from lagging, drops a minute trifle _in front_
+     of the foot of a perpendicular, because the top of the tower is
+     moving a trace faster than the bottom, by reason of the diurnal
+     rotation. But, ignoring this, a stone dropped from the lamp of a
+     railway carriage drops in the centre of the floor, whether the
+     carriage be moving steadily or standing still; a slant direction of
+     fall could only be detected if the carriage were being accelerated
+     or if the brake were applied. A body dropped from a moving carriage
+     shares the motion of the carriage, and starts with that as its
+     initial velocity. A ball dropped from a moving balloon does not
+     simply drop, but starts off in whatever direction the car was
+     moving, its motion being immediately modified by gravity, precisely
+     in the same way as that of a thrown ball is modified. This is,
+     indeed, the whole philosophy of throwing--to drop a ball from a
+     moving carriage. The carriage is the hand, and, to throw far, a run
+     is taken and the body is jerked forward; the arm is also moved as
+     rapidly as possible on the shoulder as pivot. The fore-arm can be
+     moved still faster, and the wrist-joint gives yet another motion:
+     the art of throwing is to bring all these to bear at the same
+     instant, and then just as they have all attained their maximum
+     velocity to let the ball go. It starts off with the initial
+     velocity thus imparted, and is abandoned to gravity. If the vehicle
+     were able to continue its motion steadily, as a balloon does, the
+     ball when let go from it would appear to the occupant simply to
+     drop; and it would strike the ground at a spot vertically under the
+     moving vehicle, though by no means vertically below the place where
+     it started. The resistance of the air makes observations of this
+     kind inaccurate, except when performed inside a carriage so that
+     the air shares in the motion. Otherwise a person could toss and
+     catch a ball out of a train window just as well as if he were
+     stationary; though to a spectator outside he would seem to be using
+     great skill to throw the ball in the parabola adapted to bring it
+     back to his hand.
+
+     The same circumstance enhances the apparent difficulty of the
+     circus rider's jumping feats. All he has to do is to jump up and
+     down on the horse; the forward motion which carries him through
+     hoops belongs to him by virtue of the motion of the horse, without
+     effort on his part.
+
+     Thus, then, it happens that a stone dropped sixteen feet on the
+     earth appears to fall straight down, although its real path in
+     space is a very flat trajectory of nineteen miles base and sixteen
+     feet height; nineteen miles being the distance traversed by the
+     earth every second in the course of its annual journey round the
+     sun.
+
+     No wonder that it was thought that bodies must be left behind if
+     the earth was subject to such terrific speed as this. All that
+     Copernicus could suggest on this head was that perhaps the
+     atmosphere might help to carry things forward, and enable them to
+     keep pace with the earth.
+
+There were thus several outstanding physical difficulties in the way of
+the acceptance of the Copernican theory, besides the Biblical
+difficulty.
+
+It was quite natural that the idea of the earth's motion should be
+repugnant, and take a long time to sink into the minds of men; and as
+scientific progress was vastly slower then than it is now, we find not
+only all priests but even some astronomers one hundred years afterwards
+still imagining the earth to be at rest. And among them was a very
+eminent one, Tycho Brah�.
+
+It is interesting to note, moreover, that the argument about the motion
+of the earth being contrary to Scripture appealed not only to
+ecclesiastics in those days, but to scientific men also; and Tycho
+Brah�, being a man of great piety, and highly superstitious also, was so
+much influenced by it, that he endeavoured to devise some scheme by
+which the chief practical advantages of the Copernican system could be
+retained, and yet the earth be kept still at the centre of the whole.
+This was done by making all the celestial sphere, with stars and
+everything, rotate round the earth once a day, as in the Ptolemaic
+scheme; and then besides this making all the planets revolve round the
+sun, and this to revolve round the earth. Such is the Tychonic system.
+
+So far as _relative_ motion is concerned it comes to the same thing;
+just as when you drop a book you may say either that the earth rises to
+meet the book, or that the book falls to meet the earth. Or when a fly
+buzzes round your head, you may say that you are revolving round the
+fly. But the absurdity of making the whole gigantic system of sun and
+planets and stars revolve round our insignificant earth was too great to
+be swallowed by other astronomers after they had once had a taste of the
+Copernican theory; and accordingly the Tychonic system died a speedy and
+an easy death at the same time as its inventor.
+
+Wherein then lay the magnitude of the man?--not in his theories, which
+were puerile, but in his observations, which were magnificent. He was
+the first observational astronomer, the founder of the splendid system
+of practical astronomy which has culminated in the present Greenwich
+Observatory.
+
+[Illustration: FIG. 16.--Tychonic system showing the sun with all the
+planets revolving round the earth.]
+
+Up to Tycho the only astronomical measurements had been of the rudest
+kind. Copernicus even improved upon what had gone before, with measuring
+rules made with his own hands. Ptolemy's observations could never be
+trusted to half a degree. Tycho introduced accuracy before undreamed of,
+and though his measurements, reckoned by modern ideas, are of course
+almost ludicrously rough (remember no such thing as a telescope or
+microscope was then dreamed of), yet, estimated by the era in which they
+were made, they are marvels of accuracy, and not a single mistake due
+to carelessness has ever been detected in them. In fact they may be
+depended on almost to minutes of arc, _i.e._ to sixtieths of a degree.
+
+For certain purposes connected with the proper motion of stars they are
+still appealed to, and they served as the certain and trustworthy data
+for succeeding generations of theorists to work upon. It was long,
+indeed, after Tycho's death before observations approaching in accuracy
+to his were again made.
+
+In every sense, therefore, he was a pioneer: let us proceed to trace his
+history.
+
+Born the eldest son of a noble family--"as noble and ignorant as sixteen
+undisputed quarterings could make them," as one of his biographers
+says--in a period when, even more than at present, killing and hunting
+were the only natural aristocratic pursuits, when all study was regarded
+as something only fit for monks, and when science was looked at askance
+as something unsavoury, useless, and semi-diabolic, there was little in
+his introduction to the world urging him in the direction where his
+genius lay. Of course he was destined for a soldier; but fortunately his
+uncle, George Brah�, a more educated man than his father, having no son
+of his own, was anxious to adopt him, and though not permitted to do so
+for a time, succeeded in getting his way on the birth of a second son,
+Steno--who, by the way, ultimately became Privy Councillor to the King
+of Denmark.
+
+Tycho's uncle gave him what he would never have got at home--a good
+education; and ultimately put him to study law. At the age of thirteen
+he entered the University of Copenhagen, and while there occurred the
+determining influence of his life.
+
+An eclipse of the sun in those days was not regarded with the
+cold-blooded inquisitiveness or matter-of-fact apathy, according as
+there is or is not anything to be learnt from it, with which such an
+event is now regarded. Every occurrence in the heavens was then
+believed to carry with it the destiny of nations and the fate of
+individuals, and accordingly was of surpassing interest. Ever since the
+time of Hipparchus it had been possible for some capable man here and
+there to predict the occurrence of eclipses pretty closely. The thing is
+not difficult. The prediction was not, indeed, to the minute and second,
+as it is now; but the day could usually be hit upon pretty accurately
+some time ahead, much as we now manage to hit upon the return of a
+comet--barring accidents; and the hour could be predicted as the event
+approached.
+
+Well, the boy Tycho, among others, watched for this eclipse on August
+21st, 1560; and when it appeared at its appointed time, every instinct
+for the marvellous, dormant in his strong nature, awoke to strenuous
+life, and he determined to understand for himself a science permitting
+such wonderful possibilities of prediction. He was sent to Leipzig with
+a tutor to go on with his study of law, but he seems to have done as
+little law as possible: he spent all his money on books and instruments,
+and sat up half the night studying and watching the stars.
+
+In 1563 he observed a conjunction of Jupiter and Saturn, the precursor,
+and _cause_ as he thought it, of the great plague. He found that the old
+planetary tables were as much as a month in error in fixing this event,
+and even the Copernican tables were several days out; so he formed the
+resolve to devote his life to improving astronomical tables. This
+resolve he executed with a vengeance. His first instrument was a jointed
+ruler with sights for fixing the position of planets with respect to the
+stars, and observing their stations and retrogressions. By thus
+measuring the angles between a planet and two fixed stars, its position
+can be plotted down on a celestial map or globe.
+
+[Illustration: FIG. 17.--Portrait of Tycho.]
+
+In 1565 his uncle George died, and made Tycho his heir. He returned to
+Denmark, but met with nothing but ridicule and contempt for his absurd
+drivelling away of time over useless pursuits. So he went back to
+Germany--first to Wittenberg, thence, driven by the plague, to Rostock.
+
+Here his fiery nature led him into an absurd though somewhat dangerous
+adventure. A quarrel at some feast, on a mathematical point, with a
+countryman, Manderupius, led to the fixing of a duel, and it was fought
+with swords at 7 p.m. at the end of December, when, if there was any
+light at all, it must have been of a flickering and unsatisfactory
+nature. The result of this insane performance was that Tycho got his
+nose cut clean off.
+
+He managed however to construct an artificial one, some say of gold and
+silver, some say of putty and brass; but whatever it was made of there
+is no doubt that he wore it for the rest of his life, and it is a most
+famous feature. It excited generally far more interest than his
+astronomical researches. It is said, moreover, to have very fairly
+resembled the original, but whether this remark was made by a friend or
+by an enemy I cannot say. One account says that he used to carry about
+with him a box of cement to apply whenever his nose came off, which it
+periodically did.
+
+About this time he visited Augsburg, met with some kindred and
+enlightened spirits in that town, and with much enthusiasm and spirit
+constructed a great quadrant. These early instruments were tremendous
+affairs. A great number of workmen were employed upon this quadrant, and
+it took twenty men to carry it to its place and erect it. It stood in
+the open air for five years, and then was destroyed by a storm. With it
+he made many observations.
+
+[Illustration: FIG. 18.--Early out-door quadrant of Tycho; for
+observing altitudes by help of the sights _D_, _L_ and the plumb line.]
+
+On his return to Denmark in 1571, his fame preceded him, and he was
+much better received; and in order to increase his power of constructing
+instruments he took up the study of alchemy, and like the rest of the
+persuasion tried to make gold. The precious metals were by many old
+philosophers considered to be related in some way to the heavenly
+bodies: silver to the moon, for instance--as we still see by the name
+lunar caustic applied to nitrate of silver; gold to the sun, copper to
+Mars, lead to Saturn. Hence astronomy and alchemy often went together.
+Tycho all his life combined a little alchemy with his astronomical
+labours, and he constructed a wonderful patent medicine to cure all
+disorders, which had as wide a circulation in Europe in its time as
+Holloway's pills; he gives a tremendous receipt for it, with liquid gold
+and all manner of ingredients in it; among them, however, occurs a
+little antimony--a well-known sudorific--and to this, no doubt, whatever
+efficacy the medicine possessed was due.
+
+So he might have gone on wasting his time, were it not that in November,
+1572, a new star made its appearance, as they have done occasionally
+before and since. On the average one may say that about every fifty
+years a new star of fair magnitude makes its temporary appearance. They
+are now known to be the result of some catastrophe or collision, whereby
+immense masses of incandescent gas are produced. This one seen by Tycho
+became as bright as Jupiter, and then died away in about a year and a
+half. Tycho observed all its changes, and endeavoured to measure its
+distance from the earth, with the result that it was proved to belong to
+the region of the fixed stars, at an immeasurable distance, and was not
+some nearer and more trivial phenomenon.
+
+He was asked by the University of Copenhagen to give a course of
+lectures on astronomy; but this was a step he felt some aristocratic
+aversion to, until a little friendly pressure was brought to bear upon
+him by a request from the king, and delivered they were.
+
+He now seems to have finally thrown off his aristocratic prejudices, and
+to have indulged himself in treading on the corns of nearly all the high
+and mighty people he came into contact with. In short, he became what we
+might now call a violent Radical; but he was a good-hearted man,
+nevertheless, and many are the tales told of his visits to sick
+peasants, of his consulting the stars as to their fate--all in perfect
+good faith--and of the medicines which he concocted and prescribed for
+them.
+
+The daughter of one of these peasants he married, and very happy the
+marriage seems to have been.
+
+[Illustration: FIG. 19.--Map of Denmark, showing the island of Huen.
+
+_Walker & Boutallse._]
+
+Now comes the crowning episode in Tycho's life. Frederick II., realizing
+how eminent a man they had among them, and how much he could do if only
+he had the means--for we must understand that Tycho, though of good
+family and well off, was by no means what we would call a wealthy
+man--Frederick II. made him a splendid and enlightened offer. The offer
+was this: that if Tycho would agree to settle down and make his
+astronomical observations in Denmark, he should have an estate in Norway
+settled upon him, a pension of �400 a year for life, a site for a large
+observatory, and �20,000 to build it with.
+
+[Illustration: FIG. 20.--Uraniburg.]
+
+[Illustration: FIG. 21.--Astrolabe. An old instrument with sights for
+marking the positions of the celestial bodies roughly. A sort of
+skeleton celestial globe.]
+
+[Illustration:
+
+  SEXTANS ASTRONOMICVS
+  TRIGONICVS PRO DISTANTIIS
+  rimandis.
+
+FIG. 22.--Tycho's large sextant; for measuring the angular distance
+between two bodies by direct sighting.]
+
+Well, if ever money was well spent, this was. By its means Denmark
+before long headed the nations of Europe in the matter of science--a
+thing it has not done before or since. The site granted was the island
+of Huen, between Copenhagen and Elsinore; and here the most magnificent
+observatory ever built was raised, and called Uraniburg--the castle of
+the heavens. It was built on a hill in the centre of the island, and
+included gardens, printing shops, laboratory, dwelling-houses, and four
+observatories--all furnished with the most splendid instruments that
+Tycho could devise, and that could then be constructed. It was decorated
+with pictures and sculptures of eminent men, and altogether was a most
+gorgeous place. �20,000 no doubt went far in those days, but the
+original grant was supplemented by Tycho himself, who is said to have
+spent another equal sum out of his own pocket on the place.
+
+[Illustration: QVADRANS MAXIMVS CHALIBEUS QUADRATO INCLUSUS, ET
+Horizonti Azimuthali chalybeo insistens.
+
+FIG. 23.--The Quadrant in Uraniburg; or altitude and azimuth
+instrument.]
+
+For twenty years this great temple of science was continually worked in
+by him, and he soon became the foremost scientific man in Europe.
+Philosophers, statesmen, and occasionally kings, came to visit the great
+astronomer, and to inspect his curiosities.
+
+[Illustration:
+
+  QVADRANS MVRALIS
+  SIVE TICHONICUS.
+
+FIG. 24.--Tycho's form of transit circle.
+
+The method of utilising the extremely uniform rotation of the earth by
+watching the planets and stars as they cross the meridian, and recording
+their times of transit; observing also at the same time their meridian
+altitudes (see observer _F_), was the invention of Tycho, and
+constitutes his greatest achievement. His method is followed to this day
+in all observatories.]
+
+[Illustration: FIG. 25.--A modern transit circle, showing essentially
+the same parts as in Tycho's instrument, viz. the observer watching the
+transit, the clock, the recorder of the observation, and the graduated
+circle; the latter to be read by a second observer.]
+
+And very wholesome for some of these great personages must have been the
+treatment they met with. For Tycho was no respecter of persons. His
+humbly-born wife sat at the head of the table, whoever was there; and he
+would snub and contradict a chancellor just as soon as he would a serf.
+Whatever form his pride may have taken when a youth, in his maturity it
+impelled him to ignore differences of rank not substantially justified,
+and he seemed to take a delight in exposing the ignorance of shallow
+titled persons, to whom contradiction and exposure were most unusual
+experiences.
+
+For sick peasants he would take no end of trouble, and went about
+doctoring them for nothing, till he set all the professional doctors
+against him; so that when his day of misfortune came, as come it did,
+their influence was not wanting to help to ruin one who spoilt their
+practice, and whom they derided as a quack.
+
+But some of the great ignorant folk who came to visit his temple of
+science, and to inspect its curiosities, felt themselves insulted--not
+always without reason. He kept a tame maniac in the house, named Lep,
+and he used to regard the sayings of this personage as oracular,
+presaging future events, and far better worth listening to than ordinary
+conversation. Consequently he used to have him at his banquets and feed
+him himself; and whenever Lep opened his mouth to speak, every one else
+was peremptorily ordered to hold his tongue, so that Lep's words might
+be written down. In fact it was something like an exaggerated edition of
+Betsy Trotwood and Mr. Dick.
+
+"It must have been an odd dinner party" (says Prof. Stuart), "with this
+strange, wild, terribly clever man, with his red hair and brazen nose,
+sometimes flashing with wit and knowledge, sometimes making the whole
+company, princes and servants alike, hold their peace and listen humbly
+to the ravings of a poor imbecile."
+
+To people he despised he did not show his serious instruments. He had
+other attractions, in the shape of a lot of toy machinery, little
+windmills, and queer doors, and golden globes, and all manner of
+ingenious tricks and automata, many of which he had made himself, and
+these he used to show them instead; and no doubt they were well enough
+pleased with them. Those of the visitors, however, who really cared to
+see and understand his instruments, went away enchanted with his genius
+and hospitality.
+
+I may, perhaps, be producing an unfair impression of imperiousness and
+insolence. Tycho was fiery, no doubt, but I think we should wrong him
+if we considered him insolent. Most of the nobles of his day were
+haughty persons, accustomed to deal with serfs, and very likely to sneer
+at and trample on any meek man of science whom they could easily
+despise. So Tycho was not meek; he stood up for the honour of his
+science, and paid them back in their own coin, with perhaps a little
+interest. That this behaviour was not worldly-wise is true enough, but I
+know of no commandment enjoining us to be worldly-wise.
+
+If we knew more about his so-called imbecile _prot�g�_ we should
+probably find some reason for the interest which Tycho took in him.
+Whether he was what is now called a "clairvoyant" or not, Tycho
+evidently regarded his utterances as oracular, and of course when one is
+receiving what may be a revelation from heaven it is natural to suppress
+ordinary conversation.
+
+Among the noble visitors whom he received and entertained, it is
+interesting to notice James I. of England, who spent eight days at
+Uraniburg on the occasion of his marriage with Anne of Denmark in 1590,
+and seems to have been deeply impressed by his visit.
+
+Among other gifts, James presented Tycho with a dog (depicted in Fig.
+24), and this same animal was subsequently the cause of trouble. For it
+seems that one day the Chancellor of Denmark, Walchendorf, brutally
+kicked the poor beast; and Tycho, who was very fond of animals, gave him
+a piece of his mind in no measured language. Walchendorf went home
+determined to ruin him. King Frederick, however, remained his true
+friend, doubtless partly influenced thereto by his Queen Sophia, an
+enlightened woman who paid many visits to Uraniburg, and knew Tycho
+well. But unfortunately Frederick died; and his son, a mere boy, came to
+the throne.
+
+Now was the time for the people whom Tycho had offended, for those who
+were jealous of his great fame and importance, as well as for those who
+cast longing eyes on his estate and endowments. The boy-king, too,
+unfortunately paid a visit to Tycho, and, venturing upon a decided
+opinion on some recondite subject, received a quiet setting down which
+he ill relished.
+
+Letters written by Tycho about this time are full of foreboding. He
+greatly dreads having to leave Uraniburg, with which his whole life has
+for twenty years been bound up. He tries to comfort himself with the
+thought that, wherever he is sent, he will have the same heavens and the
+same stars over his head.
+
+Gradually his Norwegian estate and his pension were taken away, and in
+five years poverty compelled him to abandon his magnificent temple, and
+to take a small house in Copenhagen.
+
+Not content with this, Walchendorf got a Royal Commission appointed to
+inquire into the value of his astronomical labours. This sapient body
+reported that his work was not only useless, but noxious; and soon after
+he was attacked by the populace in the public street.
+
+Nothing was left for him now but to leave the country, and he went into
+Germany, leaving his wife and instruments to follow him whenever he
+could find a home for them.
+
+His wanderings in this dark time--some two years--are not quite clear;
+but at last the enlightened Emperor of Bohemia, Rudolph II., invited him
+to settle in Prague. Thither he repaired, a castle was given him as an
+observatory, a house in the city, and 3000 crowns a year for life. So
+his instruments were set up once more, students flocked to hear him and
+to receive work at his hands--among them a poor youth, John Kepler, to
+whom he was very kind, and who became, as you know, a still greater man
+than his master.
+
+But the spirit of Tycho was broken, and though some good work was done
+at Prague--more observations made, and the Rudolphine tables begun--yet
+the hand of death was upon him. A painful disease seized him, attended
+with sleeplessness and temporary delirium, during the paroxysms of
+which he frequently exclaimed, _Ne frustra vixisse videar_. ("Oh that it
+may not appear that I have lived in vain!")
+
+Quietly, however, at last, and surrounded by his friends and relatives,
+this fierce, passionate soul passed away, on the 24th of October, 1601.
+
+His beloved instruments, which were almost a part of himself, were
+stored by Rudolph in a museum with scrupulous care, until the taking of
+Prague by the Elector Palatine's troops. In this disturbed time they got
+smashed, dispersed, and converted to other purposes. One thing only was
+saved--the great brass globe, which some thirty years after was
+recognized by a later king of Denmark as having belonged to Tycho, and
+deposited in the Library of the Academy of Sciences at Copenhagen, where
+I believe it is to this day.
+
+The island of Huen was overrun by the Danish nobility, and nothing now
+remains of Uraniburg but a mound of earth and two pits.
+
+As to the real work of Tycho, that has become immortal enough,--chiefly
+through the labours of his friend and scholar whose life we shall
+consider in the next lecture.
+
+
+
+
+SUMMARY OF FACTS FOR LECTURE III
+
+
+_Life and work of Kepler._ Kepler was born in December, 1571, at Weil in
+W�rtemberg. Father an officer in the duke's army, mother something of a
+virago, both very poor. Kepler was utilized as a tavern pot-boy, but
+ultimately sent to a charity school, and thence to the University of
+T�bingen. Health extremely delicate; he was liable to violent attacks
+all his life. Studied mathematics, and accepted an astronomical
+lectureship at Graz as the first post which offered. Endeavoured to
+discover some connection between the number of the planets, their times
+of revolution, and their distances from the sun. Ultimately hit upon his
+fanciful regular-solid hypothesis, and published his first book in 1597.
+In 1599 was invited by Tycho to Prague, and there appointed Imperial
+mathematician, at a handsome but seldom paid salary. Observed the new
+star of 1604. Endeavoured to find the law of refraction of light from
+Vitellio's measurements, but failed. Analyzed Tycho's observations to
+find the true law of motion of Mars. After incredible labour, through
+innumerable wrong guesses, and six years of almost incessant
+calculation, he at length emerged in his two "laws"--discoveries which
+swept away all epicycles, deferents, equants, and other remnants of the
+Greek system, and ushered in the dawn of modern astronomy.
+
+LAW I. _Planets move in ellipses, with the Sun in one focus._
+
+LAW II. _The radius vector (or line joining sun and planet) sweeps out
+equal areas in equal times._
+
+Published his second book containing these laws in 1609. Death of
+Rudolph in 1612, and subsequent increased misery and misfortune of
+Kepler. Ultimately discovered the connection between the times and
+distances of the planets for which he had been groping all his mature
+life, and announced it in 1618:--
+
+LAW III. _The square of the time of revolution (or year) of each planet
+is proportional to the cube of its mean distance from the sun._
+
+The book in which this law was published ("On Celestial Harmonies") was
+dedicated to James of England. In 1620 had to intervene to protect his
+mother from being tortured for witchcraft. Accepted a professorship at
+Linz. Published the Rudolphine tables in 1627, embodying Tycho's
+observations and his own theory. Made a last effort to overcome his
+poverty by getting the arrears of his salary paid at Prague, but was
+unsuccessful, and, contracting brain fever on the journey, died in
+November, 1630, aged 59.
+
+A man of keen imagination, indomitable perseverance, and uncompromising
+love of truth, Kepler overcame ill-health, poverty, and misfortune, and
+placed himself in the very highest rank of scientific men. His laws, so
+extraordinarily discovered, introduced order and simplicity into what
+else would have been a chaos of detailed observations; and they served
+as a secure basis for the splendid erection made on them by Newton.
+
+  _Seven planets of the Ptolemaic system--_
+      Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.
+
+  _Six planets of the Copernican system--_
+      Mercury, Venus, Earth, Mars, Jupiter, Saturn.
+
+  _The five regular solids, in appropriate order--_
+      Octahedron, Icosahedron, Dodecahedron, Tetrahedron, Cube.
+
+_Table illustrating Kepler's third law._
+
+ +---------+---------------+-----------+---------------+----------------+
+ |         | Mean distance |   Length  |  Cube of the  |  Square of the |
+ | Planet.  |   from Sun.  |  of Year. |   Distance.   |     Time.      |
+ |         |       D       |     T     |      D^3      |      T^2       |
+ +---------+---------------+-----------+---------------+----------------+
+ | Mercury |     �3871     |   �24084  |     �05801    |     �05801     |
+ | Venus   |     �7233     |   �61519  |     �37845    |     �37846     |
+ | Earth   |    1�0000     |  1�0000   |    1�0000     |    1�0000      |
+ | Mars    |    1�5237     |  1�8808   |    3�5375     |    3�5375      |
+ | Jupiter |    5�2028     | 11�862    |  140�83       |  140�70        |
+ | Saturn  |    9�5388     | 29�457    |  867�92       |  867�70        |
+ +---------+---------------+-----------+---------------+----------------+
+
+The length of the earth's year is 365�256 days; its mean distance from
+the sun, taken above as unity, is 92,000,000 miles.
+
+
+
+
+LECTURE III
+
+KEPLER AND THE LAWS OF PLANETARY MOTION
+
+
+It is difficult to imagine a stronger contrast between two men engaged
+in the same branch of science than exists between Tycho Brah�, the
+subject of last lecture, and Kepler, our subject on the present
+occasion.
+
+The one, rich, noble, vigorous, passionate, strong in mechanical
+ingenuity and experimental skill, but not above the average in
+theoretical and mathematical power.
+
+The other, poor, sickly, devoid of experimental gifts, and unfitted by
+nature for accurate observation, but strong almost beyond competition in
+speculative subtlety and innate mathematical perception.
+
+The one is the complement of the other; and from the fact of their
+following each other so closely arose the most surprising benefits to
+science.
+
+The outward life of Kepler is to a large extent a mere record of poverty
+and misfortune. I shall only sketch in its broad features, so that we
+may have more time to attend to his work.
+
+He was born (so his biographer assures us) in longitude 29� 7', latitude
+48� 54', on the 21st of December, 1571. His parents seem to have been of
+fair condition, but by reason, it is said, of his becoming surety for a
+friend, the father lost all his slender income, and was reduced to
+keeping a tavern. Young John Kepler was thereupon taken from school,
+and employed as pot-boy between the ages of nine and twelve. He was a
+sickly lad, subject to violent illnesses from the cradle, so that his
+life was frequently despaired of. Ultimately he was sent to a monastic
+school and thence to the University of T�bingen, where he graduated
+second on the list. Meanwhile home affairs had gone to rack and ruin.
+His father abandoned the home, and later died abroad. The mother
+quarrelled with all her relations, including her son John; who was
+therefore glad to get away as soon as possible.
+
+All his connection with astronomy up to this time had been the hearing
+the Copernican theory expounded in University lectures, and defending it
+in a college debating society.
+
+An astronomical lectureship at Graz happening to offer itself, he was
+urged to take it, and agreed to do so, though stipulating that it should
+not debar him from some more brilliant profession when there was a
+chance.
+
+For astronomy in those days seems to have ranked as a minor science,
+like mineralogy or meteorology now. It had little of the special dignity
+with which the labours of Kepler himself were destined so greatly to aid
+in endowing it.
+
+Well, he speedily became a thorough Copernican, and as he had a most
+singularly restless and inquisitive mind, full of appreciation of
+everything relating to number and magnitude--was a born speculator and
+thinker just as Mozart was a born musician, or Bidder a born
+calculator--he was agitated by questions such as these: Why are there
+exactly six planets? Is there any connection between their orbital
+distances, or between their orbits and the times of describing them?
+These things tormented him, and he thought about them day and night. It
+is characteristic of the spirit of the times--this questioning why there
+should be six planets. Nowadays, we should simply record the fact and
+look out for a seventh. Then, some occult property of the number six was
+groped for, such as that it was equal to 1 + 2 + 3 and likewise equal to
+1 � 2 � 3, and so on. Many fine reasons had been given for the seven
+planets of the Ptolemaic system (see, for instance, p. 106), but for
+the six planets of the Copernican system the reasons were not so cogent.
+
+Again, with respect to their successive distances from the sun, some law
+would seem to regulate their distance, but it was not known.
+(Parenthetically I may remark that it is not known even now: a crude
+empirical statement known as Bode's law--see page 294--is all that has
+been discovered.)
+
+Once more, the further the planet the slower it moved; there seemed to
+be some law connecting speed and distance. This also Kepler made
+continual attempts to discover.
+
+[Illustration: FIG. 26.--Orbits of some of the planets drawn to scale:
+showing the gap between Mars and Jupiter.]
+
+One of his ideas concerning the law of the successive distances was
+based on the inscription of a triangle in a circle. If you inscribe in a
+circle a large number of equilateral triangles, they envelop another
+circle bearing a definite ratio to the first: these might do for the
+orbits of two planets (see Fig. 27). Then try inscribing and
+circumscribing squares, hexagons, and other figures, and see if the
+circles thus defined would correspond to the several planetary orbits.
+But they would not give any satisfactory result. Brooding over this
+disappointment, the idea of trying solid figures suddenly strikes him.
+"What have plane figures to do with the celestial orbits?" he cries out;
+"inscribe the regular solids." And then--brilliant idea--he remembers
+that there are but five. Euclid had shown that there could be only five
+regular solids.[4] The number evidently corresponds to the gaps between
+the six planets. The reason of there being only six seems to be
+attained. This coincidence assures him he is on the right track, and
+with great enthusiasm and hope he "represents the earth's orbit by a
+sphere as the norm and measure of all"; round it he circumscribes a
+dodecahedron, and puts another sphere round that, which is approximately
+the orbit of Mars; round that, again, a tetrahedron, the corners of
+which mark the sphere of the orbit of Jupiter; round that sphere, again,
+he places a cube, which roughly gives the orbit of Saturn.
+
+[Illustration: FIG. 27.--Many-sided polygon or approximate circle
+enveloped by straight lines, as for instance by a number of equilateral
+triangles.]
+
+On the other hand, he inscribes in the sphere of the earth's orbit an
+icosahedron; and inside the sphere determined by that, an octahedron;
+which figures he takes to inclose the spheres of Venus and of Mercury
+respectively.
+
+The imagined discovery is purely fictitious and accidental. First of
+all, eight planets are now known; and secondly, their real distances
+agree only very approximately with Kepler's hypothesis.
+
+[Illustration: FIG. 28.--Frameworks with inscribed and circumscribed
+spheres, representing the five regular solids distributed as Kepler
+supposed them to be among the planetary orbits. (See "Summary" at
+beginning of this lecture, p. 57.)]
+
+Nevertheless, the idea gave him great delight. He says:--"The intense
+pleasure I have received from this discovery can never be told in words.
+I regretted no more the time wasted; I tired of no labour; I shunned no
+toil of reckoning, days and nights spent in calculation, until I could
+see whether my hypothesis would agree with the orbits of Copernicus, or
+whether my joy was to vanish into air."
+
+He then went on to speculate as to the cause of the planets' motion.
+The old idea was that they were carried round by angels or celestial
+intelligences. Kepler tried to establish some propelling force emanating
+from the sun, like the spokes of a windmill.
+
+This first book of his brought him into notice, and served as an
+introduction to Tycho and to Galileo.
+
+Tycho Brah� was at this time at Prague under the patronage of the
+Emperor Rudolph; and as he was known to have by far the best planetary
+observations of any man living, Kepler wrote to him to know if he might
+come and examine them so as to perfect his theory.
+
+Tycho immediately replied, "Come, not as a stranger, but as a very
+welcome friend; come and share in my observations with such instruments
+as I have with me, and as a dearly beloved associate." After this visit,
+Tycho wrote again, offering him the post of mathematical assistant,
+which after hesitation was accepted. Part of the hesitation Kepler
+expresses by saying that "for observations his sight was dull, and for
+mechanical operations his hand was awkward. He suffered much from weak
+eyes, and dare not expose himself to night air." In all this he was, of
+course, the antipodes of Tycho, but in mathematical skill he was greatly
+his superior.
+
+On his way to Prague he was seized with one of his periodical illnesses,
+and all his means were exhausted by the time he could set forward again,
+so that he had to apply for help to Tycho.
+
+It is clear, indeed, that for some time now he subsisted entirely on the
+bounty of Tycho, and he expresses himself most deeply grateful for all
+the kindness he received from that noble and distinguished man, the head
+of the scientific world at that date.
+
+To illustrate Tycho's kindness and generosity, I must read you a letter
+written to him by Kepler. It seems that Kepler, on one of his absences
+from Prague, driven half mad with poverty and trouble, fell foul of
+Tycho, whom he thought to be behaving badly in money matters to him and
+his family, and wrote him a violent letter full of reproaches and
+insults. Tycho's secretary replied quietly enough, pointing out the
+groundlessness and ingratitude of the accusation.
+
+Kepler repents instantly, and replies:--
+
+     "MOST NOBLE TYCHO," (these are the words of his letter), "how shall
+     I enumerate or rightly estimate your benefits conferred on me? For
+     two months you have liberally and gratuitously maintained me, and
+     my whole family; you have provided for all my wishes; you have done
+     me every possible kindness; you have communicated to me everything
+     you hold most dear; no one, by word or deed, has intentionally
+     injured me in anything; in short, not to your children, your wife,
+     or yourself have you shown more indulgence than to me. This being
+     so, as I am anxious to put on record, I cannot reflect without
+     consternation that I should have been so given up by God to my own
+     intemperance as to shut my eyes on all these benefits; that,
+     instead of modest and respectful gratitude, I should indulge for
+     three weeks in continual moroseness towards all your family, in
+     headlong passion and the utmost insolence towards yourself, who
+     possess so many claims on my veneration, from your noble family,
+     your extraordinary learning, and distinguished reputation. Whatever
+     I have said or written against the person, the fame, the honour,
+     and the learning of your excellency; or whatever, in any other way,
+     I have injuriously spoken or written (if they admit no other more
+     favourable interpretation), as, to my grief, I have spoken and
+     written many things, and more than I can remember; all and
+     everything I recant, and freely and honestly declare and profess to
+     be groundless, false, and incapable of proof."
+
+Tycho accepted the apology thus heartily rendered, and the temporary
+breach was permanently healed.
+
+In 1601, Kepler was appointed "Imperial mathematician," to assist Tycho
+in his calculations.
+
+The Emperor Rudolph did a good piece of work in thus maintaining these
+two eminent men, but it is quite clear that it was as astrologers that
+he valued them; and all he cared for in the planetary motions was
+limited to their supposed effect on his own and his kingdom's destiny.
+He seems to have been politically a weak and superstitious prince, who
+was letting his kingdom get into hopeless confusion, and entangling
+himself in all manner of political complications. While Bohemia
+suffered, however, the world has benefited at his hands; and the tables
+upon which Tycho was now engaged are well called the Rudolphine tables.
+
+These tables of planetary motion Tycho had always regarded as the main
+work of his life; but he died before they were finished, and on his
+death-bed he intrusted the completion of them to Kepler, who loyally
+undertook their charge.
+
+The Imperial funds were by this time, however, so taxed by wars and
+other difficulties that the tables could only be proceeded with very
+slowly, a staff of calculators being out of the question. In fact,
+Kepler could not get even his own salary paid: he got orders, and
+promises, and drafts on estates for it; but when the time came for them
+to be honoured they were worthless, and he had no power to enforce his
+claims.
+
+So everything but brooding had to be abandoned as too expensive, and he
+proceeded to study optics. He gave a very accurate explanation of the
+action of the human eye, and made many hypotheses, some of them shrewd
+and close to the mark, concerning the law of refraction of light in
+dense media: but though several minor points of interest turned up,
+nothing of the first magnitude came out of this long research.
+
+The true law of refraction was discovered some years after by a Dutch
+professor, Willebrod Snell.
+
+We must now devote a little time to the main work of Kepler's life. All
+the time he had been at Prague he had been making a severe study of the
+motion of the planet Mars, analyzing minutely Tycho's books of
+observations, in order to find out, if possible, the true theory of his
+motion. Aristotle had taught that circular motion was the only perfect
+and natural motion, and that the heavenly bodies therefore necessarily
+moved in circles.
+
+So firmly had this idea become rooted in men's minds, that no one ever
+seems to have contemplated the possibility of its being false or
+meaningless.
+
+When Hipparchus and others found that, as a matter of fact, the planets
+did _not_ revolve in simple circles, they did not try other curves, as
+we should at once do now, but they tried combinations of circles, as we
+saw in Lecture I. The small circle carried by a bigger one was called an
+Epicycle. The carrying circle was called the Deferent. If for any reason
+the earth had to be placed out of the centre, the main planetary orbit
+was called an Excentric, and so on.
+
+But although the planetary paths might be roughly represented by a
+combination of circles, their speeds could not, on the hypothesis of
+uniform motion in each circle round the earth as a fixed body. Hence was
+introduced the idea of an Equant, _i.e._ an arbitrary point, not the
+earth, about which the speed might be uniform. Copernicus, by making the
+sun the centre, had been able to simplify a good deal of this, and to
+abolish the equant.
+
+But now that Kepler had the accurate observations of Tycho to refer to,
+he found immense difficulty in obtaining the true positions of the
+planets for long together on any such theory.
+
+He specially attacked the motion of the planet Mars, because that was
+sufficiently rapid in its changes for a considerable collection of data
+to have accumulated with respect to it. He tried all manner of circular
+orbits for the earth and for Mars, placing them in all sorts of aspects
+with respect to the sun. The problem to be solved was to choose such an
+orbit and such a law of speed, for both the earth and Mars, that a line
+joining them, produced out to the stars, should always mark correctly
+the apparent position of Mars as seen from the earth. He had to arrange
+the size of the orbits that suited best, then the positions of their
+centres, both being supposed excentric with respect to the sun; but he
+could not get any such arrangement to work with uniform motion about the
+sun. So he reintroduced the equant, and thus had another variable at his
+disposal--in fact, two, for he had an equant for the earth and another
+for Mars, getting a pattern of the kind suggested in Fig. 29.
+
+The equants might divide the line in any arbitrary ratio. All sorts of
+combinations had to be tried, the relative positions of the earth and
+Mars to be worked out for each, and compared with Tycho's recorded
+observations. It was easy to get them to agree for a short time, but
+sooner or later a discrepancy showed itself.
+
+[Illustration: FIG. 29.--_S_ represents the sun; _EC_, the centre of the
+earth's orbit, to be placed as best suited; _MC_, the same for Mars;
+_EE_, the earth's equant, or point about which the earth uniformly
+revolved (_i.e._ the point determining the law of speed about the sun),
+likewise to be placed anywhere, but supposed to be in the line joining
+_S_ to _EC_; _ME_, the same thing for Mars; with _?ME_ for an
+alternative hypothesis that perhaps Mars' equant was on line joining
+_EC_ with _MC_.]
+
+I need not say that all these attempts and gropings, thus briefly
+summarized, entailed enormous labour, and required not only great
+pertinacity, but a most singularly constituted mind, that could thus
+continue groping in the dark without a possible ray of theory to
+illuminate its search. Grope he did, however, with unexampled diligence.
+
+At length he hit upon a point that seemed nearly right. He thought he
+had found the truth; but no, before long the position of the planet, as
+calculated, and as recorded by Tycho, differed by eight minutes of arc,
+or about one-eighth of a degree. Could the observation be wrong by this
+small amount? No, he had known Tycho, and knew that he was never wrong
+eight minutes in an observation.
+
+So he set out the whole weary way again, and said that with those eight
+minutes he would yet find out the law of the universe. He proceeded to
+see if by making the planet librate, or the plane of its orbit tilt up
+and down, anything could be done. He was rewarded by finding that at any
+rate the plane of the orbit did not tilt up and down: it was fixed, and
+this was a simplification on Copernicus's theory. It is not an absolute
+fixture, but the changes are very small (see Laplace, page 266).
+
+[Illustration: FIG. 30.--Excentric circle supposed to be divided into
+equal areas. The sun, _S_, being placed at a selected point, it was
+possible to represent the varying speed of a planet by saying that it
+moved from _A_ to _B_, from _B_ to _C_, and so on, in equal times.]
+
+At last he thought of giving up the idea of _uniform_ circular motion,
+and of trying _varying_ circular motion, say inversely as its distance
+from the sun. To simplify calculation, he divided the orbit into
+triangles, and tried if making the triangles equal would do. A great
+piece of luck, they did beautifully: the rate of description of areas
+(not arcs) is uniform. Over this discovery he greatly rejoices. He feels
+as though he had been carrying on a war against the planet and had
+triumphed; but his gratulation was premature. Before long fresh little
+errors appeared, and grew in importance. Thus he announces it himself:--
+
+"While thus triumphing over Mars, and preparing for him, as for one
+already vanquished, tabular prisons and equated excentric fetters, it is
+buzzed here and there that the victory is vain, and that the war is
+raging anew as violently as before. For the enemy left at home a
+despised captive has burst all the chains of the equations, and broken
+forth from the prisons of the tables."
+
+Still, a part of the truth had been gained, and was not to be abandoned
+any more. The law of speed was fixed: that which is now known as his
+second law. But what about the shape of the orbit--Was it after all
+possible that Aristotle, and every philosopher since Aristotle, had been
+wrong? that circular motion was not the perfect and natural motion, but
+that planets might move in some other closed curve?
+
+Suppose he tried an oval. Well, there are a great variety of ovals, and
+several were tried: with the result that they could be made to answer
+better than a circle, but still were not right.
+
+Now, however, the geometrical and mathematical difficulties of
+calculation, which before had been tedious and oppressive, threatened to
+become overwhelming; and it is with a rising sense of despondency that
+Kepler sees his six years' unremitting labour leading deeper and deeper
+into complication.
+
+One most disheartening circumstance appeared, viz. that when he made the
+circuit oval his law of equable description of areas broke down. That
+seemed to require the circular orbit, and yet no circular orbit was
+quite accurate.
+
+While thinking and pondering for weeks and months over this new dilemma
+and complication of difficulties, till his brain reeled, an accidental
+ray of light broke upon him in a way not now intelligible, or barely
+intelligible. Half the extreme breadth intercepted between the circle
+and oval was 429/100,000 of the radius, and he remembered that the
+"optical inequality" of Mars was also about 429/100,000. This
+coincidence, in his own words, woke him out of sleep; and for some
+reason or other impelled him instantly to try making the planet
+oscillate in the diameter of its epicycle instead of revolve round it--a
+singular idea, but Copernicus had had a similar one to explain the
+motions of Mercury.
+
+[Illustration: FIG. 31.--Mode of drawing an ellipse. The two pins _F_
+are the foci.]
+
+Away he started through his calculations again. A long course of work
+night and day was rewarded by finding that he was now able to hit off
+the motions better than before; but what a singularly complicated motion
+it was. Could it be expressed no more simply? Yes, the curve so
+described by the planet is a comparatively simple one: it is a special
+kind of oval--the ellipse. Strange that he had not thought of it before.
+It was a famous curve, for the Greek geometers had studied it as one of
+the sections of a cone, but it was not so well known in Kepler's time.
+The fact that the planets move in it has raised it to the first
+importance, and it is familiar enough to us now. But did it satisfy the
+law of speed? Could the rate of description of areas be uniform with
+it? Well, he tried the ellipse, and to his inexpressible delight he
+found that it did satisfy the condition of equable description of areas,
+if the sun was in one focus. So, moving the planet in a selected
+ellipse, with the sun in one focus, at a speed given by the equable area
+description, its position agreed with Tycho's observations within the
+limits of the error of experiment. Mars was finally conquered, and
+remains in his prison-house to this day. The orbit was found.
+
+[Illustration: FIG. 32.]
+
+In a paroxysm of delight Kepler celebrates his victory by a triumphant
+figure, sketched actually on his geometrical diagram--the diagram which
+proves that the law of equable description of areas can hold good with
+an ellipse. The above is a tracing of it.
+
+Such is a crude and bald sketch of the steps by which Kepler rose to his
+great generalizations--the two laws which have immortalized his name.
+
+All the complications of epicycle, equant, deferent, excentric, and the
+like, were swept at once away, and an orbit of striking and beautiful
+properties substituted. Well might he be called, as he was, "the
+legislator," or law interpreter, "of the heavens."
+
+[Illustration: FIG. 33.--If _S_ is the sun, a planet or comet moves from
+_P_ to _P_1_, from _P_2_ to _P_3_, and from _P_4_ to _P_5_ in
+the same time; if the shaded areas are equal.]
+
+He concludes his book on the motions of Mars with a half comic appeal to
+the Emperor to provide him with the sinews of war for an attack on
+Mars's relations--father Jupiter, brother Mercury, and the rest--but the
+death of his unhappy patron in 1612 put an end to all these schemes, and
+reduced Kepler to the utmost misery. While at Prague his salary was in
+continual arrear, and it was with difficulty that he could provide
+sustenance for his family. He had been there eleven years, but they had
+been hard years of poverty, and he could leave without regret were it
+not that he should have to leave Tycho's instruments and observations
+behind him. While he was hesitating what best to do, and reduced to the
+verge of despair, his wife, who had long been suffering from low spirits
+and despondency, and his three children, were taken ill; one of the sons
+died of small-pox, and the wife eleven days after of low fever and
+epilepsy. No money could be got at Prague, so after a short time he
+accepted a professorship at Linz, and withdrew with his two quite young
+remaining children.
+
+He provided for himself now partly by publishing a prophesying almanack,
+a sort of Zadkiel arrangement--a thing which he despised, but the
+support of which he could not afford to do without. He is continually
+attacking and throwing sarcasm at astrology, but it was the only thing
+for which people would pay him, and on it after a fashion he lived. We
+do not find that his circumstances were ever prosperous, and though
+8,000 crowns were due to him from Bohemia he could not manage to get
+them paid.
+
+About this time occurred a singular interruption to his work. His old
+mother, of whose fierce temper something has already been indicated, had
+been engaged in a law-suit for some years near their old home in
+W�rtemberg. A change of judge having in process of time occurred, the
+defendant saw his way to turn the tables on the old lady by accusing her
+of sorcery. She was sent to prison, and condemned to the torture, with
+the usual intelligent idea of extracting a "voluntary" confession.
+Kepler had to hurry from Linz to interpose. He succeeded in saving her
+from the torture, but she remained in prison for a year or so. Her
+spirit, however, was unbroken, for no sooner was she released than she
+commenced a fresh action against her accuser. But fresh trouble was
+averted by the death of the poor old dame at the age of nearly eighty.
+
+This narration renders the unflagging energy shown by her son in his
+mathematical wrestlings less surprising.
+
+Interspersed with these domestic troubles, and with harassing and
+unsuccessful attempts to get his rights, he still brooded over his old
+problem of some possible connection between the distances of the planets
+from the sun and their times of revolution, _i.e._ the length of their
+years.
+
+It might well have been that there was no connection, that it was purely
+imaginary, like his old idea of the law of the successive distances of
+the planets, and like so many others of the guesses and fancies which
+he entertained and spent his energies in probing. But fortunately this
+time there was a connection, and he lived to have the joy of discovering
+it.
+
+The connection is this, that if one compares the distance of the
+different planets from the sun with the length of time they take to go
+round him, the cube of the respective distances is proportional to the
+square of the corresponding times. In other words, the ratio of r^3
+to T^2 for every planet is the same. Or, again, the length of a
+planet's year depends on the 3/2th power of its distance from the sun.
+Or, once more, the speed of each planet in its orbit is as the inverse
+square-root of its distance from the sun. The product of the distance
+into the square of the speed is the same for each planet.
+
+This (however stated) is called Kepler's third law. It welds the planets
+together, and shows them to be one system. His rapture on detecting the
+law was unbounded, and he breaks out into an exulting rhapsody:--
+
+"What I prophesied two-and-twenty years ago, as soon as I discovered the
+five solids among the heavenly orbits--what I firmly believed long
+before I had seen Ptolemy's _Harmonies_--what I had promised my friends
+in the title of this book, which I named before I was sure of my
+discovery--what sixteen years ago, I urged as a thing to be sought--that
+for which I joined Tycho Brah�, for which I settled in Prague, for which
+I have devoted the best part of my life to astronomical contemplations,
+at length I have brought to light, and recognized its truth beyond my
+most sanguine expectations. It is not eighteen months since I got the
+first glimpse of light, three months since the dawn, very few days since
+the unveiled sun, most admirable to gaze upon, burst upon me. Nothing
+holds me; I will indulge my sacred fury; I will triumph over mankind by
+the honest confession that I have stolen the golden vases of the
+Egyptians to build up a tabernacle for my God far away from the
+confines of Egypt. If you forgive me, I rejoice; if you are angry, I can
+bear it; the die is cast, the book is written, to be read either now or
+by posterity, I care not which; it may well wait a century for a reader,
+as God has waited six thousand years for an observer."
+
+Soon after this great work his third book appeared: it was an epitome of
+the Copernican theory, a clear and fairly popular exposition of it,
+which had the honour of being at once suppressed and placed on the list
+of books prohibited by the Church, side by side with the work of
+Copernicus himself, _De Revolutionibus Orbium Coelestium_.
+
+This honour, however, gave Kepler no satisfaction--it rather occasioned
+him dismay, especially as it deprived him of all pecuniary benefit, and
+made it almost impossible for him to get a publisher to undertake
+another book.
+
+Still he worked on at the Rudolphine tables of Tycho, and ultimately,
+with some small help from Vienna, completed them; but he could not get
+the means to print them. He applied to the Court till he was sick of
+applying: they lay idle four years. At last he determined to pay for the
+type himself. What he paid it with, God knows, but he did pay it, and he
+did bring out the tables, and so was faithful to the behest of his
+friend.
+
+This great publication marks an era in astronomy. They were the first
+really accurate tables which navigators ever possessed; they were the
+precursors of our present _Nautical Almanack_.
+
+After this, the Grand Duke of Tuscany sent Kepler a golden chain, which
+is interesting inasmuch as it must really have come from Galileo, who
+was in high favour at the Italian Court at this time.
+
+Once more Kepler made a determined attempt to get his arrears of salary
+paid, and rescue himself and family from their bitter poverty. He
+travelled to Prague on purpose, attended the imperial meeting, and
+pleaded his own cause, but it was all fruitless; and exhausted by the
+journey, weakened by over-study, and disheartened by the failure, he
+caught a fever, and died in his fifty-ninth year. His body was buried at
+Ratisbon, and a century ago a proposal was made to erect a marble
+monument to his memory, but nothing was done. It matters little one way
+or the other whether Germany, having almost refused him bread during his
+life, should, a century and a half after his death, offer him a stone.
+
+[Illustration: FIG. 34.--Portrait of Kepler, older.]
+
+The contiguity of the lives of Kepler and Tycho furnishes a moral too
+obvious to need pointing out. What Kepler might have achieved had he
+been relieved of those ghastly struggles for subsistence one cannot
+tell, but this much is clear, that had Tycho been subjected to the same
+misfortune, instead of being born rich and being assisted by generous
+and enlightened patrons, he could have accomplished very little. His
+instruments, his observatory--the tools by which he did his work--would
+have been impossible for him. Frederick and Sophia of Denmark, and
+Rudolph of Bohemia, are therefore to be remembered as co-workers with
+him.
+
+Kepler, with his ill-health and inferior physical energy, was unable to
+command the like advantages. Much, nevertheless, he did; more one cannot
+but feel he might have done had he been properly helped. Besides, the
+world would have been free from the reproach of accepting the fruits of
+his bright genius while condemning the worker to a life of misery,
+relieved only by the beauty of his own thoughts and the ecstasy awakened
+in him by the harmony and precision of Nature.
+
+Concerning the method of Kepler, the mode by which he made his
+discoveries, we must remember that he gives us an account of all the
+steps, unsuccessful as well as successful, by which he travelled. He
+maps out his route like a traveller. In fact he compares himself to
+Columbus or Magellan, voyaging into unknown lands, and recording his
+wandering route. This being remembered, it will be found that his
+methods do not differ so utterly from those used by other philosophers
+in like case. His imagination was perhaps more luxuriant and was allowed
+freer play than most men's, but it was nevertheless always controlled by
+rigid examination and comparison of hypotheses with fact.
+
+Brewster says of him:--"Ardent, restless, burning to distinguish
+himself by discovery, he attempted everything; and once having obtained
+a glimpse of a clue, no labour was too hard in following or verifying
+it. A few of his attempts succeeded--a multitude failed. Those which
+failed seem to us now fanciful, those which succeeded appear to us
+sublime. But his methods were the same. When in search of what really
+existed he sometimes found it; when in pursuit of a chim�ra he could not
+but fail; but in either case he displayed the same great qualities, and
+that obstinate perseverance which must conquer all difficulties except
+those really insurmountable."
+
+To realize what he did for astronomy, it is necessary for us now to
+consider some science still in its infancy. Astronomy is so clear and so
+thoroughly explored now, that it is difficult to put oneself into a
+contemporary attitude. But take some other science still barely
+developed: meteorology, for instance. The science of the weather, the
+succession of winds and rain, sunshine and frost, clouds and fog, is now
+very much in the condition of astronomy before Kepler.
+
+We have passed through the stage of ascribing atmospheric
+disturbances--thunderstorms, cyclones, earthquakes, and the like--to
+supernatural agency; we have had our Copernican era: not perhaps brought
+about by a single individual, but still achieved. Something of the laws
+of cyclone and anticyclone are known, and rude weather predictions
+across the Atlantic are roughly possible. Barometers and thermometers
+and anemometers, and all their tribe, represent the astronomical
+instruments in the island of Huen; and our numerous meteorological
+observatories, with their continual record of events, represent the work
+of Tycho Brah�.
+
+Observation is heaped on observation; tables are compiled; volumes are
+filled with data; the hours of sunshine are recorded, the fall of rain,
+the moisture in the air, the kind of clouds, the temperature--millions
+of facts; but where is the Kepler to study and brood over them? Where
+is the man to spend his life in evolving the beginnings of law and order
+from the midst of all this chaos?
+
+Perhaps as a man he may not come, but his era will come. Through this
+stage the science must pass, ere it is ready for the commanding
+intellect of a Newton.
+
+But what a work it will be for the man, whoever he be that undertakes
+it--a fearful monotonous grind of calculation, hypothesis, hypothesis,
+calculation, a desperate and groping endeavour to reconcile theories
+with facts.
+
+A life of such labour, crowned by three brilliant discoveries, the world
+owes (and too late recognizes its obligation) to the harshly treated
+German genius, Kepler.
+
+
+
+
+SUMMARY OF FACTS FOR LECTURES IV AND V
+
+
+In 1564, Michael Angelo died and Galileo was born; in 1642, Galileo died
+and Newton was born. Milton lived from 1608 to 1674.
+
+For teaching the plurality of worlds, with other heterodox doctrines,
+and refusing to recant, Bruno, after six years' imprisonment in Rome,
+was burnt at the stake on the 16th of February, 1600 A.D. A "natural"
+death in the dungeons of the Inquisition saved Antonio de Dominis, the
+explainer of the rainbow, from the same fate, but his body and books
+were publicly burned at Rome in 1624.
+
+The persecution of Galileo began in 1615, became intense in 1632, and so
+lasted till his death and after.
+
+*      *      *      *      *
+
+Galileo Galilei, eldest son of Vincenzo de Bonajuti de Galilei, a noble
+Florentine, was born at Pisa, 18th of February, 1564. At the age of 17
+was sent to the University of Pisa to study medicine. Observed the swing
+of a pendulum and applied it to count pulse-beats. Read Euclid and
+Archimedes, and could be kept at medicine no more. At 26 was appointed
+Lecturer in Mathematics at Pisa. Read Bruno and became smitten with the
+Copernican theory. Controverted the Aristotelians concerning falling
+bodies, at Pisa. Hence became unpopular and accepted a chair at Padua,
+1592. Invented a thermometer. Wrote on astronomy, adopting the Ptolemaic
+system provisionally, and so opened up a correspondence with Kepler,
+with whom he formed a friendship. Lectured on the new star of 1604, and
+publicly renounced the old systems of astronomy. Invented a calculating
+compass or "Gunter's scale." In 1609 invented a telescope, after hearing
+of a Dutch optician's discovery. Invented the microscope soon after.
+Rapidly completed a better telescope and began a survey of the heavens.
+On the 8th of January, 1610, discovered Jupiter's satellites. Observed
+the mountains in the moon, and roughly measured their height. Explained
+the visibility of the new moon by _earth-shine_. Was invited to the
+Grand Ducal Court of Tuscany by Cosmo de Medici, and appointed
+philosopher to that personage. Discovered innumerable new stars, and the
+nebul�. Observed a triple appearance of Saturn. Discovered the phases
+of Venus predicted by Copernicus, and spots on the sun. Wrote on
+floating bodies. Tried to get his satellites utilized for determining
+longitude at sea.
+
+Went to Rome to defend the Copernican system, then under official
+discussion, and as a result was formally forbidden ever to teach it. On
+the accession of Pope Urban VIII. in 1623, Galileo again visited Rome to
+pay his respects, and was well received. In 1632 appeared his
+"Dialogues" on the Ptolemaic and Copernican systems. Summoned to Rome,
+practically imprisoned, and "rigorously questioned." Was made to recant
+22nd of June, 1633. Forbidden evermore to publish anything, or to teach,
+or receive friends. Retired to Arcetri in broken down health. Death of
+his favourite daughter, Sister Maria Celeste. Wrote and meditated on the
+laws of motion. Discovered the moon's libration. In 1637 he became
+blind. The rigour was then slightly relaxed and many visited him: among
+them John Milton. Died 8th of January, 1642, aged 78. As a prisoner of
+the Inquisition his right to make a will or to be buried in consecrated
+ground was disputed. Many of his manuscripts were destroyed.
+
+Galileo, besides being a singularly clear-headed thinker and
+experimental genius, was also something of a musician, a poet, and an
+artist. He was full of humour as well as of solid common-sense, and his
+literary style is brilliant. Of his scientific achievements those now
+reckoned most weighty, are the discovery of the Laws of Motion, and the
+laying of the foundations of Mechanics.
+
+_Particulars of Jupiter's Satellites,
+Illustrating their obedience to Kepler's third law._
+
+--------------------------------------------------------------------------
+          |       |            |  Distance|         |         |   T^2
+          |       |  Time of   |    from  |         |         |   ----
+Satellite.|Diameter revolution | Jupiter, |  T^2    |  d^3    |   d^3
+          | miles.|  in hours. |in Jovian |         |         | which is
+          | miles |   (T)      |   radii. |         |         |practically
+          |       |            |    (d)   |         |         | constant.
+----------|-------|------------|----------|---------|---------|-----------
+No. 1.    |  2437 |   42�47    |   6�049  |  1803�7 |  221�44 |  8�149
+No. 2.    |  2188 |   85�23    |   9�623  |  7264�1 |  891�11 |  8�152
+No. 3.    |  3575 |  177�72    |  15�350  | 29488�  | 3916�8  |  8�153
+No. 4.    |  3059 |  400�53    |  26�998  |160426�  |19679�   |  8�152
+--------------------------------------------------------------------------
+
+The diameter of Jupiter is 85,823 miles.
+
+
+_Falling Bodies._
+
+
+Since all bodies fall at the same rate, except for the disturbing effect
+of the resistance of the air, a statement of their rates of fall is of
+interest. In one second a freely falling body near the earth is found to
+drop 16 feet. In two seconds it drops 64 feet altogether, viz. 16 feet
+in the first, and 48 feet in the next second; because at the beginning
+of every second after the first it has the accumulated velocity of
+preceding seconds. The height fallen by a dropped body is not
+proportional to the time simply, but to what is rather absurdly called
+the square of the time, _i.e._ the time multiplied by itself.
+
+For instance, in 3 seconds it drops 9 � 16 = 144 feet; in 4 seconds 16 �
+16, or 256 feet, and so on. The distances travelled in 1, 2, 3, 4, &c.,
+seconds by a body dropped from rest and not appreciably resisted by the
+air, are 1, 4, 9, 16, 25, &c., respectively, each multiplied by the
+constant 16 feet.
+
+Another way of stating the law is to say that the heights travelled in
+successive seconds proceed in the proportion 1, 3, 5, 7, 9, &c.; again
+multiplied by 16 feet in each case.
+
+[Illustration: FIG. 35.--Curve described by a projectile, showing how it
+drops from the line of fire, _O D_, in successive seconds, the same
+distances _AP_, _BQ_, _CR_, &c., as are stated above for a dropped
+body.]
+
+All this was experimentally established by Galileo.
+
+A body takes half a second to drop 4 feet; and a quarter of a second to
+drop 1 foot. The easiest way of estimating a quarter of a second with
+some accuracy is to drop a bullet one foot.
+
+A bullet thrown or shot in any direction falls just as much as if merely
+dropped; but instead of falling from the starting-point it drops
+vertically from the line of fire. (See fig. 35).
+
+The rate of fall depends on the intensity of gravity; if it could be
+doubled, a body would fall twice as far in the same time; but to make it
+fall a given distance in half the time the intensity of gravity would
+have to be quadrupled. At a place where the intensity of gravity is
+1/3600 of what it is here, a body would fall as far in a minute as it
+now falls in a second. Such a place occurs at about the distance of the
+moon (_cf._ page 177).
+
+The fact that the height fallen through is proportional to the square
+of the time proves that the attraction of the earth or the intensity of
+gravity is sensibly constant throughout ordinary small ranges. Over
+great distances of fall, gravity cannot be considered constant; so for
+things falling through great spaces the Galilean law of the square of
+the time does not hold.
+
+The fact that things near the earth fall 16 feet in the first second
+proves that the intensity of ordinary terrestrial gravity is 32 British
+units of force per pound of matter.
+
+The fact that all bodies fall at the same rate (when the resistance of
+the air is eliminated), proves that weight is proportional to mass; or
+more explicitly, that the gravitative attraction of the earth on matter
+near its surface depends on the amount of that matter, as estimated by
+its inertia, and on nothing else.
+
+
+
+
+LECTURE IV
+
+GALILEO AND THE INVENTION OF THE TELESCOPE
+
+
+Contemporary with the life of Kepler, but overlapping it at both ends,
+comes the great and eventful life of Galileo Galilei,[5] a man whose
+influence on the development of human thought has been greater than that
+of any man whom we have yet considered, and upon whom, therefore, it is
+necessary for us, in order to carry out the plan of these lectures, to
+bestow much time. A man of great and wide culture, a so-called universal
+genius, it is as an experimental philosopher that he takes the first
+rank. In this capacity he must be placed alongside of Archimedes, and it
+is pretty certain that between the two there was no man of magnitude
+equal to either in experimental philosophy. It is perhaps too bold a
+speculation, but I venture to doubt whether in succeeding generations we
+find his equal in the domain of purely experimental science until we
+come to Faraday. Faraday was no doubt his superior, but I know of no
+other of whom the like can unhesitatingly be said. In mathematical and
+deductive science, of course, it is quite otherwise. Kepler, for
+instance, and many men before and since, have far excelled Galileo in
+mathematical skill and power, though at the same time his achievements
+in this department are by no means to be despised.
+
+Born at Pisa three centuries ago, on the very day that Michael Angelo
+lay dying in Rome, he inherited from his father a noble name, cultivated
+tastes, a keen love of truth, and an impoverished patrimony. Vincenzo de
+Galilei, a descendant of the important Bonajuti family, was himself a
+mathematician and a musician, and in a book of his still extant he
+declares himself in favour of free and open inquiry into scientific
+matters, unrestrained by the weight of authority and tradition.
+
+In all probability the son imbibed these precepts: certainly he acted on
+them.
+
+Vincenzo, having himself experienced the unremunerative character of
+scientific work, had a horror of his son's taking to it, especially as
+in his boyhood he was always constructing ingenious mechanical toys, and
+exhibiting other marks of precocity. So the son was destined for
+business--to be, in fact, a cloth-dealer. But he was to receive a good
+education first, and was sent to an excellent convent school.
+
+Here he made rapid progress, and soon excelled in all branches of
+classics and literature. He delighted in poetry, and in later years
+wrote several essays on Dante, Tasso, and Ariosto, besides composing
+some tolerable poems himself. He played skilfully on several musical
+instruments, especially on the lute, of which indeed he became a master,
+and on which he solaced himself when quite an old man. Besides this he
+seems to have had some skill as an artist, which was useful afterwards
+in illustrating his discoveries, and to have had a fine sensibility as
+an art critic, for we find several eminent painters of that day
+acknowledging the value of the opinion of the young Galileo.
+
+Perceiving all this display of ability, the father wisely came to the
+conclusion that the selling of woollen stuffs would hardly satisfy his
+aspirations for long, and that it was worth a sacrifice to send him to
+the University. So to the University of his native town he went, with
+the avowed object of studying medicine, that career seeming the most
+likely to be profitable. Old Vincenzo's horror of mathematics or science
+as a means of obtaining a livelihood is justified by the fact that while
+the University Professor of Medicine received 2,000 scudi a year, the
+Professor of Mathematics had only 60, that is �13 a year, or 7-1/2_d._ a
+day.
+
+So the son had been kept properly ignorant of such poverty-stricken
+subjects, and to study medicine he went.
+
+But his natural bent showed itself even here. For praying one day in the
+Cathedral, like a good Catholic as he was all his life, his attention
+was arrested by the great lamp which, after lighting it, the verger had
+left swinging to and fro. Galileo proceeded to time its swings by the
+only watch he possessed--viz., his own pulse. He noticed that the time
+of swing remained as near as he could tell the same, notwithstanding the
+fact that the swings were getting smaller and smaller.
+
+By subsequent experiment he verified the law, and the isochronism of the
+pendulum was discovered. An immensely important practical discovery
+this, for upon it all modern clocks are based; and Huyghens soon applied
+it to the astronomical clock, which up to that time had been a crude and
+quite untrustworthy instrument.
+
+The best clock which Tycho Brah� could get for his observatory was
+inferior to one that may now be purchased for a few shillings; and this
+change is owing to the discovery of the pendulum by Galileo. Not that he
+applied it to clocks; he was not thinking of astronomy, he was thinking
+of medicine, and wanted to count people's pulses. The pendulum served;
+and "pulsilogies," as they were called, were thus introduced to and used
+by medical practitioners.
+
+The Tuscan Court came to Pisa for the summer months, for it was then a
+seaside place, and among the suite was Ostillio Ricci, a distinguished
+mathematician and old friend of the Galileo family. The youth visited
+him, and one day, it is said, heard a lesson in Euclid being given by
+Ricci to the pages while he stood outside the door entranced. Anyhow he
+implored Ricci to help him into some knowledge of mathematics, and the
+old man willingly consented. So he mastered Euclid and passed on to
+Archimedes, for whom he acquired a great veneration.
+
+His father soon heard of this obnoxious proclivity, and did what he
+could to divert him back to medicine again. But it was no use.
+Underneath his Galen and Hippocrates were secreted copies of Euclid and
+Archimedes, to be studied at every available opportunity. Old Vincenzo
+perceived the bent of genius to be too strong for him, and at last gave
+way.
+
+[Illustration: FIG. 36.--Two forms of pulsilogy. The string is wound up
+till the swinging weight keeps time with the pulse, and the position of
+a bead or of an index connected with the string is then read on a scale
+or dial.]
+
+With prodigious rapidity the released philosopher now assimilated the
+elements of mathematics and physics, and at twenty-six we find him
+appointed for three years to the University Chair of Mathematics, and
+enjoying the paternally dreaded stipend of 7-1/2_d._ a day.
+
+Now it was that he pondered over the laws of falling bodies. He
+verified, by experiment, the fact that the velocity acquired by falling
+down any slope of given height was independent of the angle of slope.
+Also, that the height fallen through was proportional to the square of
+the time.
+
+Another thing he found experimentally was that all bodies, heavy and
+light, fell at the same rate, striking the ground at the same time.[6]
+
+Now this was clean contrary to what he had been taught. The physics of
+those days were a simple reproduction of statements in old books.
+Aristotle had asserted certain things to be true, and these were
+universally believed. No one thought of trying the thing to see if it
+really were so. The idea of making an experiment would have savoured of
+impiety, because it seemed to tend towards scepticism, and cast a doubt
+on a reverend authority.
+
+Young Galileo, with all the energy and imprudence of youth (what a
+blessing that youth has a little imprudence and disregard of
+consequences in pursuing a high ideal!), as soon as he perceived that
+his instructors were wrong on the subject of falling bodies, instantly
+informed them of the fact. Whether he expected them to be pleased or not
+is a question. Anyhow, they were not pleased, but were much annoyed by
+his impertinent arrogance.
+
+It is, perhaps, difficult for us now to appreciate precisely their
+position. These doctrines of antiquity, which had come down hoary with
+age, and the discovery of which had reawakened learning and quickened
+intellectual life, were accepted less as a science or a philosophy, than
+as a religion. Had they regarded Aristotle as a verbally inspired
+writer, they could not have received his statements with more
+unhesitating conviction. In any dispute as to a question of fact, such
+as the one before us concerning the laws of falling bodies, their method
+was not to make an experiment, but to turn over the pages of Aristotle;
+and he who could quote chapter and verse of this great writer was held
+to settle the question and raise it above the reach of controversy.
+
+It is very necessary for us to realize this state of things clearly,
+because otherwise the attitude of the learned of those days towards
+every new discovery seems stupid and almost insane. They had a
+crystallized system of truth, perfect, symmetrical--it wanted no
+novelty, no additions; every addition or growth was an imperfection, an
+excrescence, a deformity. Progress was unnecessary and undesired. The
+Church had a rigid system of dogma, which must be accepted in its
+entirety on pain of being treated as a heretic. Philosophers had a
+cast-iron system of truth to match--a system founded upon Aristotle--and
+so interwoven with the great theological dogmas that to question one was
+almost equivalent to casting doubt upon the other.
+
+In such an atmosphere true science was impossible. The life-blood of
+science is growth, expansion, freedom, development. Before it could
+appear it must throw off these old shackles of centuries. It must burst
+its old skin, and emerge, worn with the struggle, weakly and
+unprotected, but free and able to grow and to expand. The conflict was
+inevitable, and it was severe. Is it over yet? I fear not quite, though
+so nearly as to disturb science hardly at all. Then it was different; it
+was terrible. Honour to the men who bore the first shock of the battle!
+
+Now Aristotle had said that bodies fell at rates depending on their
+weight.
+
+A 5 lb. weight would fall five times as quick as a 1 lb. weight; a 50
+lb. weight fifty times as quick, and so on.
+
+Why he said so nobody knows. He cannot have tried. He was not above
+trying experiments, like his smaller disciples; but probably it never
+occurred to him to doubt the fact. It seems so natural that a heavy body
+should fall quicker than a light one; and perhaps he thought of a stone
+and a feather, and was satisfied.
+
+Galileo, however, asserted that the weight did not matter a bit, that
+everything fell at the same rate (even a stone and a feather, but for
+the resistance of the air), and would reach the ground in the same time.
+
+And he was not content to be pooh-poohed and snubbed. He knew he was
+right, and he was determined to make every one see the facts as he saw
+them. So one morning, before the assembled University, he ascended the
+famous leaning tower, taking with him a 100 lb. shot and a 1 lb. shot.
+He balanced them on the edge of the tower, and let them drop together.
+Together they fell, and together they struck the ground.
+
+The simultaneous clang of those two weights sounded the death-knell of
+the old system of philosophy, and heralded the birth of the new.
+
+But was the change sudden? Were his opponents convinced? Not a jot.
+Though they had seen with their eyes, and heard with their ears, the
+full light of heaven shining upon them, they went back muttering and
+discontented to their musty old volumes and their garrets, there to
+invent occult reasons for denying the validity of the observation, and
+for referring it to some unknown disturbing cause.
+
+They saw that if they gave way on this one point they would be letting
+go their anchorage, and henceforward would be liable to drift along with
+the tide, not knowing whither. They dared not do this. No; they _must_
+cling to the old traditions; they could not cast away their rotting
+ropes and sail out on to the free ocean of God's truth in a spirit of
+fearless faith.
+
+[Illustration: FIG. 37.--Tower of Pisa.]
+
+Yet they had received a shock: as by a breath of fresh salt breeze and
+a dash of spray in their faces, they had been awakened out of their
+comfortable lethargy. They felt the approach of a new era.
+
+Yes, it was a shock; and they hated the young Galileo for giving it
+them--hated him with the sullen hatred of men who fight for a lost and
+dying cause.
+
+We need scarcely blame these men; at least we need not blame them
+overmuch. To say that they acted as they did is to say that they were
+human, were narrow-minded, and were the apostles of a lost cause. But
+_they_ could not know this; _they_ had no experience of the past to
+guide them; the conditions under which they found themselves were novel,
+and had to be met for the first time. Conduct which was excusable then
+would be unpardonable now, in the light of all this experience to guide
+us. Are there any now who practically repeat their error, and resist new
+truth? who cling to any old anchorage of dogma, and refuse to rise with
+the tide of advancing knowledge? There may be some even now.
+
+Well, the unpopularity of Galileo smouldered for a time, until, by
+another noble imprudence, he managed to offend a semi-royal personage,
+Giovanni de Medici, by giving his real opinion, when consulted, about a
+machine which de Medici had invented for cleaning out the harbour of
+Leghorn. He said it was as useless as it in fact turned out to be.
+Through the influence of the mortified inventor he lost favour at Court;
+and his enemies took advantage of the fact to render his chair
+untenable. He resigned before his three years were up, and retired to
+Florence.
+
+His father at this time died, and the family were left in narrow
+circumstances. He had a brother and three sisters to provide for.
+
+He was offered a professorship at Padua for six years by the Senate of
+Venice, and willingly accepted it.
+
+Now began a very successful career. His introductory address was marked
+by brilliant eloquence, and his lectures soon acquired fame. He wrote
+for his pupils on the laws of motion, on fortifications, on sundials, on
+mechanics, and on the celestial globe: some of these papers are now
+lost, others have been printed during the present century.
+
+Kepler sent him a copy of his new book, _Mysterium Cosmographicum_, and
+Galileo in thanking him for it writes him the following letter:--[7]
+
+     "I count myself happy, in the search after truth, to have so great
+     an ally as yourself, and one who is so great a friend of the truth
+     itself. It is really pitiful that there are so few who seek truth,
+     and who do not pursue a perverse method of philosophising. But this
+     is not the place to mourn over the miseries of our times, but to
+     congratulate you on your splendid discoveries in confirmation of
+     truth. I shall read your book to the end, sure of finding much that
+     is excellent in it. I shall do so with the more pleasure, because
+     _I have been for many years an adherent of the Copernican system_,
+     and it explains to me the causes of many of the appearances of
+     nature which are quite unintelligible on the commonly accepted
+     hypothesis. _I have collected many arguments for the purpose of
+     refuting the latter_; but I do not venture to bring them to the
+     light of publicity, for fear of sharing the fate of our master,
+     Copernicus, who, although he has earned immortal fame with some,
+     yet with very many (so great is the number of fools) has become an
+     object of ridicule and scorn. I should certainly venture to publish
+     my speculations if there were more people like you. But this not
+     being the case, I refrain from such an undertaking."
+
+Kepler urged him to publish his arguments in favour of the Copernican
+theory, but he hesitated for the present, knowing that his declaration
+would be received with ridicule and opposition, and thinking it wiser to
+get rather more firmly seated in his chair before encountering the
+storm of controversy.
+
+The six years passed away, and the Venetian Senate, anxious not to lose
+so bright an ornament, renewed his appointment for another six years at
+a largely increased salary.
+
+Soon after this appeared a new star, the stella nova of 1604, not the
+one Tycho had seen--that was in 1572--but the same that Kepler was so
+much interested in.
+
+Galileo gave a course of three lectures upon it to a great audience. At
+the first the theatre was over-crowded, so he had to adjourn to a hall
+holding 1000 persons. At the next he had to lecture in the open air.
+
+He took occasion to rebuke his hearers for thronging to hear about an
+ephemeral novelty, while for the much more wonderful and important
+truths about the permanent stars and facts of nature they had but deaf
+ears.
+
+But the main point he brought out concerning the new star was that it
+upset the received Aristotelian doctrine of the immutability of the
+heavens. According to that doctrine the heavens were unchangeable,
+perfect, subject neither to growth nor to decay. Here was a body, not a
+meteor but a real distant star, which had not been visible and which
+would shortly fade away again, but which meanwhile was brighter than
+Jupiter.
+
+The staff of petrified professorial wisdom were annoyed at the
+appearance of the star, still more at Galileo's calling public attention
+to it; and controversy began at Padua. However, he accepted it; and now
+boldly threw down the gauntlet in favour of the Copernican theory,
+utterly repudiating the old Ptolemaic system which up to that time he
+had taught in the schools according to established custom.
+
+The earth no longer the only world to which all else in the firmament
+were obsequious attendants, but a mere insignificant speck among the
+host of heaven! Man no longer the centre and cynosure of creation, but,
+as it were, an insect crawling on the surface of this little speck! All
+this not set down in crabbed Latin in dry folios for a few learned
+monks, as in Copernicus's time, but promulgated and argued in rich
+Italian, illustrated by analogy, by experiment, and with cultured wit;
+taught not to a few scholars here and there in musty libraries, but
+proclaimed in the vernacular to the whole populace with all the energy
+and enthusiasm of a recent convert and a master of language! Had a
+bombshell been exploded among the fossilized professors it had been less
+disturbing.
+
+But there was worse in store for them.
+
+A Dutch optician, Hans Lippershey by name, of Middleburg, had in his
+shop a curious toy, rigged up, it is said, by an apprentice, and made
+out of a couple of spectacle lenses, whereby, if one looked through it,
+the weather-cock of a neighbouring church spire was seen nearer and
+upside down.
+
+The tale goes that the Marquis Spinola, happening to call at the shop,
+was struck with the toy and bought it. He showed it to Prince Maurice of
+Nassau, who thought of using it for military reconnoitring. All this is
+trivial. What is important is that some faint and inaccurate echo of
+this news found its way to Padua, and into the ears of Galileo.
+
+The seed fell on good soil. All that night he sat up and pondered. He
+knew about lenses and magnifying glasses. He had read Kepler's theory of
+the eye, and had himself lectured on optics. Could he not hit on the
+device and make an instrument capable of bringing the heavenly bodies
+nearer? Who knew what marvels he might not so perceive! By morning he
+had some schemes ready to try, and one of them was successful.
+Singularly enough it was not the same plan as the Dutch optician's, it
+was another mode of achieving the same end.
+
+He took an old small organ pipe, jammed a suitably chosen spectacle
+glass into either end, one convex the other concave, and behold, he had
+the half of a wretchedly bad opera glass capable of magnifying three
+times. It was better than the Dutchman's, however; it did not invert.
+
+     It is easy to understand the general principle of a telescope. A
+     general knowledge of the common magnifying glass may be assumed.
+     Roger Bacon knew about lenses; and the ancients often refer to
+     them, though usually as burning glasses. The magnifying power of
+     globes of water must have been noticed soon after the discovery of
+     glass and the art of working it.
+
+     A magnifying glass is most simply thought of as an additional lens
+     to the eye. The eye has a lens by which ordinary vision is
+     accomplished, an extra glass lens strengthens it and enables
+     objects to be seen nearer and therefore apparently bigger. But to
+     apply a magnifying glass to distant objects is impossible. In order
+     to magnify distant objects, another function of lenses has also to
+     be employed, viz., their power of forming real images, the power on
+     which their use as burning-glasses depends: for the best focus is
+     an image of the sun. Although the object itself is inaccessible,
+     the image of it is by no means so, and to the image a magnifier can
+     be applied. This is exactly what is done in the telescope; the
+     object glass or large lens forms an image, which is then looked at
+     through a magnifying glass or eye-piece.
+
+     Of course the image is nothing like so big as the object. For
+     astronomical objects it is almost infinitely less; still it is an
+     exact representation at an accessible place, and no one expects a
+     telescope to show distant bodies as big as they really are. All it
+     does is to show them bigger than they could be seen without it.
+
+     But if the objects are not distant, the same principle may still be
+     applied, and two lenses may be used, one to form an image, the
+     other to magnify it; only if the object can be put where we please,
+     we can easily place it so that its image is already much bigger
+     than the object even before magnification by the eye lens. This is
+     the compound microscope, the invention of which soon followed the
+     telescope. In fact the two instruments shade off into one another,
+     so that the reading telescope or reading microscope of a laboratory
+     (for reading thermometers, and small divisions generally) goes by
+     either name at random.
+
+     The arrangement so far described depicts things on the retina the
+     unaccustomed way up. By using a concave glass instead of a convex,
+     and placing it so as to prevent any image being formed, except on
+     the retina direct, this inconvenience is avoided.
+
+[Illustration: FIG. 38.--View of the half-moon in small telescope. The
+darker regions, or plains, used to be called "seas."]
+
+Such a thing as Galileo made may now be bought at a toy-shop for I
+suppose half a crown, and yet what a potentiality lay in that "glazed
+optic tube," as Milton called it. Away he went with it to Venice and
+showed it to the Signoria, to their great astonishment. "Many noblemen
+and senators," says Galileo, "though of advanced age, mounted to the top
+of one of the highest towers to watch the ships, which were visible
+through my glass two hours before they were seen entering the harbour,
+for it makes a thing fifty miles off as near and clear as if it were
+only five." Among the people too the instrument excited the greatest
+astonishment and interest, so that he was nearly mobbed. The Senate
+hinted to him that a present of the instrument would not be
+unacceptable, so Galileo took the hint and made another for them.
+
+[Illustration: FIG. 39.--Portion of the lunar surface more highly
+magnified, showing the shadows of a mountain range, deep pits, and other
+details.]
+
+They immediately doubled his salary at Padua, making it 1000 florins,
+and confirmed him in the enjoyment of it for life.
+
+He now eagerly began the construction of a larger and better instrument.
+Grinding the lenses with his own hands with consummate skill, he
+succeeded in making a telescope magnifying thirty times. Thus equipped
+he was ready to begin a survey of the heavens.
+
+[Illustration: FIG. 40.--Another portion of the lunar surface, showing a
+so-called crater or vast lava pool and other evidences of ancient heat
+unmodified by water.]
+
+The first object he carefully examined was naturally the moon. He found
+there everything at first sight very like the earth, mountains and
+valleys, craters and plains, rocks, and apparently seas. You may imagine
+the hostility excited among the Aristotelian philosophers, especially no
+doubt those he had left behind at Pisa, on the ground of his spoiling
+the pure, smooth, crystalline, celestial face of the moon as they had
+thought it, and making it harsh and rugged and like so vile and ignoble
+a body as the earth.
+
+[Illustration: FIG. 41.--Lunar landscape showing earth. The earth would
+be a stationary object in the moon's sky: its only apparent motion being
+a slow oscillation as of a pendulum (the result of the moon's
+libration).]
+
+He went further, however, into heterodoxy than this--he not only made
+the moon like the earth, but he made the earth shine like the moon. The
+visibility of "the old moon in the new moon's arms" he explained by
+earth-shine. Leonardo had given the same explanation a century before.
+Now one of the many stock arguments against Copernican theory of the
+earth being a planet like the rest was that the earth was dull and dark
+and did not shine. Galileo argued that it shone just as much as the moon
+does, and in fact rather more--especially if it be covered with clouds.
+One reason of the peculiar brilliancy of Venus is that she is a very
+cloudy planet.[8] Seen from the moon the earth would look exactly as the
+moon does to us, only a little brighter and sixteen times as big (four
+times the diameter).
+
+[Illustration: FIG. 42.--Galileo's method of estimating the height of
+lunar mountain.
+
+_AB'BC_ is the illuminated half of the moon. _SA_ is a solar ray just
+catching the peak of the mountain _M_. Then by geometry, as _MN_ is to
+_MA_, so is _MA_ to _MB'_; whence the height of the mountain, _MN_, can
+be determined. The earth and spectator are supposed to be somewhere in
+the direction _BA_ produced, _i.e._ towards the top of the page.]
+
+     Galileo made a very good estimate of the height of lunar mountains,
+     of which many are five miles high and some as much as seven. He did
+     this simply by measuring from the half-moon's straight edge the
+     distance at which their peaks caught the rising or setting sun. The
+     above simple diagram shows that as this distance is to the diameter
+     of the moon, so is the height of the sun-tipped mountain to the
+     aforesaid distance.
+
+Wherever Galileo turned his telescope new stars appeared. The Milky Way,
+which had so puzzled the ancients, was found to be composed of stars.
+Stars that appeared single to the eye were some of them found to be
+double; and at intervals were found hazy nebulous wisps, some of which
+seemed to be star clusters, while others seemed only a fleecy cloud.
+
+[Illustration: FIG. 43.--Some clusters and nebul�.]
+
+[Illustration: FIG. 44.--Jupiter's satellites, showing the stages of
+their discovery.]
+
+Now we come to his most brilliant, at least his most sensational,
+discovery. Examining Jupiter minutely on January 7, 1610, he noticed
+three little stars near it, which he noted down as fixing its then
+position. On the following night Jupiter had moved to the other side of
+the three stars. This was natural enough, but was it moving the right
+way? On examination it appeared not. Was it possible the tables were
+wrong? The next evening was cloudy, and he had to curb his feverish
+impatience. On the 10th there were only two, and those on the other
+side. On the 11th two again, but one bigger than the other. On the 12th
+the three re-appeared, and on the 13th there were four. No more
+appeared.
+
+Jupiter then had moons like the earth, four of them in fact, and they
+revolved round him in periods which were soon determined.
+
+     The reason why they were not all visible at first, and why their
+     visibility so rapidly changes, is because they revolve round him
+     almost in the plane of our vision, so that sometimes they are in
+     front and sometimes behind him, while again at other times they
+     plunge into his shadow and are thus eclipsed from the light of the
+     sun which enables us to see them. A large modern telescope will
+     show the moons when in front of Jupiter, but small telescopes will
+     only show them when clear of the disk and shadow. Often all four
+     can be thus seen, but three or two is a very common amount of
+     visibility. Quite a small telescope, such as a ship's telescope, if
+     held steadily, suffices to show the satellites of Jupiter, and very
+     interesting objects they are. They are of habitable size, and may
+     be important worlds for all we know to the contrary.
+
+The news of the discovery soon spread and excited the greatest interest
+and astonishment. Many of course refused to believe it. Some there were
+who having been shown them refused to believe their eyes, and asserted
+that although the telescope acted well enough for terrestrial objects,
+it was altogether false and illusory when applied to the heavens. Others
+took the safer ground of refusing to look through the glass. One of
+these who would not look at the satellites happened to die soon
+afterwards. "I hope," says Galileo, "that he saw them on his way to
+heaven."
+
+The way in which Kepler received the news is characteristic, though by
+adding four to the supposed number of planets it might have seemed to
+upset his notions about the five regular solids.
+
+     He says,[9] "I was sitting idle at home thinking of you, most
+     excellent Galileo, and your letters, when the news was brought me
+     of the discovery of four planets by the help of the double
+     eye-glass. Wachenfels stopped his carriage at my door to tell me,
+     when such a fit of wonder seized me at a report which seemed so
+     very absurd, and I was thrown into such agitation at seeing an old
+     dispute between us decided in this way, that between his joy, my
+     colouring, and the laughter of us both, confounded as we were by
+     such a novelty, we were hardly capable, he of speaking, or I of
+     listening....
+
+     "On our separating, I immediately fell to thinking how there could
+     be any addition to the number of planets without overturning my
+     _Mysterium Cosmographicon_, published thirteen years ago, according
+     to which Euclid's five regular solids do not allow more than six
+     planets round the sun.
+
+     "But I am so far from disbelieving the existence of the four
+     circumjovial planets that I long for a telescope to anticipate you
+     if possible in discovering two round Mars (as the proportion seems
+     to me to require) six or eight round Saturn, and one each round
+     Mercury and Venus."
+
+[Illustration: FIG. 45.--Eclipses of Jupiter's satellites. The diagram
+shows the first (_i.e._ the nearest) moon in Jupiter's shadow, the
+second as passing between earth and Jupiter, and appearing to transit
+his disk, the third as on the verge of entering his shadow, and the
+fourth quite plainly and separately visible.]
+
+As an illustration of the opposite school, I will take the following
+extract from Francesco Sizzi, a Florentine astronomer, who argues
+against the discovery thus:--
+
+     "There are seven windows in the head, two nostrils, two eyes, two
+     ears, and a mouth; so in the heavens there are two favourable
+     stars, two unpropitious, two luminaries, and Mercury alone
+     undecided and indifferent. From which and many other similar
+     phenomena of nature, such as the seven metals, &c., which it were
+     tedious to enumerate, we gather that the number of planets is
+     necessarily seven.
+
+     "Moreover, the satellites are invisible to the naked eye, and
+     therefore can have no influence on the earth, and therefore would
+     be useless, and therefore do not exist.
+
+     "Besides, the Jews and other ancient nations as well as modern
+     Europeans have adopted the division of the week into seven days,
+     and have named them from the seven planets: now if we increase the
+     number of the planets this whole system falls to the ground."
+
+To these arguments Galileo replied that whatever their force might be as
+a reason for believing beforehand that no more than seven planets would
+be discovered, they hardly seemed of sufficient weight to destroy the
+new ones when actually seen.
+
+Writing to Kepler at this time, Galileo ejaculates:
+
+     "Oh, my dear Kepler, how I wish that we could have one hearty laugh
+     together! Here, at Padua, is the principal professor of philosophy
+     whom I have repeatedly and urgently requested to look at the moon
+     and planets through my glass, which he pertinaciously refuses to
+     do. Why are you not here? What shouts of laughter we should have at
+     this glorious folly! And to hear the professor of philosophy at
+     Pisa labouring before the grand duke with logical arguments, as if
+     with magical incantations, to charm the new planets out of the
+     sky."
+
+A young German _prot�g�_ of Kepler, Martin Horkey, was travelling in
+Italy, and meeting Galileo at Bologna was favoured with a view through
+his telescope. But supposing that Kepler must necessarily be jealous of
+such great discoveries, and thinking to please him, he writes, "I cannot
+tell what to think about these observations. They are stupendous, they
+are wonderful, but whether they are true or false I cannot tell." He
+concludes, "I will never concede his four new planets to that Italian
+from Padua though I die for it." So he published a pamphlet asserting
+that reflected rays and optical illusions were the sole cause of the
+appearance, and that the only use of the imaginary planets was to
+gratify Galileo's thirst for gold and notoriety.
+
+When after this performance he paid a visit to his old instructor
+Kepler, he got a reception which astonished him. However, he pleaded so
+hard to be forgiven that Kepler restored him to partial favour, on this
+condition, that he was to look again at the satellites, and this time to
+see them and own that they were there.
+
+By degrees the enemies of Galileo were compelled to confess to the truth
+of the discovery, and the next step was to outdo him. Scheiner counted
+five, Rheiter nine, and others went as high as twelve. Some of these
+were imaginary, some were fixed stars, and four satellites only are
+known to this day.[10]
+
+Here, close to the summit of his greatness, we must leave him for a
+time. A few steps more and he will be on the brow of the hill; a short
+piece of table-land, and then the descent begins.
+
+
+
+
+LECTURE V
+
+GALILEO AND THE INQUISITION
+
+
+One sinister event occurred while Galileo was at Padua, some time before
+the era we have now arrived at, before the invention of the
+telescope--two years indeed after he had first gone to Padua; an event
+not directly concerning Galileo, but which I must mention because it
+must have shadowed his life both at the time and long afterwards. It was
+the execution of Giordano Bruno for heresy. This eminent philosopher had
+travelled largely, had lived some time in England, had acquired new and
+heterodox views on a variety of subjects, and did not hesitate to
+propound them even after he had returned to Italy.
+
+The Copernican doctrine of the motion of the earth was one of his
+obnoxious heresies. Being persecuted to some extent by the Church, Bruno
+took refuge in Venice--a free republic almost independent of the
+Papacy--where he felt himself safe. Galileo was at Padua hard by: the
+University of Padua was under the government of the Senate of Venice:
+the two men must in all probability have met.
+
+Well, the Inquisition at Rome sent messengers to Venice with a demand
+for the extradition of Bruno--they wanted him at Rome to try him for
+heresy.
+
+In a moment of miserable weakness the Venetian republic gave him up, and
+Bruno was taken to Rome. There he was tried, and cast into the dungeons
+for six years, and because he entirely refused to recant, was at length
+delivered over to the secular arm and burned at the stake on 16th
+February, Anno Domini 1600.
+
+This event could not but have cast a gloom over the mind of lovers and
+expounders of truth, and the lesson probably sank deep into Galileo's
+soul.
+
+In dealing with these historic events will you allow me to repudiate
+once for all the slightest sectarian bias or meaning. I have nothing to
+do with Catholic or Protestant as such. I have nothing to do with the
+Church of Rome as such. I am dealing with the history of science. But
+historically at one period science and the Church came into conflict. It
+was not specially one Church rather than another--it was the Church in
+general, the only one that then existed in those countries.
+Historically, I say, they came into conflict, and historically the
+Church was the conqueror. It got its way; and science, in the persons of
+Bruno, Galileo, and several others, was vanquished.
+
+Such being the facts, there is no help but to mention them in dealing
+with the history of science.
+
+Doubtless _now_ the Church regards it as an unhappy victory, and gladly
+would ignore this painful struggle. This, however, is impossible. With
+their creed the Churchmen of that day could act in no other way. They
+were bound to prosecute heresy, and they were bound to conquer in the
+struggle or be themselves shattered.
+
+But let me insist on the fact that no one accuses the ecclesiastical
+courts of crime or evil motives. They attacked heresy after their
+manner, as the civil courts attacked witchcraft after _their_ manner.
+Both erred grievously, but both acted with the best intentions.
+
+We must remember, moreover, that his doctrines were scientifically
+heterodox, and the University Professors of that day were probably quite
+as ready to condemn them as the Church was. To realise the position we
+must think of some subjects which _to-day_ are scientifically
+heterodox, and of the customary attitude adopted towards them by
+persons of widely differing creeds.
+
+If it be contended now, as it is, that the ecclesiastics treated Galileo
+well, I admit it freely: they treated him as well as they possibly
+could. They overcame him, and he recanted; but if he had not recanted,
+if he had persisted in his heresy, they would--well, they would still
+have treated his soul well, but they would have set fire to his body.
+Their mistake consisted not in cruelty, but in supposing themselves the
+arbiters of eternal truth; and by no amount of slurring and glossing
+over facts can they evade the responsibility assumed by them on account
+of this mistaken attitude.
+
+I am not here attacking the dogma of Papal Infallibility: it is
+historically, I believe, quite unaffected by the controversy respecting
+the motion of the earth, no Papal edict _ex cathedr�_ having been
+promulgated on the subject.
+
+We left Galileo standing at his telescope and beginning his survey of
+the heavens. We followed him indeed through a few of his first great
+discoveries--the discovery of the mountains and other variety of surface
+in the moon, of the nebul� and a multitude of faint stars, and lastly of
+the four satellites of Jupiter.
+
+This latter discovery made an immense sensation, and contributed its
+share to his removal from Padua, which quickly followed it, as I shall
+shortly narrate; but first I think it will be best to continue our
+survey of his astronomical discoveries without regard to the place
+whence they were made.
+
+Before the end of the year Galileo had made another discovery--this time
+on Saturn. But to guard against the host of plagiarists and impostors,
+he published it in the form of an anagram, which, at the request of the
+Emperor Rudolph (a request probably inspired by Kepler), he interpreted;
+it ran thus: The furthest planet is triple.
+
+Very soon after he found that Venus was changing from a full moon to a
+half moon appearance. He announced this also by an anagram, and waited
+till it should become a crescent, which it did.
+
+This was a dreadful blow to the anti-Copernicans, for it removed the
+last lingering difficulty to the reception of the Copernican doctrine.
+
+[Illustration: FIG. 46.--Old drawings of Saturn by different observers,
+with the imperfect instruments of that day. The first is Galileo's idea
+of what he saw.]
+
+Copernicus had predicted, indeed, a hundred years before, that, if ever
+our powers of sight were sufficiently enhanced, Venus and Mercury would
+be seen to have phases like the moon. And now Galileo with his
+telescope verifies the prediction to the letter.
+
+Here was a triumph for the grand old monk, and a bitter morsel for his
+opponents.
+
+     Castelli writes: "This must now convince the most obstinate." But
+     Galileo, with more experience, replies:--"You almost make me laugh
+     by saying that these clear observations are sufficient to convince
+     the most obstinate; it seems you have yet to learn that long ago
+     the observations were enough to convince those who are capable of
+     reasoning, and those who wish to learn the truth; but that to
+     convince the obstinate, and those who care for nothing beyond the
+     vain applause of the senseless vulgar, not even the testimony of
+     the stars would suffice, were they to descend on earth to speak for
+     themselves. Let us, then, endeavour to procure some knowledge for
+     ourselves, and rest contented with this sole satisfaction; but of
+     advancing in popular opinion, or of gaining the assent of the
+     book-philosophers, let us abandon both the hope and the desire."
+
+[Illustration: FIG. 47.--Phases of Venus. Showing also its apparent
+variations in size by reason of its varying distance from the earth.
+When fully illuminated it is necessarily most distant. It looks
+brightest to us when a broad crescent.]
+
+What a year's work it had been!
+
+In twelve months observational astronomy had made such a bound as it has
+never made before or since.
+
+Why did not others make any of these observations? Because no one could
+make telescopes like Galileo.
+
+He gathered pupils round him however, and taught them how to work the
+lenses, so that gradually these instruments penetrated Europe, and
+astronomers everywhere verified his splendid discoveries.
+
+But still he worked on, and by March in the very next year, he saw
+something still more hateful to the Aristotelian philosophers, viz.
+spots on the sun.
+
+[Illustration: FIG. 48.]
+
+If anything was pure and perfect it was the sun, they said. Was this
+impostor going to blacken its face too?
+
+Well, there they were. They slowly formed and changed, and by moving all
+together showed him that the sun rotated about once a month.
+
+Before taking leave of Galileo's astronomical researches, I must
+mention an observation made at the end of 1612, that the apparent
+triplicity of Saturn (Fig. 46) had vanished.
+
+[Illustration: FIG. 49.--A portion of the sun's disk as seen in a
+powerful modern telescope.]
+
+     "Looking on Saturn within these few days, I found it solitary,
+     without the assistance of its accustomed stars, and in short
+     perfectly round and defined, like Jupiter, and such it still
+     remains. Now what can be said of so strange a metamorphosis? Are
+     perhaps the two smaller stars consumed like spots on the sun? Have
+     they suddenly vanished and fled? Or has Saturn devoured his own
+     children? Or was the appearance indeed fraud and illusion, with
+     which the glasses have so long time mocked me and so many others
+     who have so often observed with me? Now perhaps the time is come to
+     revive the withering hopes of those, who, guided by more profound
+     contemplations, have fathomed all the fallacies of the new
+     observations and recognized their impossibility! I cannot resolve
+     what to say in a chance so strange, so new, so unexpected. The
+     shortness of time, the unexampled occurrence, the weakness of my
+     intellect, the terror of being mistaken, have greatly confounded
+     me."
+
+However, he plucked up courage, and conjectured that the two attendants
+would reappear, by revolving round the planet.
+
+[Illustration: FIG. 50.--Saturn and his rings, as seen under the most
+favourable circumstances.]
+
+The real reason of their disappearance is well known to us now. The
+plane of Saturn's rings oscillates slowly about our line of sight, and
+so we sometimes see them edgeways and sometimes with a moderate amount
+of obliquity. The rings are so thin that, when turned precisely
+edgeways, they become invisible. The two imaginary attendants were the
+most conspicuous portions of the ring, subsequently called _ans�_.
+
+I have thought it better not to interrupt this catalogue of brilliant
+discoveries by any biographical details; but we must now retrace our
+steps to the years 1609 and 1610, the era of the invention of the
+telescope.
+
+By this time Galileo had been eighteen years at Padua, and like many
+another man in like case, was getting rather tired of continual
+lecturing. Moreover, he felt so full of ideas that he longed to have a
+better opportunity of following them up, and more time for thinking them
+out.
+
+Now in the holidays he had been accustomed to return to his family home
+at Pisa, and there to come a good deal into contact with the Grand-Ducal
+House of Tuscany. Young Cosmo di Medici became in fact his pupil, and
+arrived at man's estate with the highest opinion of the philosopher.
+This young man had now come to the throne as Cosmo II., and to him
+Galileo wrote saying how much he should like more time and leisure, how
+full he was of discoveries if he only had the chance of a reasonable
+income without the necessity of consuming so large a portion of his time
+in elementary teaching, and practically asking to be removed to some
+position in the Court. Nothing was done for a time, but negotiations
+proceeded, and soon after the discovery of Jupiter's satellites Cosmo
+wrote making a generous offer, which Galileo gladly and enthusiastically
+accepted, and at once left Padua for Florence. All his subsequent
+discoveries date from Florence.
+
+Thus closed his brilliant and happy career as a professor at the
+University of Padua. He had been treated well: his pay had become larger
+than that of any Professor of Mathematics up to that time; and, as you
+know, immediately after his invention of the telescope the Venetian
+Senate, in a fit of enthusiasm, had doubled it and secured it to him for
+life wherever he was. To throw up his chair and leave the place the very
+next year scarcely seems a strictly honourable procedure. It was legal
+enough no doubt, and it is easy for small men to criticize a great one,
+but nevertheless I think we must admit that it is a step such as a man
+with a keen sense of honour would hardly have taken.
+
+One quite feels and sympathizes with the temptation. Not emolument, but
+leisure; freedom from harassing engagements and constant teaching, and
+liberty to prosecute his studies day and night without interference:
+this was the golden prospect before him. He yielded, but one cannot help
+wishing he had not.
+
+As it turned out it was a false step--the first false step of his public
+career. When made it was irretrievable, and it led to great misery.
+
+At first it seemed brilliant enough. The great philosopher of the Tuscan
+Court was courted and flattered by princes and nobles, he enjoyed a
+world-wide reputation, lived as luxuriously as he cared for, had his
+time all to himself, and lectured but very seldom, on great occasions or
+to a few crowned heads.
+
+His position was in fact analogous to that of Tycho Brah� in his island
+of Huen.
+
+Misfortune overtook both. In Tycho's case it arose mainly from the death
+of his patron. In Galileo's it was due to a more insidious cause, to
+understand which cause aright we must remember the political divisions
+of Italy at that date.
+
+Tuscany was a Papal State, and thought there was by no means free.
+Venice was a free republic, and was even hostile to the Papacy. In 1606
+the Pope had placed it under an interdict. In reply it had ejected every
+Jesuit.
+
+Out of this atmosphere of comparative enlightenment and freedom into
+that hotbed of medi�valism and superstition went Galileo with his eyes
+open. Keen was the regret of his Paduan and Venetian friends; bitter
+were their remonstrances and exhortations. But he was determined to go,
+and, not without turning some of his old friends into enemies, he went.
+
+Seldom has such a man made so great a mistake: never, I suppose, has one
+been so cruelly punished for it.
+
+[Illustration: FIG. 51.--Map of Italy.]
+
+We must remember, however, that Galileo, though by no means a saint, was
+yet a really religious man, a devout Catholic and thorough adherent of
+the Church, so that he would have no dislike to place himself under her
+sway. Moreover, he had been born a Tuscan, his family had lived at
+Florence or Pisa, and it felt like going home. His theological attitude
+is worthy of notice, for he was not in the least a sceptic. He quite
+acquiesces in the authority of the Bible, especially in all matters
+concerning faith and conduct; as to its statements in scientific
+matters, he argues that we are so liable to misinterpret their meaning
+that it is really easier to examine Nature for truth in scientific
+matters, and that when direct observation and Scripture seem to clash,
+it is because of our fallacious interpretation of one or both of them.
+He is, in fact, what one now calls a "reconciler."
+
+It is curious to find such a man prosecuted for heresy, when to-day his
+opinions are those of the orthodox among the orthodox. But so it ever
+is, and the heresy of one generation becomes the commonplace of the
+next.
+
+He accepts Joshua's miracle, for instance, not as a striking poem, but
+as a literal fact; and he points out how much more simply it could be
+done on the Copernican system by stopping the earth's rotation for a
+short time, than by stopping the sun and moon and all the host of heaven
+as on the old Ptolemaic system, or again by stopping only the sun and
+not any of the other bodies, and so throwing astronomy all wrong.
+
+This reads to us like satire, but no doubt it was his genuine opinion.
+
+These Scriptural reconciliations of his, however, angered the religious
+authorities still more. They said it was bad enough for this heretic to
+try and upset old _scientific_ beliefs, and to spoil the face of
+_Nature_ with his infidel discoveries, but at least he might leave the
+Bible alone; and they addressed an indignant remonstrance to Rome, to
+protect it from the hands of ignorant laymen.
+
+Thus, wherever he turned he encountered hostility. Of course he had many
+friends--some of them powerful like Cosmo, all of them faithful and
+sincere. But against the power of Rome what could they do? Cosmo dared
+no more than remonstrate, and ultimately his successor had to refrain
+from even this, so enchained and bound was the spirit of the rulers of
+those days; and so when his day of tribulation came he stood alone and
+helpless in the midst of his enemies.
+
+You may wonder, perhaps, why this man should excite so much more
+hostility than many another man who was suffered to believe and teach
+much the same doctrines unmolested. But no other man had made such
+brilliant and exciting discoveries. No man stood so prominently forward
+in the eyes of all Christendom as the champion of the new doctrines. No
+other man stated them so clearly and forcibly, nor drove them home with
+such brilliant and telling illustrations.
+
+And again, there was the memory of his early conflict with the
+Aristotelians at Pisa, of his scornful and successful refutation of
+their absurdities. All this made him specially obnoxious to the
+Aristotelian Jesuits in their double capacity both of priests and of
+philosophers, and they singled him out for relentless official
+persecution.
+
+Not yet, however, is he much troubled by them. The chief men at Rome
+have not yet moved. Messages, however, keep going up from Tuscany to
+Rome respecting the teachings of this man, and of the harm he is doing
+by his pertinacious preaching of the Copernican doctrine that the earth
+moves.
+
+At length, in 1615, Pope Paul V. wrote requesting him to come to Rome to
+explain his views. He went, was well received, made a special friend of
+Cardinal Barberino--an accomplished man in high position, who became in
+fact the next Pope. Galileo showed cardinals and others his telescope,
+and to as many as would look through it he showed Jupiter's satellites
+and his other discoveries. He had a most successful visit. He talked, he
+harangued, he held forth in the midst of fifteen or twenty disputants at
+once, confounding his opponents and putting them to shame.
+
+His method was to let the opposite arguments be stated as fully and
+completely as possible, himself aiding, and often adducing the most
+forcible and plausible arguments against his own views; and then, all
+having been well stated, he would proceed to utterly undermine and
+demolish the whole fabric, and bring out the truth in such a way as to
+convince all honest minds. It was this habit that made him such a
+formidable antagonist. He never shrank from meeting an opposing
+argument, never sought to ignore it, or cloak it in a cloud of words.
+Every hostile argument he seemed to delight in, as a foe to be crushed,
+and the better and stronger they sounded the more he liked them. He knew
+many of them well, he invented a number more, and had he chosen could
+have out-argued the stoutest Aristotelian on his own grounds. Thus did
+he lead his adversaries on, almost like Socrates, only to ultimately
+overwhelm them in a more hopeless rout. All this in Rome too, in the
+heart of the Catholic world. Had he been worldly-wise, he would
+certainly have kept silent and unobtrusive till he had leave to go away
+again. But he felt like an apostle of the new doctrines, whose mission
+it was to proclaim them even in this centre of the world and of the
+Church.
+
+Well, he had an audience with the Pope--a chat an hour long--and the two
+parted good friends, mutually pleased with each other.
+
+He writes that he is all right now, and might return home when he liked.
+But the question began to be agitated whether the whole system of
+Copernicus ought not to be condemned as impious and heretical. This view
+was persistently urged upon the Pope and College of Cardinals, and it
+was soon to be decided upon.
+
+Had Galileo been unfaithful to the Church he could have left them to
+stultify themselves in any way they thought proper, and himself have
+gone; but he felt supremely interested in the result, and he stayed. He
+writes:--
+
+     "So far as concerns the clearing of my own character, I might
+     return home immediately; but although this new question regards me
+     no more than all those who for the last eighty years have supported
+     those opinions both in public and private, yet, as perhaps I may be
+     of some assistance in that part of the discussion which depends on
+     the knowledge of truths ascertained by means of the sciences which
+     I profess, I, as a zealous and Catholic Christian, neither can nor
+     ought to withhold that assistance which my knowledge affords, and
+     this business keeps me sufficiently employed."
+
+It is possible that his stay was the worst thing for the cause he had at
+heart. Anyhow, the result was that the system was condemned, and both
+the book of Copernicus and the epitome of it by Kepler were placed on
+the forbidden list,[11] and Galileo himself was formally ordered never
+to teach or to believe the motion of the earth.
+
+He quitted Rome in disgust, which before long broke out in satire. The
+only way in which he could safely speak of these views now was as if
+they were hypothetical and uncertain, and so we find him writing to the
+Archduke Leopold, with a presentation copy of his book on the tides, the
+following:--
+
+     "This theory occurred to me when in Rome whilst the theologians
+     were debating on the prohibition of Copernicus's book, and of the
+     opinion maintained in it of the motion of the earth, which I at
+     that time believed: until it pleased those gentlemen to suspend the
+     book, and declare the opinion false and repugnant to the Holy
+     Scriptures. Now, as I know how well it becomes me to obey and
+     believe the decisions of my superiors, which proceed out of more
+     knowledge than the weakness of my intellect can attain to, this
+     theory which I send you, which is founded on the motion of the
+     earth, I now look upon as a fiction and a dream, and beg your
+     highness to receive it as such. But as poets often learn to prize
+     the creations of their fancy, so in like manner do I set some value
+     on this absurdity of mine. It is true that when I sketched this
+     little work I did hope that Copernicus would not, after eighty
+     years, be convicted of error; and I had intended to develop and
+     amplify it further, but a voice from heaven suddenly awakened me,
+     and at once annihilated all my confused and entangled fancies."
+
+This sarcasm, if it had been in print, would probably have been
+dangerous. It was safe in a private letter, but it shows us his real
+feelings.
+
+However, he was left comparatively quiet for a time. He was getting an
+old man now, and passed the time studiously enough, partly at his house
+in Florence, partly at his villa in Arcetri, a mile or so out of the
+town.
+
+Here was a convent, and in it his two daughters were nuns. One of them,
+who passed under the name of Sister Maria Celeste, seems to have been a
+woman of considerable capacity--certainly she was of a most affectionate
+disposition--and loved and honoured her father in the most dutiful way.
+
+This was a quiet period of his life, spoiled only by occasional fits of
+illness and severe rheumatic pains, to which the old man was always
+liable. Many little circumstances are known of this peaceful time. For
+instance, the convent clock won't go, and Galileo mends it for them. He
+is always doing little things for them, and sending presents to the Lady
+Superior and his two daughters.
+
+He was occupied now with problems in hydrostatics, and on other matters
+unconnected with astronomy: a large piece of work which I must pass
+over. Most interesting and acute it is, however.
+
+In 1623, when the old Pope died, there was elected to the Papal throne,
+as Urban VIII., Cardinal Barberino, a man of very considerable
+enlightenment, and a personal friend of Galileo's, so that both he and
+his daughters rejoice greatly, and hope that things will come all right,
+and the forbidding edict be withdrawn.
+
+The year after this election he manages to make another journey to Rome
+to compliment his friend on his elevation to the Pontifical chair. He
+had many talks with Urban, and made himself very agreeable.
+
+Urban wrote to the Grand Duke Ferdinand, son of Cosmo:--
+
+     "For We find in him not only literary distinction but also love of
+     piety, and he is strong in those qualities by which Pontifical good
+     will is easily obtainable. And now, when he has been brought to
+     this city to congratulate Us on Our elevation, We have very
+     lovingly embraced him; nor can We suffer him to return to the
+     country whither your liberality recalls him without an ample
+     provision of Pontifical love. And that you may know how dear he is
+     to Us, We have willed to give him this honourable testimonial of
+     virtue and piety. And We further signify that every benefit which
+     you shall confer upon him, imitating or even surpassing your
+     father's liberality, will conduce to Our gratification."
+
+Encouraged, doubtless, by these marks of approbation, and reposing too
+much confidence in the individual good will of the Pope, without heeding
+the crowd of half-declared enemies who were seeking to undermine his
+reputation, he set about, after his return to Florence, his greatest
+literary and most popular work, _Dialogues on the Ptolemaic and
+Copernican Systems_. This purports to be a series of four conversations
+between three characters: Salviati, a Copernican philosopher; Sagredo, a
+wit and scholar, not specially learned, but keen and critical, and who
+lightens the talk with chaff; Simplicio, an Aristotelian philosopher,
+who propounds the stock absurdities which served instead of arguments to
+the majority of men.
+
+The conversations are something between Plato's _Dialogues_ and Sir
+Arthur Helps's _Friends in Council_. The whole is conducted with great
+good temper and fairness; and, discreetly enough, no definite conclusion
+is arrived at, the whole being left in abeyance as if for a fifth and
+decisive dialogue, which, however, was never written, and perhaps was
+only intended in case the reception was favourable.
+
+The preface also sets forth that the object of the writer is to show
+that the Roman edict forbidding the Copernican doctrine was not issued
+in ignorance of the facts of the case, as had been maliciously reported,
+and that he wishes to show how well and clearly it was all known
+beforehand. So he says the dialogue on the Copernican side takes up the
+question purely as a mathematical hypothesis or speculative figment, and
+gives it every artificial advantage of which the theory is capable.
+
+This piece of caution was insufficient to blind the eyes of the
+Cardinals; for in it the arguments in favour of the earth's motion are
+so cogent and unanswerable, and are so popularly stated, as to do more
+in a few years to undermine the old system than all that he had written
+and spoken before. He could not get it printed for two years after he
+had written it, and then only got consent through a piece of
+carelessness or laziness on the part of the ecclesiastical censor
+through whose hands the manuscript passed--for which he was afterwards
+dismissed.
+
+However, it did appear, and was eagerly read; the more, perhaps, as the
+Church at once sought to suppress it.
+
+The Aristotelians were furious, and represented to the Pope that he
+himself was the character intended by Simplicio, the philosopher whose
+opinions get alternately refuted and ridiculed by the other two, till he
+is reduced to an abject state of impotence.
+
+The idea that Galileo had thus cast ridicule upon his friend and patron
+is no doubt a gratuitous and insulting libel: there is no telling
+whether or not Urban believed it, but certainly his countenance changed
+to Galileo henceforward, and whether overruled by his Cardinals, or
+actuated by some other motive, his favour was completely withdrawn.
+
+The infirm old man was instantly summoned to Rome. His friends pleaded
+his age--he was now seventy--his ill-health, the time of year, the state
+of the roads, the quarantine existing on account of the plague. It was
+all of no avail, to Rome he must go, and on the 14th of February he
+arrived.
+
+[Illustration: FIG. 52.--Portrait of Galileo.]
+
+His daughter at Arcetri was in despair; and anxiety and fastings and
+penances self-inflicted on his account, dangerously reduced her health.
+
+At Rome he was not imprisoned, but he was told to keep indoors, and show
+himself as little as possible. He was allowed, however, to stay at the
+house of the Tuscan Ambassador instead of in gaol.
+
+By April he was removed to the chambers of the Inquisition, and examined
+several times. Here, however, the anxiety was too much, and his health
+began to give way seriously; so, before long, he was allowed to return
+to the Ambassador's house; and, after application had been made, was
+allowed to drive in the public garden in a half-closed carriage. Thus in
+every way the Inquisition dealt with him as leniently as they could. He
+was now their prisoner, and they might have cast him into their
+dungeons, as many another had been cast. By whatever they were
+influenced--perhaps the Pope's old friendship, perhaps his advanced age
+and infirmities--he was not so cruelly used.
+
+Still, they had their rules; he _must_ be made to recant and abjure his
+heresy; and, if necessary, torture must be applied. This he knew well
+enough, and his daughter knew it, and her distress may be imagined.
+Moreover, it is not as if they had really been heretics, as if they
+hated or despised the Church of Rome. On the contrary, they loved and
+honoured the Church. They were sincere and devout worshippers, and only
+on a few scientific matters did Galileo presume to differ from his
+ecclesiastical superiors: his disagreement with them occasioned him real
+sorrow; and his dearest hope was that they could be brought to his way
+of thinking and embrace the truth.
+
+Every time he was sent for by the Inquisition he was in danger of
+torture unless he recanted. All his friends urged him repeatedly to
+submit. They said resistance was hopeless and fatal. Within the memory
+of men still young, Giordano Bruno had been burnt alive for a similar
+heresy. This had happened while Galileo was at Padua. Venice was full of
+it. And since that, only eight years ago indeed, Antonio de Dominis,
+Archbishop of Salpetria, had been sentenced to the same fate: "to be
+handed over to the secular arm to be dealt with as mercifully as
+possible without the shedding of blood." So ran the hideous formula
+condemning a man to the stake. After his sentence, this unfortunate man
+died in the dungeons in which he had been incarcerated six years--died
+what is called a "natural" death; but the sentence was carried out,
+notwithstanding, on his lifeless body and his writings. His writings for
+which he had been willing to die!
+
+These were the tender mercies of the Inquisition; and this was the kind
+of meaning lurking behind many of their well-sounding and merciful
+phrases. For instance, what they call "rigorous examination," we call
+"torture." Let us, however, remember in our horror at this mode of
+compelling a prisoner to say anything they wished, that they were a
+legally constituted tribunal; that they acted with well established
+rules, and not in passion; and that torture was a recognized mode of
+extracting evidence, not only in ecclesiastical but in civil courts, at
+that date.
+
+All this, however, was but poor solace to the pitiable old philosopher,
+thus ruthlessly haled up and down, questioned and threatened, threatened
+and questioned, receiving agonizing letters from his daughter week by
+week, and trying to keep up a little spirit to reply as happily and
+hopefully as he could.
+
+This condition of things could not go on. From February to June the
+suspense lasted. On the 20th of June he was summoned again, and told he
+would be wanted all next day for a rigorous examination. Early in the
+morning of the 21st he repaired thither, and the doors were shut. Out of
+those chambers of horror he did not reappear till the 24th. What went on
+all those three days no one knows. He himself was bound to secrecy. No
+outsider was present. The records of the Inquisition are jealously
+guarded. That he was technically tortured is certain; that he actually
+underwent the torment of the rack is doubtful. Much learning has been
+expended upon the question, especially in Germany. Several eminent
+scholars have held the fact of actual torture to be indisputable
+(geometrically certain, one says), and they confirm it by the hernia
+from which he afterwards suffered, this being a well-known and frequent
+consequence.
+
+Other equally learned commentators, however, deny that the last stage
+was reached. For there are five stages all laid down in the rules of the
+Inquisition, and steadily adhered to in a rigorous examination, at each
+stage an opportunity being given for recantation, every utterance,
+groan, or sigh being strictly recorded. The recantation so given has to
+be confirmed a day or two later, under pain of a precisely similar
+ordeal.
+
+The five stages are:--1st. The official threat in the court. 2nd. The
+taking to the door of the torture chamber and renewing the official
+threat. 3rd. The taking inside and showing the instruments. 4th.
+Undressing and binding upon the rack. 5th. _Territio realis._
+
+Through how many of these ghastly acts Galileo passed I do not know. I
+hope and believe not the last.
+
+There are those who lament that he did not hold out, and accept the
+crown of martyrdom thus offered to him. Had he done so we know his
+fate--a few years' languishing in the dungeons, and then the flames.
+
+Whatever he ought to have done, he did not hold out--he gave way. At one
+stage or another of the dread ordeal he said: "I am in your hands. I
+will say whatever you wish." Then was he removed to a cell while his
+special form of perjury was drawn up.
+
+The next day, clothed as a penitent, the venerable old man was taken to
+the Convent of Minerva, where the Cardinals and prelates were assembled
+for the purpose of passing judgment upon him.
+
+The text of the judgment I have here, but it is too long to read. It
+sentences him--1st. To the abjuration. 2nd. To formal imprisonment for
+life. 3rd. To recite the seven penitential psalms every week.
+
+Ten Cardinals were present; but, to their honour be it said, three
+refused to sign; and this blasphemous record of intolerance and bigoted
+folly goes down the ages with the names of seven Cardinals immortalized
+upon it.
+
+This having been read, he next had to read word for word the abjuration
+which had been drawn up for him, and then sign it.
+
+
+THE ABJURATION OF GALILEO.
+
+     "I, Galileo Galilei, son of the late Vincenzo Galilei, of Florence,
+     aged seventy years, being brought personally to judgment, and
+     kneeling before you Most Eminent and Most Reverend Lords Cardinals,
+     General Inquisitors of the universal Christian republic against
+     heretical depravity, having before my eyes the Holy Gospels, which
+     I touch with my own hands, swear that I have always believed, and
+     now believe, and with the help of God will in future believe, every
+     article which the Holy Catholic and Apostolic Church of Rome holds,
+     teaches, and preaches. But because I have been enjoined by this
+     Holy Office altogether to abandon the false opinion which maintains
+     that the sun is the centre and immovable, and forbidden to hold,
+     defend, or teach the said false doctrine in any manner, and after
+     it hath been signified to me that the said doctrine is repugnant
+     with the Holy Scripture, I have written and printed a book, in
+     which I treat of the same doctrine now condemned, and adduce
+     reasons with great force in support of the same, without giving any
+     solution, and therefore have been judged grievously suspected of
+     heresy; that is to say, that I held and believed that the sun is
+     the centre of the universe and is immovable, and that the earth is
+     not the centre and is movable; willing, therefore, to remove from
+     the minds of your Eminences, and of every Catholic Christian, this
+     vehement suspicion rightfully entertained towards me, with a
+     sincere heart and unfeigned faith, I abjure, curse, and detest the
+     said errors and heresies, and generally every other error and sect
+     contrary to Holy Church; and I swear that I will never more in
+     future say or assert anything verbally, or in writing, which may
+     give rise to a similar suspicion of me; but if I shall know any
+     heretic, or any one suspected of heresy, that I will denounce him
+     to this Holy Office, or to the Inquisitor or Ordinary of the place
+     where I may be; I swear, moreover, and promise, that I will fulfil
+     and observe fully, all the penances which have been or shall be
+     laid on me by this Holy Office. But if it shall happen that I
+     violate any of my said promises, oaths, and protestations (which
+     God avert!), I subject myself to all the pains and punishments
+     which have been decreed and promulgated by the sacred canons, and
+     other general and particular constitutions, against delinquents of
+     this description. So may God help me, and his Holy Gospels which I
+     touch with my own hands. I, the above-named Galileo Galilei, have
+     abjured, sworn, promised, and bound myself as above, and in witness
+     thereof with my own hand have subscribed this present writing of my
+     abjuration, which I have recited word for word. At Rome, in the
+     Convent of Minerva, 22nd June, 1633. I, Galileo Galilei, have
+     abjured as above with my own hand."
+
+Those who believe the story about his muttering to a friend, as he rose
+from his knees, "e pur si muove," do not realize the scene.
+
+1st. There was no friend in the place.
+
+2nd. It would have been fatally dangerous to mutter anything before such
+an assemblage.
+
+3rd. He was by this time an utterly broken and disgraced old man;
+wishful, of all things, to get away and hide himself and his miseries
+from the public gaze; probably with his senses deadened and stupefied by
+the mental sufferings he had undergone, and no longer able to think or
+care about anything--except perhaps his daughter,--certainly not about
+any motion of this wretched earth.
+
+Far and wide the news of the recantation spread. Copies of the
+abjuration were immediately sent to all Universities, with instructions
+to the professors to read it publicly.
+
+At Florence, his home, it was read out in the Cathedral church, all his
+friends and adherents being specially summoned to hear it.
+
+For a short time more he was imprisoned in Rome; but at length was
+permitted to depart, never more of his own will to return.
+
+He was allowed to go to Siena. Here his daughter wrote consolingly,
+rejoicing at his escape, and saying how joyfully she already recited the
+penitential psalms for him, and so relieved him of that part of his
+sentence.
+
+But the poor girl was herself, by this time, ill--thoroughly worn out
+with anxiety and terror; she lay, in fact, on what proved to be her
+death-bed. Her one wish was to see her dearest lord and father, so she
+calls him, once more. The wish was granted. His prison was changed, by
+orders from Rome, from Siena to Arcetri, and once more father and
+daughter embraced. Six days after this she died.
+
+The broken-hearted old man now asks for permission to go to live in
+Florence, but is met with the stern answer that he is to stay at
+Arcetri, is not to go out of the house, is not to receive visitors, and
+that if he asks for more favours, or transgresses the commands laid upon
+him, he is liable to be haled back to Rome and cast into a dungeon.
+These harsh measures were dictated, not by cruelty, but by the fear of
+his still spreading heresy by conversation, and so he was to be kept
+isolated.
+
+Idle, however, he was not and could not be. He often complains that his
+head is too busy for his body. In the enforced solitude of Arcetri he
+was composing those dialogues on motion which are now reckoned his
+greatest and most solid achievement. In these the true laws of motion
+are set forth for the first time (see page 167). One more astronomical
+discovery also he was to make--that of the moon's libration.
+
+And then there came one more crushing blow. His eyes became inflamed and
+painful--the sight of one of them failed, the other soon went; he
+became totally blind. But this, being a heaven-sent infliction, he could
+bear with resignation, though it must have been keenly painful to a
+solitary man of his activity. "Alas!" says he, in one of his letters,
+"your dear friend and servant is totally blind. Henceforth this heaven,
+this universe, which by wonderful observations I had enlarged a hundred
+and a thousand times beyond the conception of former ages, is shrunk for
+me into the narrow space which I myself fill in it. So it pleases God;
+it shall therefore please me also."
+
+He was now allowed an amanuensis, and the help of his pupils Torricelli,
+Castelli, and Viviani, all devotedly attached to him, and Torricelli
+very famous after him. Visitors also were permitted, after approval by a
+Jesuit supervisor; and under these circumstances many visited him, among
+them a man as immortal as himself--John Milton, then only twenty-nine,
+travelling in Italy. Surely a pathetic incident, this meeting of these
+two great men--the one already blind, the other destined to become so.
+No wonder that, as in his old age he dictated his masterpiece, the
+thoughts of the English poet should run on the blind sage of Tuscany,
+and the reminiscence of their conversation should lend colour to the
+poem.
+
+Well, it were tedious to follow the petty annoyances and troubles to
+which Galileo was still subject--how his own son was set to see that no
+unauthorized procedure took place, and that no heretic visitors were
+admitted; how it was impossible to get his new book printed till long
+afterwards; and how one form of illness after another took possession of
+him. The merciful end came at last, and at the age of seventy-eight he
+was released from the Inquisition.
+
+They wanted to deny him burial--they did deny him a monument; they
+threatened to cart his bones away from Florence if his friends attempted
+one. And so they hoped that he and his work might be forgotten.
+
+Poor schemers! Before the year was out an infant was born in
+Lincolnshire, whose destiny it was to round and complete and carry
+forward the work of their victim, so that, until man shall cease from
+the planet, neither the work nor its author shall have need of a
+monument.
+
+*      *      *      *      *
+
+Here might I end, were it not that the same kind of struggle as went on
+fiercely in the seventeenth century is still smouldering even now. Not
+in astronomy indeed, as then; nor yet in geology, as some fifty years
+ago; but in biology mainly--perhaps in other subjects. I myself have
+heard Charles Darwin spoken of as an atheist and an infidel, the theory
+of evolution assailed as unscriptural, and the doctrine of the ascent of
+man from a lower state of being, as opposed to the fall of man from some
+higher condition, denied as impious and un-Christian.
+
+Men will not learn by the past; still they brandish their feeble weapons
+against the truths of Nature, as if assertions one way or another could
+alter fact, or make the thing other than it really is. As Galileo said
+before his spirit was broken, "In these and other positions certainly no
+man doubts but His Holiness the Pope hath always an absolute power of
+admitting or condemning them; but it is not in the power of any creature
+to make them to be true or false, or otherwise than of their own nature
+and in fact they are."
+
+I know nothing of the views of any here present; but I have met educated
+persons who, while they might laugh at the men who refused to look
+through a telescope lest they should learn something they did not like,
+yet also themselves commit the very same folly. I have met persons who
+utterly refuse to listen to any view concerning the origin of man other
+than that of a perfect prim�val pair in a garden, and I am constrained
+to say this much: Take heed lest some prophet, after having excited your
+indignation at the follies and bigotry of a bygone generation, does not
+turn upon you with the sentence, "Thou art the man."
+
+
+
+
+SUMMARY OF FACTS FOR LECTURE VI
+
+_Science before Newton_
+
+
+_Dr. Gilbert_, of Colchester, Physician to Queen Elizabeth, was an
+excellent experimenter, and made many discoveries in magnetism and
+electricity. He was contemporary with Tycho Brah�, and lived from 1540
+to 1603.
+
+_Francis Bacon_, Lord Verulam, 1561-1626, though a brilliant writer, is
+not specially important as regards science. He was not a scientific man,
+and his rules for making discoveries, or methods of induction, have
+never been consciously, nor often indeed unconsciously, followed by
+discoverers. They are not in fact practical rules at all, though they
+were so intended. His really strong doctrines are that phenomena must be
+studied direct, and that variations in the ordinary course of nature
+must be induced by aid of experiment; but he lacked the scientific
+instinct for pursuing these great truths into detail and special cases.
+He sneered at the work and methods of both Gilbert and Galileo, and
+rejected the Copernican theory as absurd. His literary gifts have
+conferred on him an artificially high scientific reputation, especially
+in England; at the same time his writings undoubtedly helped to make
+popular the idea of there being new methods for investigating Nature,
+and, by insisting on the necessity for freedom from preconceived ideas
+and opinions, they did much to release men from the bondage of
+Aristotelian authority and scholastic tradition.
+
+The greatest name between Galileo and Newton is that of Descartes.
+
+_Ren� Descartes_ was born at La Haye in Touraine, 1596, and died at
+Stockholm in 1650. He did important work in mathematics, physics,
+anatomy, and philosophy. Was greatest as a philosopher and
+mathematician. At the age of twenty-one he served as a volunteer under
+Prince Maurice of Nassau, but spent most of his later life in Holland.
+His famous _Discourse on Method_ appeared at Leyden in 1637, and his
+_Principia_ at Amsterdam in 1644; great pains being taken to avoid the
+condemnation of the Church.
+
+Descartes's main scientific achievement was the application of algebra
+to geometry; his most famous speculation was the "theory of vortices,"
+invented to account for the motion of planets. He also made many
+discoveries in optics and physiology. His best known immediate pupils
+were the Princess Elizabeth of Bohemia, and Christina, Queen of Sweden.
+
+He founded a distinct school of thought (the Cartesian), and was the
+precursor of the modern mathematical method of investigating science,
+just as Galileo and Gilbert were the originators of the modern
+experimental method.
+
+
+
+
+LECTURE VI
+
+DESCARTES AND HIS THEORY OF VORTICES
+
+
+After the dramatic life we have been considering in the last two
+lectures, it is well to have a breathing space, to look round on what
+has been accomplished, and to review the state of scientific thought,
+before proceeding to the next great era. For we are still in the early
+morning of scientific discovery: the dawn of the modern period, faintly
+heralded by Copernicus, brought nearer by the work of Tycho and Kepler,
+and introduced by the discoveries of Galileo--the dawn has occurred, but
+the sun is not yet visible. It is hidden by the clouds and mists of the
+long night of ignorance and prejudice. The light is sufficient, indeed,
+to render these earth-born vapours more visible: it is not sufficient to
+dispel them. A generation of slow and doubtful progress must pass,
+before the first ray of sunlight can break through the eastern clouds
+and the full orb of day itself appear.
+
+It is this period of hesitating progress and slow leavening of men's
+ideas that we have to pass through in this week's lecture. It always
+happens thus: the assimilation of great and new ideas is always a slow
+and gradual process: there is no haste either here or in any other
+department of Nature. _Die Zeit ist unendlich lang._ Steadily the forces
+work, sometimes seeming to accomplish nothing; sometimes even the
+motion appears retrograde; but in the long run the destined end is
+reached, and the course, whether of a planet or of men's thoughts about
+the universe, is permanently altered. Then, the controversy was about
+the _earth's_ place in the universe; now, if there be any controversy of
+the same kind, it is about _man's_ place in the universe; but the
+process is the same: a startling statement by a great genius or prophet,
+general disbelief, and, it may be, an attitude of hostility, gradual
+acceptance by a few, slow spreading among the many, ending in universal
+acceptance and faith often as unquestioning and unreasoning as the old
+state of unfaith had been. Now the process is comparatively speedy:
+twenty years accomplishes a great deal: then it was tediously slow, and
+a century seemed to accomplish very little. Periodical literature may be
+responsible for some waste of time, but it certainly assists the rapid
+spread of ideas. The rate with which ideas are assimilated by the
+general public cannot even now be considered excessive, but how much
+faster it is than it was a few centuries ago may be illustrated by the
+attitude of the public to Darwinism now, twenty-five years after _The
+Origin of Species_, as compared with their attitude to the Copernican
+system a century after _De Revolutionibus_. By the way, it is, I know,
+presumptuous for me to have an opinion, but I cannot hear Darwin
+compared to or mentioned along with Newton without a shudder. The stage
+in which he found biology seems to me far more comparable with the
+Ptolemaic era in astronomy, and he himself to be quite fairly comparable
+to Copernicus.
+
+Let us proceed to summarize the stage at which the human race had
+arrived at the epoch with which we are now dealing.
+
+The Copernican view of the solar system had been stated, restated,
+fought, and insisted on; a chain of brilliant telescopic discoveries had
+made it popular and accessible to all men of any intelligence:
+henceforth it must be left to slowly percolate and sink into the minds
+of the people. For the nations were waking up now, and were accessible
+to new ideas. England especially was, in some sort, at the zenith of its
+glory; or, if not at the zenith, was in that full flush of youth and
+expectation and hope which is stronger and more prolific of great deeds
+and thoughts than a maturer period.
+
+A common cause against a common and detested enemy had roused in the
+hearts of Englishmen a passion of enthusiasm and patriotism; so that the
+mean elements of trade, their cheating yard-wands, were forgotten for a
+time; the Armada was defeated, and the nation's true and conscious adult
+life began. Commerce was now no mere struggle for profit and hard
+bargains; it was full of the spirit of adventure and discovery; a new
+world had been opened up; who could tell what more remained unexplored?
+Men awoke to the splendour of their inheritance, and away sailed Drake
+and Frobisher and Raleigh into the lands of the West.
+
+For literature, you know what a time it was. The author of _Hamlet_ and
+_Othello_ was alive: it is needless to say more. And what about science?
+The atmosphere of science is a more quiet and less stirring one; it
+thrives best when the fever of excitement is allayed; it is necessarily
+a later growth than literature. Already, however, our second great man
+of science was at work in a quiet country town--second in point of time,
+I mean, Roger Bacon being the first. Dr. Gilbert, of Colchester, was the
+second in point of time, and the age was ripening for the time when
+England was to be honoured with such a galaxy of scientific
+luminaries--Hooke and Boyle and Newton--as the world had not yet known.
+
+Yes, the nations were awake. "In all directions," as Draper says,
+"Nature was investigated: in all directions new methods of examination
+were yielding unexpected and beautiful results. On the ruins of its
+ivy-grown cathedrals Ecclesiasticism [or Scholasticism], surprised and
+blinded by the breaking day, sat solemnly blinking at the light and life
+about it, absorbed in the recollection of the night that had passed,
+dreaming of new phantoms and delusions in its wished-for return, and
+vindictively striking its talons at any derisive assailant who
+incautiously approached too near."
+
+Of the work of Gilbert there is much to say; so there is also of Roger
+Bacon, whose life I am by no means sure I did right in omitting. But
+neither of them had much to do with astronomy, and since it is in
+astronomy that the most startling progress was during these centuries
+being made, I have judged it wiser to adhere mainly to the pioneers in
+this particular department.
+
+Only for this reason do I pass Gilbert with but slight mention. He knew
+of the Copernican theory and thoroughly accepted it (it is convenient to
+speak of it as the Copernican theory, though you know that it had been
+considerably improved in detail since the first crude statement by
+Copernicus), but he made in it no changes. He was a cultivated
+scientific man, and an acute experimental philosopher; his main work lay
+in the domain of magnetism and electricity. The phenomena connected with
+the mariner's compass had been studied somewhat by Roger Bacon; and they
+were now examined still more thoroughly by Gilbert, whose treatise _De
+Magnete_, marks the beginning of the science of magnetism.
+
+As an appendix to that work he studied the phenomenon of amber, which
+had been mentioned by Thales. He resuscitated this little fact after its
+burial of 2,200 years, and greatly extended it. He it was who invented
+the name electricity--I wish it had been a shorter one. Mankind invents
+names much better than do philosophers. What can be better than "heat,"
+"light," "sound"? How favourably they compare with electricity,
+magnetism, galvanism, electro-magnetism, and magneto-electricity! The
+only long-established monosyllabic name I know invented by a philosopher
+is "gas"--an excellent attempt, which ought to be imitated.[12]
+
+Of Lord Bacon, who flourished about the same time (a little later), it
+is necessary to say something, because many persons are under the
+impression that to him and his _Novum Organon_ the reawakening of the
+world, and the overthrow of Aristotelian tradition, are mainly due. His
+influence, however, has been exaggerated. I am not going to enter into a
+discussion of the _Novum Organon_, and the mechanical methods which he
+propounded as certain to evolve truth if patiently pursued; for this is
+what he thought he was doing--giving to the world an infallible recipe
+for discovering truth, with which any ordinarily industrious man could
+make discoveries by means of collection and discrimination of instances.
+You will take my statement for what it is worth, but I assert this: that
+many of the methods which Bacon lays down are not those which the
+experience of mankind has found to be serviceable; nor are they such as
+a scientific man would have thought of devising.
+
+True it is that a real love and faculty for science are born in a man,
+and that to the man of scientific capacity rules of procedure are
+unnecessary; his own intuition is sufficient, or he has mistaken his
+vocation,--but that is not my point. It is not that Bacon's methods are
+useless because the best men do not need them; if they had been founded
+on a careful study of the methods actually employed, though it might be
+unconsciously employed, by scientific men--as the methods of induction,
+stated long after by John Stuart Mill, were founded--then, no doubt,
+their statement would have been a valuable service and a great thing to
+accomplish. But they were not this. They are the ideas of a brilliant
+man of letters, writing in an age when scientific research was almost
+unknown, about a subject in which he was an amateur. I confess I do not
+see how he, or John Stuart Mill, or any one else, writing in that age,
+could have formulated the true rules of philosophizing; because the
+materials and information were scarcely to hand. Science and its methods
+were only beginning to grow. No doubt it was a brilliant attempt. No
+doubt also there are many good and true points in the statement,
+especially in his insistence on the attitude of free and open candour
+with which the investigation of Nature should be approached. No doubt
+there was much beauty in his allegories of the errors into which men
+were apt to fall--the _idola_ of the market-place, of the tribe, of the
+theatre, and of the den; but all this is literature, and on the solid
+progress of science may be said to have had little or no effect.
+Descartes's _Discourse on Method_ was a much more solid production.
+
+You will understand that I speak of Bacon purely as a scientific man. As
+a man of letters, as a lawyer, a man of the world, and a statesman, he
+is beyond any criticism of mine. I speak only of the purely scientific
+aspect of the _Novum Organon_. _The Essays_ and _The Advancement of
+Learning_ are masterly productions; and as a literary man he takes high
+rank.
+
+The over-praise which, in the British Isles, has been lavished upon his
+scientific importance is being followed abroad by what may be an
+unnecessary amount of detraction. This is always the worst of setting up
+a man on too high a pinnacle; some one has to undertake the ungrateful
+task of pulling him down again. Justus von Liebig addressed himself to
+this task with some vigour in his _Reden und Abhandlung_ (Leipzig,
+1874), where he quotes from Bacon a number of suggestions for absurd
+experimentation.[13]
+
+The next paragraph I read, not because I endorse it, but because it is
+always well to hear both sides of a question. You have probably been
+long accustomed to read over-estimates of Bacon's importance, and
+extravagant laudation of his writings as making an epoch in science;
+hear what Draper says on the opposite side:--[14]
+
+     "The more closely we examine the writings of Lord Bacon, the more
+     unworthy does he seem to have been of the great reputation which
+     has been awarded to him. The popular delusion to which he owes so
+     much originated at a time when the history of science was unknown.
+     They who first brought him into notice knew nothing of the old
+     school of Alexandria. This boasted founder of a new philosophy
+     could not comprehend, and would not accept, the greatest of all
+     scientific doctrines when it was plainly set before his eyes.
+
+     "It has been represented that the invention of the true method of
+     physical science was an amusement of Bacon's hours of relaxation
+     from the more laborious studies of law, and duties of a Court.
+
+     "His chief admirers have been persons of a literary turn, who have
+     an idea that scientific discoveries are accomplished by a
+     mechanico-mental operation. Bacon never produced any great
+     practical result himself, no great physicist has ever made any use
+     of his method. He has had the same to do with the development of
+     modern science that the inventor of the orrery has had to do with
+     the discovery of the mechanism of the world. Of all the important
+     physical discoveries, there is not one which shows that its author
+     made it by the Baconian instrument.
+
+     "Newton never seems to have been aware that he was under any
+     obligation to Bacon. Archimedes, and the Alexandrians, and the
+     Arabians, and Leonardo da Vinci did very well before he was born;
+     the discovery of America by Columbus and the circumnavigation by
+     Magellan can hardly be attributed to him, yet they were the
+     consequences of a truly philosophical reasoning. But the
+     investigation of Nature is an affair of genius, not of rules. No
+     man can invent an _organon_ for writing tragedies and epic poems.
+     Bacon's system is, in its own terms, an idol of the theatre. It
+     would scarcely guide a man to a solution of the riddle of �lia
+     L�lia Crispis, or to that of the charade of Sir Hilary.
+
+     "Few scientific pretenders have made more mistakes than Lord Bacon.
+     He rejected the Copernican system, and spoke insolently of its
+     great author; he undertook to criticize adversely Gilbert's
+     treatise _De Magnete_; he was occupied in the condemnation of any
+     investigation of final causes, while Harvey was deducing the
+     circulation of the blood from Aquapendente's discovery of the
+     valves in the veins; he was doubtful whether instruments were of
+     any advantage, while Galileo was investigating the heavens with the
+     telescope. Ignorant himself of every branch of mathematics, he
+     presumed that they were useless in science but a few years before
+     Newton achieved by their aid his immortal discoveries.
+
+     "It is time that the sacred name of philosophy should be severed
+     from its long connection with that of one who was a pretender in
+     science, a time-serving politician, an insidious lawyer, a corrupt
+     judge, a treacherous friend, a bad man."
+
+This seems to me a depreciation as excessive as are the eulogies
+commonly current. The truth probably lies somewhere between the two
+extremes. It is unfair to judge Bacon's methods by thinking of physical
+science in its present stage. To realise his position we must think of a
+subject still in its very early infancy, one in which the advisability
+of applying experimental methods is still doubted; one which has been
+studied by means of books and words and discussion of normal instances,
+instead of by collection and observation of the unusual and irregular,
+and by experimental production of variety. If we think of a subject
+still in this infantile and almost pre-scientific stage, Bacon's words
+and formul� are far from inapplicable; they are, within their
+limitations, quite necessary and wholesome. A subject in this stage,
+strange to say, exists,--psychology; now hesitatingly beginning to
+assume its experimental weapons amid a stifling atmosphere of distrust
+and suspicion. Bacon's lack of the modern scientific instinct must be
+admitted, but he rendered humanity a powerful service in directing it
+from books to nature herself, and his genius is indubitable. A judicious
+account of his life and work is given by Prof. Adamson, in the
+_Encyclop�dia Britannica_, and to this article I now refer you.
+
+*      *      *      *      *
+
+Who, then, was the man of first magnitude filling up the gap in
+scientific history between the death of Galileo and the maturity of
+Newton? Unknown and mysterious are the laws regulating the appearance of
+genius. We have passed in review a Pole, a Dane, a German, and an
+Italian,--the great man is now a Frenchman, Ren� Descartes, born in
+Touraine, on the 31st of March, 1596.
+
+His mother died at his birth; the father was of no importance, save as
+the owner of some landed property. The boy was reared luxuriously, and
+inherited a fair fortune. Nearly all the men of first rank, you notice,
+were born well off. Genius born to poverty might, indeed, even then
+achieve name and fame--as we see in the case of Kepler--but it was
+terribly handicapped. Handicapped it is still, but far less than of old;
+and we may hope it will become gradually still less so as enlightenment
+proceeds, and the tremendous moment of great men to a nation is more
+clearly and actively perceived.
+
+It is possible for genius, when combined with strong character, to
+overcome all obstacles, and reach the highest eminence, but the
+struggle must be severe; and the absence of early training and
+refinement during the receptive years of youth must be a lifelong
+drawback.
+
+Descartes had none of these drawbacks; life came easily to him, and, as
+a consequence perhaps, he never seems to have taken it quite seriously.
+Great movements and stirring events were to him opportunities for the
+study of men and manners; he was not the man to court persecution, nor
+to show enthusiasm for a losing or struggling cause.
+
+In this, as in many other things, he was imbued with a very modern
+spirit, a cynical and sceptical spirit, which, to an outside and
+superficial observer like myself, seems rather rife just now.
+
+He was also imbued with a phase of scientific spirit which you sometimes
+still meet with, though I believe it is passing away, viz. an uncultured
+absorption in his own pursuits, and some feeling of contempt for
+classical and literary and �sthetic studies.
+
+In politics, art, and history he seems to have had no interest. He was a
+spectator rather than an actor on the stage of the world; and though he
+joined the army of that great military commander Prince Maurice of
+Nassau, he did it not as a man with a cause at heart worth fighting for,
+but precisely in the spirit in which one of our own gilded youths would
+volunteer in a similar case, as a good opportunity for frolic and for
+seeing life.
+
+He soon tired of it and withdrew--at first to gay society in Paris. Here
+he might naturally have sunk into the gutter with his companions, but
+for a great mental shock which became the main epoch and turning-point
+of his life, the crisis which diverted him from frivolity to
+seriousness. It was a purely intellectual emotion, not excited by
+anything in the visible or tangible world; nor could it be called
+conversion in the common acceptation of that term. He tells us that on
+the 10th of November, 1619, at the age of twenty-four, a brilliant idea
+flashed upon him--the first idea, namely, of his great and powerful
+mathematical method, of which I will speak directly; and in the flush of
+it he foresaw that just as geometers, starting with a few simple and
+evident propositions or axioms, ascend by a long and intricate ladder of
+reasoning to propositions more and more abstruse, so it might be
+possible to ascend from a few data, to all the secrets and facts of the
+universe, by a process of mathematical reasoning.
+
+"Comparing the mysteries of Nature with the laws of mathematics, he
+dared to hope that the secrets of both could be unlocked with the same
+key."
+
+That night he lapsed gradually into a state of enthusiasm, in which he
+saw three dreams or visions, which he interpreted at the time, even
+before waking, to be revelations from the Spirit of Truth to direct his
+future course, as well as to warn him from the sins he had already
+committed.
+
+His account of the dreams is on record, but is not very easy to follow;
+nor is it likely that a man should be able to convey to others any
+adequate idea of the deepest spiritual or mental agitation which has
+shaken him to his foundations.
+
+His associates in Paris were now abandoned, and he withdrew, after some
+wanderings, to Holland, where he abode the best part of his life and did
+his real work.
+
+Even now, however, he took life easily. He recommends idleness as
+necessary to the production of good mental work. He worked and meditated
+but a few hours a day: and most of those in bed. He used to think best
+in bed, he said. The afternoon he devoted to society and recreation.
+After supper he wrote letters to various persons, all plainly intended
+for publication, and scrupulously preserved. He kept himself free from
+care, and was most cautious about his health, regarding himself, no
+doubt, as a subject of experiment, and wishful to see how long he could
+prolong his life. At one time he writes to a friend that he shall be
+seriously disappointed if he does not manage to see 100 years.
+
+[Illustration: FIG. 53.--Descartes.]
+
+This plan of not over-working himself, and limiting the hours devoted to
+serious thought, is one that might perhaps advantageously be followed by
+some over-laborious students of the present day. At any rate it conveys
+a lesson; for the amount of ground covered by Descartes, in a life not
+very long, is extraordinary. He must, however, have had a singular
+aptitude for scientific work; and the judicious leaven of selfishness
+whereby he was able to keep himself free from care and embarrassments
+must have been a great help to him.
+
+And what did his versatile genius accomplish during his fifty-four years
+of life?
+
+In philosophy, using the term as meaning mental or moral philosophy and
+metaphysics, as opposed to natural philosophy or physics, he takes a
+very high rank, and it is on this that perhaps his greatest fame rests.
+(He is the author, you may remember, of the famous aphorism, "_Cogito,
+ergo sum_.")
+
+In biology I believe he may be considered almost equally great:
+certainly he spent a great deal of time in dissecting, and he made out a
+good deal of what is now known of the structure of the body, and of the
+theory of vision. He eagerly accepted the doctrine of the circulation of
+the blood, then being taught by Harvey, and was an excellent anatomist.
+
+You doubtless know Professor Huxley's article on Descartes in the _Lay
+Sermons_, and you perceive in what high estimation he is there held.
+
+He originated the hypothesis that animals are automata, for which indeed
+there is much to be said from some points of view; but he unfortunately
+believed that they were unconscious and non-sentient automata, and this
+belief led his disciples into acts of abominable cruelty. Professor
+Huxley lectured on this hypothesis and partially upheld it not many
+years since. The article is included in his volume called _Science and
+Culture_.
+
+Concerning his work in mathematics and physics I can speak with more
+confidence. He is the author of the Cartesian system of algebraic or
+analytic geometry, which has been so powerful an engine of research, far
+easier to wield than the old synthetic geometry. Without it Newton could
+never have written the _Principia_, or made his greatest discoveries.
+He might indeed have invented it for himself, but it would have consumed
+some of his life to have brought it to the necessary perfection.
+
+     The principle of it is the specification of the position of a point
+     in a plane by two numbers, indicating say its distance from two
+     lines of reference in the plane; like the latitude and longitude of
+     a place on the globe. For instance, the two lines of reference
+     might be the bottom edge and the left-hand vertical edge of a wall;
+     then a point on the wall, stated as being for instance 6 feet along
+     and 2 feet up, is precisely determined. These two distances are
+     called co-ordinates; horizontal ones are usually denoted by _x_,
+     and vertical ones by _y_.
+
+     If, instead of specifying two things, only one statement is made,
+     such as _y_ = 2, it is satisfied by a whole row of points, all the
+     points in a horizontal line 2 feet above the ground. Hence _y_ = 2
+     may be said to represent that straight line, and is called the
+     equation to that straight line. Similarly _x_ = 6 represents a
+     vertical straight line 6 feet (or inches or some other unit) from
+     the left-hand edge. If it is asserted that _x_ = 6 and _y_ = 2,
+     only one point can be found to satisfy both conditions, viz. the
+     crossing point of the above two straight lines.
+
+     Suppose an equation such as _x_ = _y_ to be given. This also is
+     satisfied by a row of points, viz. by all those that are
+     equidistant from bottom and left-hand edges. In other words, _x_ =
+     _y_ represents a straight line slanting upwards at 45�. The
+     equation _x_ = 2_y_ represents another straight line with a
+     different angle of slope, and so on. The equation x^2 + y^2
+     = 36 represents a circle of radius 6. The equation 3x^2 +
+     4y^2 = 25 represents an ellipse; and in general every algebraic
+     equation that can be written down, provided it involve only two
+     variables, _x_ and _y_, represents some curve in a plane; a curve
+     moreover that can be drawn, or its properties completely
+     investigated without drawing, from the equation. Thus algebra is
+     wedded to geometry, and the investigation of geometric relations by
+     means of algebraic equations is called analytical geometry, as
+     opposed to the old Euclidian or synthetic mode of treating the
+     subject by reasoning consciously directed to the subject by help of
+     figures.
+
+     If there be three variables--_x_, _y_, and _z_,--instead of only
+     two, an equation among them represents not a curve in a plane but a
+     surface in space; the three variables corresponding to the three
+     dimensions of space: length, breadth, and thickness.
+
+     An equation with four variables usually requires space of four
+     dimensions for its geometrical interpretation, and so on.
+
+     Thus geometry can not only be reasoned about in a more mechanical
+     and therefore much easier, manner, but it can be extended into
+     regions of which we have and can have no direct conception, because
+     we are deficient in sense organs for accumulating any kind of
+     experience in connexion with such ideas.
+
+[Illustration: FIG. 54.--The eye diagram. [From Descartes' _Principia_.]
+Three external points are shown depicted on the retina: the image being
+appreciated by a representation of the brain.]
+
+In physics proper Descartes' tract on optics is of considerable
+historical interest. He treats all the subjects he takes up in an able
+and original manner.
+
+In Astronomy he is the author of that famous and long upheld theory, the
+doctrine of vortices.
+
+He regarded space as a plenum full of an all-pervading fluid. Certain
+portions of this fluid were in a state of whirling motion, as in a
+whirlpool or eddy of water; and each planet had its own eddy, in which
+it was whirled round and round, as a straw is caught and whirled in a
+common whirlpool. This idea he works out and elaborates very fully,
+applying it to the system of the world, and to the explanation of all
+the motions of the planets.
+
+[Illustration: FIG. 55.--Descartes's diagram of vortices, from his
+_Principia_.]
+
+This system evidently supplied a void in men's minds, left vacant by the
+overthrow of the Ptolemaic system, and it was rapidly accepted. In the
+English Universities it held for a long time almost undisputed sway; it
+was in this faith that Newton was brought up.
+
+Something was felt to be necessary to keep the planets moving on their
+endless round; the _primum mobile_ of Ptolemy had been stopped; an angel
+was sometimes assigned to each planet to carry it round, but though a
+widely diffused belief, this was a fantastic and not a serious
+scientific one. Descartes's vortices seemed to do exactly what was
+wanted.
+
+It is true they had no connexion with the laws of Kepler. I doubt
+whether he knew about the laws of Kepler; he had not much opinion of
+other people's work; he read very little--found it easier to think. (He
+travelled through Florence once when Galileo was at the height of his
+renown without calling upon or seeing him.) In so far as the motion of a
+planet was not circular, it had to be accounted for by the jostling and
+crowding and distortion of the vortices.
+
+Gravitation he explained by a settling down of bodies toward the centre
+of each vortex; and cohesion by an absence of relative motion tending to
+separate particles of matter. He "can imagine no stronger cement."
+
+The vortices, as Descartes imagined them, are not now believed in. Are
+we then to regard the system as absurd and wholly false? I do not see
+how we can do this, when to this day philosophers are agreed in
+believing space to be completely full of fluid, which fluid is certainly
+capable of vortex motion, and perhaps everywhere does possess that
+motion. True, the now imagined vortices are not the large whirls of
+planetary size, they are rather infinitesimal whirls of less than atomic
+dimensions; still a whirling fluid is believed in to this day, and many
+are seeking to deduce all the properties of matter (rigidity,
+elasticity, cohesion gravitation, and the rest) from it.
+
+Further, although we talk glibly about gravitation and magnetism, and so
+on, we do not really know what they are. Progress is being made, but we
+do not yet properly know. Much, overwhelmingly much, remains to be
+discovered, and it ill-behoves us to reject any well-founded and
+long-held theory as utterly and intrinsically false and absurd. The more
+one gets to know, the more one perceives a kernel of truth even in the
+most singular statements; and scientific men have learned by experience
+to be very careful how they lop off any branch of the tree of knowledge,
+lest as they cut away the dead wood they lose also some green shoot,
+some healthy bud of unperceived truth.
+
+However, it may be admitted that the idea of a Cartesian vortex in
+connexion with the solar system applies, if at all, rather to an
+earlier--its nebulous--stage, when the whole thing was one great whirl,
+ready to split or shrink off planetary rings at their appropriate
+distances.
+
+Soon after he had written his great work, the _Principia Mathematica_,
+and before he printed it, news reached him of the persecution and
+recantation of Galileo. "He seems to have been quite thunderstruck at
+the tidings," says Mr. Mahaffy, in his _Life of Descartes_.[15] "He had
+started on his scientific journeys with the firm determination to enter
+into no conflict with the Church, and to carry out his system of pure
+mathematics and physics without ever meddling with matters of faith. He
+was rudely disillusioned as to the possibility of this severance. He
+wrote at once--apparently, November 20th, 1633--to Mersenne to say he
+would on no account publish his work--nay, that he had at first resolved
+to burn all his papers, for that he would never prosecute philosophy at
+the risk of being censured by his Church. 'I could hardly have
+believed,' he says, 'that an Italian, and in favour with the Pope as I
+hear, could be considered criminal for nothing else than for seeking to
+establish the earth's motion; though I know it has formerly been
+censured by some Cardinals. But I thought I had heard that since then it
+was constantly being taught, even at Rome; and I confess that if the
+opinion of the earth's movement is false, all the foundations of my
+philosophy are so also, because it is demonstrated clearly by them. It
+is so bound up with every part of my treatise that I could not sever it
+without making the remainder faulty; and although I consider all my
+conclusions based on very certain and clear demonstrations, I would not
+for all the world sustain them against the authority of the Church.'"
+
+Ten years later, however, he did publish the book, for he had by this
+time hit on an ingenious compromise. He formally denied that the earth
+moved, and only asserted that it was carried along with its water and
+air in one of those larger motions of the celestial ether which produce
+the diurnal and annual revolutions of the solar system. So, just as a
+passenger on the deck of a ship might be called stationary, so was the
+earth. He gives himself out therefore as a follower of Tycho rather than
+of Copernicus, and says if the Church won't accept this compromise he
+must return to the Ptolemaic system; but he hopes they won't compel him
+to do that, seeing that it is manifestly untrue.
+
+This elaborate deference to the powers that be did not indeed save the
+work from being ultimately placed upon the forbidden list by the Church,
+but it saved himself, at any rate, from annoying persecution. He was
+not, indeed, at all willing to be persecuted, and would no doubt have at
+once withdrawn anything they wished. I should be sorry to call him a
+time-server, but he certainly had plenty of that worldly wisdom in which
+some of his predecessors had been so lamentably deficient. Moreover, he
+was really a sceptic, and cared nothing at all about the Church or its
+dogmas. He knew the Church's power, however, and the advisability of
+standing well with it: he therefore professed himself a Catholic, and
+studiously kept his science and his Christianity distinct.
+
+In saying that he was a sceptic you must not understand that he was in
+the least an atheist. Very few men are; certainly Descartes never
+thought of being one. The term is indeed ludicrously inapplicable to
+him, for a great part of his philosophy is occupied with what he
+considers a rigorous proof of the existence of the Deity.
+
+At the age of fifty-three he was sent for to Stockholm by Christina,
+Queen of Sweden, a young lady enthusiastically devoted to study of all
+kinds and determined to surround her Court with all that was most famous
+in literature and science. Thither, after hesitation, Descartes went. He
+greatly liked royalty, but he dreaded the cold climate. Born in
+Touraine, a Swedish winter was peculiarly trying to him, especially as
+the energetic Queen would have lessons given her at five o'clock in the
+morning. She intended to treat him well, and was immensely taken with
+him; but this getting up at five o'clock on a November morning, to a man
+accustomed all his life to lie in bed till eleven, was a cruel hardship.
+He was too much of a courtier, however, to murmur, and the early morning
+audience continued. His health began to break down: he thought of
+retreating, but suddenly he gave way and became delirious. The Queen's
+physician attended him, and of course wanted to bleed him. This, knowing
+all he knew of physiology, sent him furious, and they could do nothing
+with him. After some days he became quiet, was bled twice, and gradually
+sank, discoursing with great calmness on his approaching death, and duly
+fortified with all the rites of the Catholic Church.
+
+His general method of research was as nearly as possible a purely
+deductive one:--_i.e._, after the manner of Euclid he starts with a few
+simple principles, and then, by a chain of reasoning, endeavours to
+deduce from them their consequences, and so to build up bit by bit an
+edifice of connected knowledge. In this he was the precursor of Newton.
+This method, when rigorously pursued, is the most powerful and
+satisfactory of all, and results in an ordered province of science far
+superior to the fragmentary conquests of experiment. But few indeed are
+the men who can handle it safely and satisfactorily: and none without
+continual appeals to experiment for verification. It was through not
+perceiving the necessity for verification that he erred. His importance
+to science lies not so much in what he actually discovered as in his
+anticipation of the right conditions for the solution of problems in
+physical science. He in fact made the discovery that Nature could after
+all be interrogated mathematically--a fact that was in great danger of
+remaining unknown. For, observe, that the mathematical study of Nature,
+the discovery of truth with a piece of paper and a pen, has a perilous
+similarity at first sight to the straw-thrashing subtleties of the
+Greeks, whose methods of investigating nature by discussing the meaning
+of words and the usage of language and the necessities of thought, had
+proved to be so futile and unproductive.
+
+A reaction had set in, led by Galileo, Gilbert, and the whole modern
+school of experimental philosophers, lasting down to the present
+day:--men who teach that the only right way of investigating Nature is
+by experiment and observation.
+
+It is indeed a very right and an absolutely necessary way; but it is not
+the only way. A foundation of experimental fact there must be; but upon
+this a great structure of theoretical deduction can be based, all
+rigidly connected together by pure reasoning, and all necessarily as
+true as the premises, provided no mistake is made. To guard against the
+possibility of mistake and oversight, especially oversight, all
+conclusions must sooner or later be brought to the test of experiment;
+and if disagreeing therewith, the theory itself must be re-examined,
+and the flaw discovered, or else the theory must be abandoned.
+
+Of this grand method, quite different from the gropings in the dark of
+Kepler--this method, which, in combination with experiment, has made
+science what it now is--this which in the hands of Newton was to lead to
+such stupendous results, we owe the beginning and early stages to Ren�
+Descartes.
+
+
+
+
+SUMMARY OF FACTS FOR LECTURES VII AND VIII
+
+  Otto Guericke          1602-1686
+  Hon. Robert Boyle      1626-1691
+  Huyghens               1629-1695
+  Christopher Wren       1632-1723
+  Robert Hooke           1635-1702
+  NEWTON                 1642-1727
+  Edmund Halley          1656-1742
+  James Bradley          1692-1762
+
+_Chronology of Newton's Life._
+
+
+Isaac Newton was born at Woolsthorpe, near Grantham, Lincolnshire, on
+Christmas Day, 1642. His father, a small freehold farmer, also named
+Isaac, died before his birth. His mother, _n�e_ Hannah Ayscough, in two
+years married a Mr. Smith, rector of North Witham, but was again left a
+widow in 1656. His uncle, W. Ayscough, was rector of a near parish and a
+graduate of Trinity College, Cambridge. At the age of fifteen Isaac was
+removed from school at Grantham to be made a farmer of, but as it seemed
+he would not make a good one his uncle arranged for him to return to
+school and thence to Cambridge, where he entered Trinity College as a
+sub-sizar in 1661. Studied Descartes's geometry. Found out a method of
+infinite series in 1665, and began the invention of Fluxions. In the
+same year and the next he was driven from Cambridge by the plague. In
+1666, at Woolsthorpe, the apple fell. In 1667 he was elected a fellow of
+his college, and in 1669 was specially noted as possessing an
+unparalleled genius by Dr. Barrow, first Lucasian Professor of
+Mathematics. The same year Dr. Barrow retired from his chair in favour
+of Newton, who was thus elected at the age of twenty-six. He lectured
+first on optics with great success. Early in 1672 he was elected a
+Fellow of the Royal Society, and communicated his researches in optics,
+his reflecting telescope, and his discovery of the compound nature of
+white light. Annoying controversies arose; but he nevertheless
+contributed a good many other most important papers in optics, including
+observations in diffraction, and colours of thin plates. He also
+invented the modern sextant. In 1672 a letter from Paris was read at the
+Royal Society concerning a new and accurate determination of the size of
+the earth by Picard. When Newton heard of it he began the _Principia_,
+working in silence. In 1684 arose a discussion between Wren, Hooke, and
+Halley concerning the law of inverse square as applied to gravity and
+the path it would cause the planets to describe. Hooke asserted that he
+had a solution, but he would not produce it. After waiting some time for
+it Halley went to Cambridge to consult Newton on the subject, and thus
+discovered the existence of the first part of the _Principia_, wherein
+all this and much more was thoroughly worked out. On his representations
+to the Royal Society the manuscript was asked for, and when complete was
+printed and published in 1687 at Halley's expense. While it was being
+completed Newton and seven others were sent to uphold the dignity of the
+University, before the Court of High Commission and Judge Jeffreys,
+against a high-handed action of James II. In 1682 he was sent to
+Parliament, and was present at the coronation of William and Mary. Made
+friends with Locke. In 1694 Montague, Lord Halifax, made him Warden, and
+in 1697 Master, of the Mint. Whiston succeeded him as Lucasian
+Professor. In 1693 the method of fluxions was published. In 1703 Newton
+was made President of the Royal Society, and held the office to the end
+of his life. In 1705 he was knighted by Anne. In 1713 Cotes helped him
+to bring out a new edition of the _Principia_, completed as we now have
+it. On the 20th of March 1727, he died: having lived from Charles I. to
+George II.
+
+
+THE LAWS OF MOTION, DISCOVERED BY GALILEO, STATED BY NEWTON.
+
+_Law 1._--If no force acts on a body in motion, it continues to move
+uniformly in a straight line.
+
+_Law 2._--If force acts on a body, it produces a change of motion
+proportional to the force and in the same direction.
+
+_Law 3._--When one body exerts force on another, that other reacts with
+equal force upon the one.
+
+
+
+
+LECTURE VII
+
+SIR ISAAC NEWTON
+
+
+The little hamlet of Woolsthorpe lies close to the village of
+Colsterworth, about six miles south of Grantham, in the county of
+Lincoln. In the manor house of Woolsthorpe, on Christmas Day, 1642, was
+born to a widowed mother a sickly infant who seemed not long for this
+world. Two women who were sent to North Witham to get some medicine for
+him scarcely expected to find him alive on their return. However, the
+child lived, became fairly robust, and was named Isaac, after his
+father. What sort of a man this father was we do not know. He was what
+we may call a yeoman, that most wholesome and natural of all classes. He
+owned the soil he tilled, and his little estate had already been in the
+family for some hundred years. He was thirty-six when he died, and had
+only been married a few months.
+
+Of the mother, unfortunately, we know almost as little. We hear that she
+was recommended by a parishioner to the Rev. Barnabas Smith, an old
+bachelor in search of a wife, as "the widow Newton--an extraordinary
+good woman:" and so I expect she was, a thoroughly sensible, practical,
+homely, industrious, middle-class, Mill-on-the-Floss sort of woman.
+However, on her second marriage she went to live at North Witham, and
+her mother, old Mrs. Ayscough, came to superintend the farm at
+Woolsthorpe, and take care of young Isaac.
+
+By her second marriage his mother acquired another piece of land, which
+she settled on her first son; so Isaac found himself heir to two little
+properties, bringing in a rental of about �80 a year.
+
+[Illustration: FIG. 56.--Manor-house of Woolsthorpe.]
+
+He had been sent to a couple of village schools to acquire the ordinary
+accomplishments taught at those places, and for three years to the
+grammar school at Grantham, then conducted by an old gentleman named Mr.
+Stokes. He had not been very industrious at school, nor did he feel
+keenly the fascinations of the Latin Grammar, for he tells us that he
+was the last boy in the lowest class but one. He used to pay much more
+attention to the construction of kites and windmills and waterwheels,
+all of which he made to work very well. He also used to tie paper
+lanterns to the tail of his kite, so as to make the country folk fancy
+they saw a comet, and in general to disport himself as a boy should.
+
+It so happened, however, that he succeeded in thrashing, in fair fight,
+a bigger boy who was higher in the school, and who had given him a
+kick. His success awakened a spirit of emulation in other things than
+boxing, and young Newton speedily rose to be top of the school.
+
+Under these circumstances, at the age of fifteen, his mother, who had
+now returned to Woolsthorpe, which had been rebuilt, thought it was time
+to train him for the management of his land, and to make a farmer and
+grazier of him. The boy was doubtless glad to get away from school, but
+he did not take kindly to the farm--especially not to the marketing at
+Grantham. He and an old servant were sent to Grantham every week to buy
+and sell produce, but young Isaac used to leave his old mentor to do all
+the business, and himself retire to an attic in the house he had lodged
+in when at school, and there bury himself in books.
+
+After a time he didn't even go through the farce of visiting Grantham at
+all; but stopped on the road and sat under a hedge, reading or making
+some model, until his companion returned.
+
+We hear of him now in the great storm of 1658, the storm on the day
+Cromwell died, measuring the force of the wind by seeing how far he
+could jump with it and against it. He also made a water-clock and set it
+up in the house at Grantham, where it kept fairly good time so long as
+he was in the neighbourhood to look after it occasionally.
+
+At his own home he made a couple of sundials on the side of the wall (he
+began by marking the position of the sun by the shadow of a peg driven
+into the wall, but this gradually developed into a regular dial) one of
+which remained of use for some time; and was still to be seen in the
+same place during the first half of the present century, only with the
+gnomon gone. In 1844 the stone on which it was carved was carefully
+extracted and presented to the Royal Society, who preserve it in their
+library. The letters WTON roughly carved on it are barely visible.
+
+All these pursuits must have been rather trying to his poor mother, and
+she probably complained to her brother, the rector of Burton Coggles:
+at any rate this gentleman found master Newton one morning under a hedge
+when he ought to have been farming. But as he found him working away at
+mathematics, like a wise man he persuaded his sister to send the boy
+back to school for a short time, and then to Cambridge. On the day of
+his finally leaving school old Mr. Stokes assembled the boys, made them
+a speech in praise of Newton's character and ability, and then dismissed
+him to Cambridge.
+
+At Trinity College a new world opened out before the country-bred lad.
+He knew his classics passably, but of mathematics and science he was
+ignorant, except through the smatterings he had picked up for himself.
+He devoured a book on logic, and another on Kepler's Optics, so fast
+that his attendance at lectures on these subjects became unnecessary. He
+also got hold of a Euclid and of Descartes's Geometry. The Euclid seemed
+childishly easy, and was thrown aside, but the Descartes baffled him for
+a time. However, he set to it again and again and before long mastered
+it. He threw himself heart and soul into mathematics, and very soon made
+some remarkable discoveries. First he discovered the binomial theorem:
+familiar now to all who have done any algebra, unintelligible to others,
+and therefore I say nothing about it. By the age of twenty-one or two he
+had begun his great mathematical discovery of infinite series and
+fluxions--now known by the name of the Differential Calculus. He wrote
+these things out and must have been quite absorbed in them, but it never
+seems to have occurred to him to publish them or tell any one about
+them.
+
+In 1664 he noticed some halos round the moon, and, as his manner was, he
+measured their angles--the small ones 3 and 5 degrees each, the larger
+one 22��35. Later he gave their theory.
+
+     Small coloured halos round the moon are often seen, and are said to
+     be a sign of rain. They are produced by the action of minute
+     globules of water or cloud particles upon light, and are brightest
+     when the particles are nearly equal in size. They are not like the
+     rainbow, every part of which is due to light that has entered a
+     raindrop, and been refracted and reflected with prismatic
+     separation of colours; a halo is caused by particles so small as to
+     be almost comparable with the size of waves of light, in a way
+     which is explained in optics under the head "diffraction." It may
+     be easily imitated by dusting an ordinary piece of window-glass
+     over with lycopodium, placing a candle near it, and then looking at
+     the candle-flame through the dusty glass from a fair distance. Or
+     you may look at the image of a candle in a dusted looking-glass.
+     Lycopodium dust is specially suitable, for its granules are
+     remarkably equal in size. The large halo, more rarely seen, of
+     angular radius 22��35, is due to another cause again, and is a
+     prismatic effect, although it exhibits hardly any colour. The angle
+     22-1/2� is characteristic of refraction in crystals with angles of
+     60� and refractive index about the same as water; in other words
+     this halo is caused by ice crystals in the higher regions of the
+     atmosphere.
+
+He also the same year observed a comet, and sat up so late watching it
+that he made himself ill. By the end of the year he was elected to a
+scholarship and took his B.A. degree. The order of merit for that year
+never existed or has not been kept. It would have been interesting, not
+as a testimony to Newton, but to the sense or non-sense of the
+examiners. The oldest Professorship of Mathematics at the University of
+Cambridge, the Lucasian, had not then been long founded, and its first
+occupant was Dr. Isaac Barrow, an eminent mathematician, and a kind old
+man. With him Newton made good friends, and was helpful in preparing a
+treatise on optics for the press. His help is acknowledged by Dr. Barrow
+in the preface, which states that he had corrected several errors and
+made some capital additions of his own. Thus we see that, although the
+chief part of his time was devoted to mathematics, his attention was
+already directed to both optics and astronomy. (Kepler, Descartes,
+Galileo, all combined some optics with astronomy. Tycho and the old ones
+combined alchemy; Newton dabbled in this also.)
+
+Newton reached the age of twenty-three in 1665, the year of the Great
+Plague. The plague broke out in Cambridge as well as in London, and the
+whole college was sent down. Newton went back to Woolsthorpe, his mind
+teeming with ideas, and spent the rest of this year and part of the next
+in quiet pondering. Somehow or other he had got hold of the notion of
+centrifugal force. It was six years before Huyghens discovered and
+published the laws of centrifugal force, but in some quiet way of his
+own Newton knew about it and applied the idea to the motion of the
+planets.
+
+We can almost follow the course of his thoughts as he brooded and
+meditated on the great problem which had taxed so many previous
+thinkers,--What makes the planets move round the sun? Kepler had
+discovered how they moved, but why did they so move, what urged them?
+
+Even the "how" took a long time--all the time of the Greeks, through
+Ptolemy, the Arabs, Copernicus, Tycho: circular motion, epicycles, and
+excentrics had been the prevailing theory. Kepler, with his marvellous
+industry, had wrested from Tycho's observations the secret of their
+orbits. They moved in ellipses with the sun in one focus. Their rate of
+description of area, not their speed, was uniform and proportional to
+time.
+
+Yes, and a third law, a mysterious law of unintelligible import, had
+also yielded itself to his penetrating industry--a law the discovery of
+which had given him the keenest delight, and excited an outburst of
+rapture--viz. that there was a relation between the distances and the
+periodic times of the several planets. The cubes of the distances were
+proportional to the squares of the times for the whole system. This law,
+first found true for the six primary planets, he had also extended,
+after Galileo's discovery, to the four secondary planets, or satellites
+of Jupiter (p. 81).
+
+But all this was working in the dark--it was only the first step--this
+empirical discovery of facts; the facts were so, but how came they so?
+What made the planets move in this particular way? Descartes's vortices
+was an attempt, a poor and imperfect attempt, at an explanation. It had
+been hailed and adopted throughout Europe for want of a better, but it
+did not satisfy Newton. No, it proceeded on a wrong tack, and Kepler had
+proceeded on a wrong tack in imagining spokes or rays sticking out from
+the sun and driving the planets round like a piece of mechanism or mill
+work. For, note that all these theories are based on a wrong idea--the
+idea, viz., that some force is necessary to maintain a body in motion.
+But this was contrary to the laws of motion as discovered by Galileo.
+You know that during his last years of blind helplessness at Arcetri,
+Galileo had pondered and written much on the laws of motion, the
+foundation of mechanics. In his early youth, at Pisa, he had been
+similarly occupied; he had discovered the pendulum, he had refuted the
+Aristotelians by dropping weights from the leaning tower (which we must
+rejoice that no earthquake has yet injured), and he had returned to
+mechanics at intervals all his life; and now, when his eyes were useless
+for astronomy, when the outer world has become to him only a prison to
+be broken by death, he returns once more to the laws of motion, and
+produces the most solid and substantial work of his life.
+
+For this is Galileo's main glory--not his brilliant exposition of the
+Copernican system, not his flashes of wit at the expense of a moribund
+philosophy, not his experiments on floating bodies, not even his
+telescope and astronomical discoveries--though these are the most taking
+and dazzling at first sight. No; his main glory and title to immortality
+consists in this, that he first laid the foundation of mechanics on a
+firm and secure basis of experiment, reasoning, and observation. He
+first discovered the true Laws of Motion.
+
+I said little of this achievement in my lecture on him; for the work was
+written towards the end of his life, and I had no time then. But I knew
+I should have to return to it before we came to Newton, and here we are.
+
+You may wonder how the work got published when so many of his
+manuscripts were destroyed. Horrible to say, Galileo's own son destroyed
+a great bundle of his father's manuscripts, thinking, no doubt, thereby
+to save his own soul. This book on mechanics was not burnt, however. The
+fact is it was rescued by one or other of his pupils, Toricelli or
+Viviani, who were allowed to visit him in his last two or three years;
+it was kept by them for some time, and then published surreptitiously in
+Holland. Not that there is anything in it bearing in any visible way on
+any theological controversy; but it is unlikely that the Inquisition
+would have suffered it to pass notwithstanding.
+
+I have appended to the summary preceding this lecture (p. 160) the three
+axioms or laws of motion discovered by Galileo. They are stated by
+Newton with unexampled clearness and accuracy, and are hence known as
+Newton's laws, but they are based on Galileo's work. The first is the
+simplest; though ignorance of it gave the ancients a deal of trouble. It
+is simply a statement that force is needed to change the motion of a
+body; _i.e._ that if no force act on a body it will continue to move
+uniformly both in speed and direction--in other words, steadily, in a
+straight line. The old idea had been that some force was needed to
+maintain motion. On the contrary, the first law asserts, some force is
+needed to destroy it. Leave a body alone, free from all friction or
+other retarding forces, and it will go on for ever. The planetary motion
+through empty space therefore wants no keeping up; it is not the motion
+that demands a force to maintain it, it is the curvature of the path
+that needs a force to produce it continually. The motion of a planet is
+approximately uniform so far as speed is concerned, but it is not
+constant in direction; it is nearly a circle. The real force needed is
+not a propelling but a deflecting force.
+
+The second law asserts that when a force acts, the motion changes,
+either in speed or in direction, or both, at a pace proportional to the
+magnitude of the force, and in the same direction as that in which the
+force acts. Now since it is almost solely in direction that planetary
+motion alters, a deflecting force only is needed; a force at right
+angles to the direction of motion, a force normal to the path.
+Considering the motion as circular, a force along the radius, a radial
+or centripetal force, must be acting continually. Whirl a weight round
+and round by a bit of elastic, the elastic is stretched; whirl it
+faster, it is stretched more. The moving mass pulls at the elastic--that
+is its centrifugal force; the hand at the centre pulls also--that is
+centripetal force.
+
+The third law asserts that these two forces are equal, and together
+constitute the tension in the elastic. It is impossible to have one
+force alone, there must be a pair. You can't push hard against a body
+that offers no resistance. Whatever force you exert upon a body, with
+that same force the body must react upon you. Action and reaction are
+always equal and opposite.
+
+Sometimes an absurd difficulty is felt with respect to this, even by
+engineers. They say, "If the cart pulls against the horse with precisely
+the same force as the horse pulls the cart, why should the cart move?"
+Why on earth not? The cart moves because the horse pulls it, and because
+nothing else is pulling it back. "Yes," they say, "the cart is pulling
+back." But what is it pulling back? Not itself, surely? "No, the horse."
+Yes, certainly the cart is pulling at the horse; if the cart offered no
+resistance what would be the good of the horse? That is what he is for,
+to overcome the pull-back of the cart; but nothing is pulling the cart
+back (except, of course, a little friction), and the horse is pulling it
+forward, hence it goes forward. There is no puzzle at all when once you
+realise that there are two bodies and two forces acting, and that one
+force acts on each body.[16]
+
+If, indeed, two balanced forces acted on one body that would be in
+equilibrium, but the two equal forces contemplated in the third law act
+on two different bodies, and neither is in equilibrium.
+
+So much for the third law, which is extremely simple, though it has
+extraordinarily far-reaching consequences, and when combined with a
+denial of "action at a distance," is precisely the principle of the
+Conservation of Energy. Attempts at perpetual motion may all be regarded
+as attempts to get round this "third law."
+
+[Illustration: FIG. 57.]
+
+     On the subject of the _second_ law a great deal more has to be said
+     before it can be in any proper sense even partially appreciated,
+     but a complete discussion of it would involve a treatise on
+     mechanics. It is _the_ law of mechanics. One aspect of it we must
+     attend to now in order to deal with the motion of the planets, and
+     that is the fact that the change of motion of a body depends solely
+     and simply on the force acting, and not at all upon what the body
+     happens to be doing at the time it acts. It may be stationary, or
+     it may be moving in any direction; that makes no difference.
+
+     Thus, referring back to the summary preceding Lecture IV, it is
+     there stated that a dropped body falls 16 feet in the first second,
+     that in two seconds it falls 64 feet, and so on, in proportion to
+     the square of the time. So also will it be the case with a thrown
+     body, but the drop must be reckoned from its line of motion--the
+     straight line which, but for gravity, it would describe.
+
+     Thus a stone thrown from _O_ with the velocity _OA_ would in one
+     second find itself at _A_, in two seconds at _B_, in three seconds
+     at _C_, and so on, in accordance with the first law of motion, if
+     no force acted. But if gravity acts it will have fallen 16 feet by
+     the time it would have got to _A_, and so will find itself at _P_.
+     In two seconds it will be at _Q_, having fallen a vertical height
+     of 64 feet; in three seconds it will be at _R_, 144 feet below _C_;
+     and so on. Its actual path will be a curve, which in this case is a
+     parabola. (Fig. 57.)
+
+     If a cannon is pointed horizontally over a level plain, the cannon
+     ball will be just as much affected by gravity as if it were
+     dropped, and so will strike the plain at the same instant as
+     another which was simply dropped where it started. One ball may
+     have gone a mile and the other only dropped a hundred feet or so,
+     but the time needed by both for the vertical drop will be the same.
+     The horizontal motion of one is an extra, and is due to the powder.
+
+     As a matter of fact the path of a projectile in vacuo is only
+     approximately a parabola. It is instructive to remember that it is
+     really an ellipse with one focus very distant, but not at infinity.
+     One of its foci is the centre of the earth. A projectile is really
+     a minute satellite of the earth's, and in vacuo it accurately obeys
+     all Kepler's laws. It happens not to be able to complete its orbit,
+     because it was started inconveniently close to the earth, whose
+     bulk gets in its way; but in that respect the earth is to be
+     reckoned as a gratuitous obstruction, like a target, but a target
+     that differs from most targets in being hard to miss.
+
+[Illustration: FIG. 58.]
+
+     Now consider circular motion in the same way, say a ball whirled
+     round by a string. (Fig. 58.)
+
+     Attending to the body at _O_, it is for an instant moving towards
+     _A_, and if no force acted it would get to _A_ in a time which for
+     brevity we may call a second. But a force, the pull of the string,
+     is continually drawing it towards _S_, and so it really finds
+     itself at _P_, having described the circular arc _OP_, which may
+     be considered to be compounded of, and analyzable into the
+     rectilinear motion _OA_ and the drop _AP_. At _P_ it is for an
+     instant moving towards _B_, and the same process therefore carries
+     it to _Q_; in the third second it gets to _R_; and so on: always
+     falling, so to speak, from its natural rectilinear path, towards
+     the centre, but never getting any nearer to the centre.
+
+     The force with which it has thus to be constantly pulled in towards
+     the centre, or, which is the same thing, the force with which it is
+     tugging at whatever constraint it is that holds it in, is
+     _mv^2/r_; where _m_ is the mass of the particle, _v_ its
+     velocity, and _r_ the radius of its circle of movement. This is the
+     formula first given by Huyghens for centrifugal force.
+
+     We shall find it convenient to express it in terms of the time of
+     one revolution, say _T_. It is easily done, since plainly T =
+     circumference/speed = _2[pi]r/v_; so the above expression for
+     centrifugal force becomes _4[pi]^2mr/T^2_.
+
+     As to the fall of the body towards the centre every microscopic
+     unit of time, it is easily reckoned. For by Euclid III. 36, and
+     Fig. 58, _AP.AA' = AO^2_. Take _A_ very near _O_, then _OA = vt_,
+     and _AA' = 2r_; so _AP = v^2t^2/2r = 2[pi]^2r
+     t^2/T^2_; or the fall per second is _2[pi]^2r/T^2_,
+     _r_ being its distance from the centre, and _T_ its time of going
+     once round.
+
+     In the case of the moon for instance, _r_ is 60 earth radii; more
+     exactly 60�2; and _T_ is a lunar month, or more precisely 27 days,
+     7 hours, 43 minutes, and 11-1/2 seconds. Hence the moon's
+     deflection from the tangential or rectilinear path every minute
+     comes out as very closely 16 feet (the true size of the earth being
+     used).
+
+Returning now to the case of a small body revolving round a big one, and
+assuming a force directly proportional to the mass of both bodies, and
+inversely proportional to the square of the distance between them:
+_i.e._ assuming the known force of gravity, it is
+
+  _V Mm/r^2_
+
+where _V_ is a constant, called the gravitation constant, to be
+determined by experiment.
+
+If this is the centripetal force pulling a planet or satellite in, it
+must be equal to the centrifugal force of this latter, viz. (see above).
+
+  _4[pi]^2mr/T^2
+
+Equate the two together, and at once we get
+
+  _r^3/T^2 = V/4[pi]^2M;_
+
+or, in words, the cube of the distance divided by the square of the
+periodic time for every planet or satellite of the system under
+consideration, will be constant and proportional to the mass of the
+central body.
+
+This is Kepler's third law, with a notable addition. It is stated above
+for circular motion only, so as to avoid geometrical difficulties, but
+even so it is very instructive. The reason of the proportion between
+_r^3_ and _T^2_ is at once manifest; and as soon as the constant _V_
+became known, _the mass of the central body_, the sun in the case of a
+planet, the earth in the case of the moon, Jupiter in the case of his
+satellites, was at once determined.
+
+Newton's reasoning at this time might, however, be better displayed
+perhaps by altering the order of the steps a little, as thus:--
+
+The centrifugal force of a body is proportional to _r^3/T^2_, but by
+Kepler's third law _r^3/T^2_ is constant for all the planets,
+reckoning _r_ from the sun. Hence the centripetal force needed to hold
+in all the planets will be a single force emanating from the sun and
+varying inversely with the square of the distance from that body.
+
+Such a force is at once necessary and sufficient. Such a force would
+explain the motion of the planets.
+
+But then all this proceeds on a wrong assumption--that the planetary
+motion is circular. Will it hold for elliptic orbits? Will an inverse
+square law of force keep a body moving in an elliptic orbit about the
+sun in one focus? This is a far more difficult question. Newton solved
+it, but I do not believe that even he could have solved it, except that
+he had at his disposal two mathematical engines of great power--the
+Cartesian method of treating geometry, and his own method of Fluxions.
+One can explain the elliptic motion now mathematically, but hardly
+otherwise; and I must be content to state that the double fact is
+true--viz., that an inverse square law will move the body in an ellipse
+or other conic section with the sun in one focus, and that if a body so
+moves it _must_ be acted on by an inverse square law.
+
+[Illustration: FIG. 59.]
+
+This then is the meaning of the first and third laws of Kepler. What
+about the second? What is the meaning of the equable description of
+areas? Well, that rigorously proves that a planet is acted on by a force
+directed to the centre about which the rate of description of areas is
+equable. It proves, in fact, that the sun is the attracting body, and
+that no other force acts.
+
+     For first of all if the first law of motion is obeyed, _i.e._ if no
+     force acts, and if the path be equally subdivided to represent
+     equal times, and straight lines be drawn from the divisions to any
+     point whatever, all these areas thus enclosed will be equal,
+     because they are triangles on equal base and of the same height
+     (Euclid, I). See Fig. 59; _S_ being any point whatever, and _A_,
+     _B_, _C_, successive positions of a body.
+
+     Now at each of the successive instants let the body receive a
+     sudden blow in the direction of that same point _S_, sufficient to
+     carry it from _A_ to _D_ in the same time as it would have got to
+     _B_ if left alone. The result will be that there will be a
+     compromise, and it will really arrive at _P_, travelling along the
+     diagonal of the parallelogram _AP_. The area its radius vector
+     sweeps out is therefore _SAP_, instead of what it would have been,
+     _SAB_. But then these two areas are equal, because they are
+     triangles on the same base _AS_, and between the same parallels
+     _BP_, _AS_; for by the parallelogram law _BP_ is parallel to _AD_.
+     Hence the area that would have been described is described, and as
+     all the areas were equal in the case of no force, they remain equal
+     when the body receives a blow at the end of every equal interval of
+     time, _provided_ that every blow is actually directed to _S_, the
+     point to which radii vectores are drawn.
+
+[Illustration: FIG. 60.]
+
+[Illustration: FIG. 61.]
+
+     It is instructive to see that it does not hold if the blow is any
+     otherwise directed; for instance, as in Fig. 61, when the blow is
+     along _AE_, the body finds itself at _P_ at the end of the second
+     interval, but the area _SAP_ is by no means equal to _SAB_, and
+     therefore not equal to _SOA_, the area swept out in the first
+     interval.
+
+     In order to modify Fig. 60 so as to represent continuous motion and
+     steady forces, we have to take the sides of the polygon _OAPQ_,
+     &c., very numerous and very small; in the limit, infinitely
+     numerous and infinitely small. The path then becomes a curve, and
+     the series of blows becomes a steady force directed towards _S_.
+     About whatever point therefore the rate of description of areas is
+     uniform, that point and no other must be the centre of all the
+     force there is. If there be no force, as in Fig. 59, well and good,
+     but if there be any force however small not directed towards _S_,
+     then the rate of description of areas about _S_ cannot be uniform.
+     Kepler, however, says that the rate of description of areas of each
+     planet about the sun is, by Tycho's observations, uniform; hence
+     the sun is the centre of all the force that acts on them, and there
+     is no other force, not even friction. That is the moral of Kepler's
+     second law.
+
+     We may also see from it that gravity does not travel like light, so
+     as to take time on its journey from sun to planet; for, if it did,
+     there would be a sort of aberration, and the force on its arrival
+     could no longer be accurately directed to the centre of the sun.
+     (See _Nature_, vol. xlvi., p. 497.) It is a matter for accuracy of
+     observation, therefore, to decide whether the minutest trace of
+     such deviation can be detected, _i.e._ within what limits of
+     accuracy Kepler's second law is now known to be obeyed.
+
+     I will content myself by saying that the limits are extremely
+     narrow. [Reference may be made also to p. 208.]
+
+Thus then it became clear to Newton that the whole solar system depended
+on a central force emanating from the sun, and varying inversely with
+the square of the distance from him: for by that hypothesis all the laws
+of Kepler concerning these motions were completely accounted for; and,
+in fact, the laws necessitated the hypothesis and established it as a
+theory.
+
+Similarly the satellites of Jupiter were controlled by a force emanating
+from Jupiter and varying according to the same law. And again our moon
+must be controlled by a force from the earth, decreasing with the
+distance according to the same law.
+
+Grant this hypothetical attracting force pulling the planets towards
+the sun, pulling the moon towards the earth, and the whole mechanism of
+the solar system is beautifully explained.
+
+If only one could be sure there was such a force! It was one thing to
+calculate out what the effects of such a force would be: it was another
+to be able to put one's finger upon it and say, this is the force that
+actually exists and is known to exist. We must picture him meditating in
+his garden on this want--an attractive force towards the earth.
+
+If only such an attractive force pulling down bodies to the earth
+existed. An apple falls from a tree. Why, it does exist! There is
+gravitation, common gravity that makes bodies fall and gives them their
+weight.
+
+Wanted, a force tending towards the centre of the earth. It is to hand!
+
+It is common old gravity that had been known so long, that was perfectly
+familiar to Galileo, and probably to Archimedes. Gravity that regulates
+the motion of projectiles. Why should it only pull stones and apples?
+Why should it not reach as high as the moon? Why should it not be the
+gravitation of the sun that is the central force acting on all the
+planets?
+
+Surely the secret of the universe is discovered! But, wait a bit; is it
+discovered? Is this force of gravity sufficient for the purpose? It must
+vary inversely with the square of the distance from the centre of the
+earth. How far is the moon away? Sixty earth's radii. Hence the force of
+gravity at the moon's distance can only be 1/3600 of what it is on the
+earth's surface. So, instead of pulling it 16 ft. per second, it should
+pull it 16/3600 ft. per second, or 16 ft. a minute.[17] How can one
+decide whether such a force is able to pull the moon the actual amount
+required? To Newton this would seem only like a sum in arithmetic. Out
+with a pencil and paper and reckon how much the moon falls toward the
+earth in every second of its motion. Is it 16/3600? That is what it
+ought to be: but is it? The size of the earth comes into the
+calculation. Sixty miles make a degree, 360 degrees a circumference.
+This gives as the earth's diameter 6,873 miles; work it out.
+
+The answer is not 16 feet a minute, it is 13�9 feet.
+
+Surely a mistake of calculation?
+
+No, it is no mistake: there is something wrong in the theory, gravity is
+too strong.
+
+Instead of falling toward the earth 5-1/3 hundredths of an inch every
+second, as it would under gravity, the moon only falls 4-2/3 hundredths
+of an inch per second.
+
+With such a discovery in his grasp at the age of twenty-three he is
+disappointed--the figures do not agree, and he cannot make them agree.
+Either gravity is not the force in action, or else something interferes
+with it. Possibly, gravity does part of the work, and the vortices of
+Descartes interfere with it.
+
+He must abandon the fascinating idea for the time. In his own words, "he
+laid aside at that time any further thought of the matter."
+
+So far as is known, he never mentioned his disappointment to a soul. He
+might, perhaps, if he had been at Cambridge, but he was a shy and
+solitary youth, and just as likely he might not. Up in Lincolnshire, in
+the seventeenth century, who was there for him to consult?
+
+True, he might have rushed into premature publication, after our
+nineteenth century fashion, but that was not his method. Publication
+never seemed to have occurred to him.
+
+His reticence now is noteworthy, but later on it is perfectly
+astonishing. He is so absorbed in making discoveries that he actually
+has to be reminded to tell any one about them, and some one else always
+has to see to the printing and publishing for him.
+
+I have entered thus fully into what I conjecture to be the stages of
+this early discovery of the law of gravitation, as applicable to the
+heavenly bodies, because it is frequently and commonly misunderstood. It
+is sometimes thought that he discovered the force of gravity; I hope I
+have made it clear that he did no such thing. Every educated man long
+before his time, if asked why bodies fell, would reply just as glibly as
+they do now, "Because the earth attracts them," or "because of the force
+of gravity."
+
+His discovery was that the motions of the solar system were due to the
+action of a central force, directed to the body at the centre of the
+system, and varying inversely with the square of the distance from it.
+This discovery was based upon Kepler's laws, and was clear and certain.
+It might have been published had he so chosen.
+
+But he did not like hypothetical and unknown forces; he tried to see
+whether the known force of gravity would serve. This discovery at that
+time he failed to make, owing to a wrong numerical datum. The size of
+the earth he only knew from the common doctrine of sailors that 60 miles
+make a degree; and that threw him out. Instead of falling 16 feet a
+minute, as it ought under gravity, it only fell 13�9 feet, so he
+abandoned the idea. We do not find that he returned to it for sixteen
+years.
+
+
+
+
+LECTURE VIII
+
+NEWTON AND THE LAW OF GRAVITATION
+
+
+We left Newton at the age of twenty-three on the verge of discovering
+the mechanism of the solar system, deterred therefrom only by an error
+in the then imagined size of the earth. He had proved from Kepler's laws
+that a centripetal force directed to the sun, and varying as the inverse
+square of the distance from that body, would account for the observed
+planetary motions, and that a similar force directed to the earth would
+account for the lunar motion; and it had struck him that this force
+might be the very same as the familiar force of gravitation which gave
+to bodies their weight: but in attempting a numerical verification of
+this idea in the case of the moon he was led by the then received notion
+that sixty miles made a degree on the earth's surface into an erroneous
+estimate of the size of the moon's orbit. Being thus baffled in
+obtaining such verification, he laid the matter aside for a time.
+
+The anecdote of the apple we learn from Voltaire, who had it from
+Newton's favourite niece, who with her husband lived and kept house for
+him all his later life. It is very like one of those anecdotes which are
+easily invented and believed in, and very often turn out on scrutiny to
+have no foundation. Fortunately this anecdote is well authenticated, and
+moreover is intrinsically probable; I say fortunately, because it is
+always painful to have to give up these child-learnt anecdotes, like
+Alfred and the cakes and so on. This anecdote of the apple we need not
+resign. The tree was blown down in 1820 and part of its wood is
+preserved.
+
+I have mentioned Voltaire in connection with Newton's philosophy. This
+acute critic at a later stage did a good deal to popularise it
+throughout Europe and to overturn that of his own countryman Descartes.
+Cambridge rapidly became Newtonian, but Oxford remained Cartesian for
+fifty years or more. It is curious what little hold science and
+mathematics have ever secured in the older and more ecclesiastical
+University. The pride of possessing Newton has however no doubt been the
+main stimulus to the special pursuits of Cambridge.
+
+He now began to turn his attention to optics, and, as was usual with
+him, his whole mind became absorbed in this subject as if nothing else
+had ever occupied him. His cash-book for this time has been discovered,
+and the entries show that he is buying prisms and lenses and polishing
+powder at the beginning of 1667. He was anxious to improve telescopes by
+making more perfect lenses than had ever been used before. Accordingly
+he calculated out their proper curves, just as Descartes had also done,
+and then proceeded to grind them as near as he could to those figures.
+But the images did not please him; they were always blurred and rather
+indistinct.
+
+At length, it struck him that perhaps it was not the lenses but the
+light which was at fault. Perhaps light was so composed that it _could_
+not be focused accurately to a sharp and definite point. Perhaps the law
+of refraction was not quite accurate, but only an approximation. So he
+bought a prism to try the law. He let in sunlight through a small round
+hole in a window shutter, inserted the prism in the light, and received
+the deflected beam on a white screen; turning the prism about till it
+was deviated as little as possible. The patch on the screen was not a
+round disk, as it would have been without the prism, but was an
+elongated oval and was coloured at its extremities. Evidently
+refraction was not a simple geometrical deflection of a ray, there was a
+spreading out as well.
+
+[Illustration: FIG. 63.--A prism not only _deviates_ a beam of sunlight,
+but also spreads it out or _disperses_ it.]
+
+Why did the image thus spread out? If it were due to irregularities in
+the glass a second prism should rather increase them, but a second prism
+when held in appropriate position was able to neutralise the dispersion
+and to reproduce the simple round white spot without deviation.
+Evidently the spreading out of the beam was connected in some definite
+way with its refraction. Could it be that the light particles after
+passing through the prism travelled in variously curved lines, as
+spinning racquet balls do? To examine this he measured the length of the
+oval patch when the screen was at different distances from the prism,
+and found that the two things were directly proportional to each other.
+Doubling the distance of the screen doubled the length of the patch.
+Hence the rays travelled in straight lines from the prism, and the
+spreading out was due to something that occurred within its substance.
+Could it be that white light was compound, was a mixture of several
+constituents, and that its different constituents were differently bent?
+No sooner thought than tried. Pierce the screen to let one of the
+constituents through and interpose a second prism in its path. If the
+spreading out depended on the prism only it should spread out just as
+much as before, but if it depended on the complex character of white
+light, this isolated simple constituent should be able to spread out no
+more. It did not spread out any more: a prism had no more dispersive
+power over it; it was deflected by the appropriate amount, but it was
+not analysed into constituents. It differed from sunlight in being
+simple. With many ingenious and beautifully simple experiments, which
+are quoted in full in several books on optics, he clinched the argument
+and established his discovery. White light was not simple but compound.
+It could be sorted out by a prism into an infinite number of constituent
+parts which were differently refracted, and the most striking of which
+Newton named violet, indigo, blue, green, yellow, orange, and red.
+
+[Illustration: FIG. 64.--A single constituent of white light, obtained
+by the use of perforated screens is capable of no more dispersion.]
+
+At once the true nature of colour became manifest. Colour resided not in
+the coloured object as had till now been thought, but in the light which
+illuminated it. Red glass for instance adds nothing to sunlight. The
+light does not get dyed red by passing through the glass; all that the
+red glass does is to stop and absorb a large part of the sunlight; it is
+opaque to the larger portion, but it is transparent to that particular
+portion which affects our eyes with the sensation of red. The prism acts
+like a sieve sorting out the different kinds of light. Coloured media
+act like filters, stopping certain kinds but allowing the rest to go
+through. Leonardo's and all the ancient doctrines of colour had been
+singularly wrong; colour is not in the object but in the light.
+
+Goethe, in his _Farbenlehre_, endeavoured to controvert Newton, and to
+reinstate something more like the old views; but his failure was
+complete.
+
+Refraction analysed out the various constituents of white light and
+displayed them in the form of a series of overlapping images of the
+aperture, each of a different colour; this series of images we call a
+spectrum, and the operation we now call spectrum analysis. The reason of
+the defect of lenses was now plain: it was not so much a defect of the
+lens as a defect of light. A lens acts by refraction and brings rays to
+a focus. If light be simple it acts well, but if ordinary white light
+fall upon a lens, its different constituents have different foci; every
+bright object is fringed with colour, and nothing like a clear image can
+be obtained.
+
+[Illustration: FIG. 65.--Showing the boundary rays of a parallel beam
+passing through a lens.]
+
+A parallel beam passing through a lens becomes conical; but instead of a
+single cone it is a sheaf or nest of cones, all having the edge of the
+lens as base, but each having a different vertex. The violet cone is
+innermost, near the lens, the red cone outermost, while the others lie
+between. Beyond the crossing point or focus the order of cones is
+reversed, as the above figure shows. Only the two marginal rays of the
+beam are depicted.
+
+If a screen be held anywhere nearer the lens than the place marked 1
+there will be a whitish centre to the patch of light and a red and
+orange fringe or border. Held anywhere beyond the region 2, the border
+of the patch will be blue and violet. Held about 3 the colour will be
+less marked than elsewhere, but nowhere can it be got rid of. Each point
+of an object will be represented in the image not by a point but by a
+coloured patch: a fact which amply explains the observed blurring and
+indistinctness.
+
+Newton measured and calculated the distance between the violet and red
+foci--VR in the diagram--and showed that it was 1/50th the diameter of
+the lens. To overcome this difficulty (called chromatic aberration)
+telescope glasses were made small and of very long focus: some of them
+so long that they had no tube, all of them egregiously cumbrous. Yet it
+was with such instruments that all the early discoveries were made. With
+such an instrument, for instance, Huyghens discovered the real shape of
+Saturn's ring.
+
+The defects of refractors seemed irremediable, being founded in the
+nature of light itself. So he gave up his "glass works"; and proceeded
+to think of reflexion from metal specula. A concave mirror forms an
+image just as a lens does, but since it does so without refraction or
+transmission through any substance, there is no accompanying dispersion
+or chromatic aberration.
+
+The first reflecting telescope he made was 1 in. diameter and 6 in.
+long, and magnified forty times. It acted as well as a three or four
+feet refractor of that day, and showed Jupiter's moons. So he made a
+larger one, now in the library of the Royal Society, London, with an
+inscription:
+
+"The first reflecting telescope, invented by Sir Isaac Newton, and made
+with his own hands."
+
+This has been the parent of most of the gigantic telescopes of the
+present day. Fifty years elapsed before it was much improved on, and
+then, first by Hadley and afterwards by Herschel and others, large and
+good reflectors were constructed.
+
+The largest telescope ever made, that of Lord Rosse, is a Newtonian
+reflector, fifty feet long, six feet diameter, with a mirror weighing
+four tons. The sextant, as used by navigators, was also invented by
+Newton.
+
+The year after the plague, in 1667, Newton returned to Trinity College,
+and there continued his experiments on optics. It is specially to be
+noted that at this time, at the age of twenty-four, Newton had laid the
+foundations of all his greatest discoveries:--
+
+[Illustration: FIG. 66.--Newton's telescope.]
+
+The Theory of Fluxions; or, the Differential Calculus.
+
+The Law of Gravitation; or, the complete theory of astronomy.
+
+The compound nature of white light; or, the beginning of Spectrum
+Analysis.
+
+[Illustration: FIG. 67.--The sextant, as now made.]
+
+His later life was to be occupied in working these incipient discoveries
+out. But the most remarkable thing is that no one knew about any one of
+them. However, he was known as an accomplished young mathematician, and
+was made a fellow of his college. You remember that he had a friend
+there in the person of Dr. Isaac Barrow, first Lucasian Professor of
+Mathematics in the University. It happened, about 1669, that a
+mathematical discovery of some interest was being much discussed, and
+Dr. Barrow happened to mention it to Newton, who said yes, he had worked
+out that and a few other similar things some time ago. He accordingly
+went and fetched some papers to Dr. Barrow, who forwarded them to other
+distinguished mathematicians, and it thus appeared that Newton had
+discovered theorems much more general than this special case that was
+exciting so much interest. Dr. Barrow, being anxious to devote his time
+more particularly to theology, resigned his chair the same year in
+favour of Newton, who was accordingly elected to the Lucasian
+Professorship, which he held for thirty years. This chair is now the
+most famous in the University, and it is commonly referred to as the
+chair of Newton.
+
+Still, however, his method of fluxions was unknown, and still he did not
+publish it. He lectured first on optics, giving an account of his
+experiments. His lectures were afterwards published both in Latin and
+English, and are highly valued to this day.
+
+The fame of his mathematical genius came to the ears of the Royal
+Society, and a motion was made to get him elected a fellow of that body.
+The Royal Society, the oldest and most famous of all scientific
+societies with a continuous existence, took its origin in some private
+meetings, got up in London by the Hon. Robert Boyle and a few scientific
+friends, during all the trouble of the Commonwealth.
+
+After the restoration, Charles II. in 1662 incorporated it under Royal
+Charter; among the original members being Boyle, Hooke, Christopher
+Wren, and other less famous names. Boyle was a great experimenter, a
+worthy follower of Dr. Gilbert. Hooke began as his assistant, but being
+of a most extraordinary ingenuity he rapidly rose so as to exceed his
+master in importance. Fate has been a little unkind to Hooke in placing
+him so near to Newton; had he lived in an ordinary age he would
+undoubtedly have shone as a star of the first magnitude. With great
+ingenuity, remarkable scientific insight, and consummate experimental
+skill, he stands in many respects almost on a level with Galileo. But it
+is difficult to see stars even of the first magnitude when the sun is
+up, and thus it happens that the name and fame of this brilliant man are
+almost lost in the blaze of Newton. Of Christopher Wren I need not say
+much. He is well known as an architect, but he was a most accomplished
+all-round man, and had a considerable taste and faculty for science.
+
+These then were the luminaries of the Royal Society at the time we are
+speaking of, and to them Newton's first scientific publication was
+submitted. He communicated to them an account of his reflecting
+telescope, and presented them with the instrument.
+
+Their reception of it surprised him; they were greatly delighted with
+it, and wrote specially thanking him for the communication, and assuring
+him that all right should be done him in the matter of the invention.
+The Bishop of Salisbury (Bishop Burnet) proposed him for election as a
+fellow, and elected he was.
+
+In reply, he expressed his surprise at the value they set on the
+telescope, and offered, if they cared for it, to send them an account of
+a discovery which he doubts not will prove much more grateful than the
+communication of that instrument, "being in my judgment the oddest, if
+not the most considerable detection that has recently been made into the
+operations of Nature."
+
+So he tells them about his optical researches and his discovery of the
+nature of white light, writing them a series of papers which were long
+afterwards incorporated and published as his _Optics_. A magnificent
+work, which of itself suffices to place its author in the first rank of
+the world's men of science.
+
+The nature of white light, the true doctrine of colour, and the
+differential calculus! besides a good number of minor results--binomial
+theorem, reflecting telescope, sextant, and the like; one would think it
+enough for one man's life-work, but the masterpiece remains still to be
+mentioned. It is as when one is considering Shakspeare: _King Lear_,
+_Macbeth_, _Othello_,--surely a sufficient achievement,--but the
+masterpiece remains.
+
+Comparisons in different departments are but little help perhaps,
+nevertheless it seems to me that in his own department, and considered
+simply as a man of science, Newton towers head and shoulders over, not
+only his contemporaries--that is a small matter--but over every other
+scientific man who has ever lived, in a way that we can find no parallel
+for in other departments. Other nations admit his scientific
+pre-eminence with as much alacrity as we do.
+
+Well, we have arrived at the year 1672 and his election to the Royal
+Society. During the first year of his membership there was read at one
+of the meetings a paper giving an account of a very careful
+determination of the length of a degree (_i.e._ of the size of the
+earth), which had been made by Picard near Paris. The length of the
+degree turned out to be not sixty miles, but nearly seventy miles. How
+soon Newton heard of this we do not learn--probably not for some
+years,--Cambridge was not so near London then as it is now, but
+ultimately it was brought to his notice. Armed with this new datum, his
+old speculation concerning gravity occurred to him. He had worked out
+the mechanics of the solar system on a certain hypothesis, but it had
+remained a hypothesis somewhat out of harmony with apparent fact. What
+if it should turn out to be true after all!
+
+He took out his old papers and began again the calculation. If gravity
+were the force keeping the moon in its orbit, it would fall toward the
+earth sixteen feet every minute. How far did it fall? The newly known
+size of the earth would modify the figures: with intense excitement he
+runs through the working, his mind leaps before his hand, and as he
+perceives the answer to be coming out right, all the infinite meaning
+and scope of his mighty discovery flashes upon him, and he can no longer
+see the paper. He throws down the pen; and the secret of the universe
+is, to one man, known.
+
+But of course it had to be worked out. The meaning might flash upon him,
+but its full detail required years of elaboration; and deeper and deeper
+consequences revealed themselves to him as he proceeded.
+
+For two years he devoted himself solely to this one object. During
+those years he lived but to calculate and think, and the most ludicrous
+stories are told concerning his entire absorption and inattention to
+ordinary affairs of life. Thus, for instance, when getting up in a
+morning he would sit on the side of the bed half-dressed, and remain
+like that till dinner time. Often he would stay at home for days
+together, eating what was taken to him, but without apparently noticing
+what he was doing.
+
+One day an intimate friend, Dr. Stukely, called on him and found on the
+table a cover laid for his solitary dinner. After waiting a long time,
+Dr. Stukely removed the cover and ate the chicken underneath it,
+replacing and covering up the bones again. At length Newton appeared,
+and after greeting his friend, sat down to dinner, but on lifting the
+cover he said in surprise, "Dear me, I thought I had not dined, but I
+see I have."
+
+It was by this continuous application that the _Principia_ was
+accomplished. Probably nothing of the first magnitude can be
+accomplished without something of the same absorbed unconsciousness and
+freedom from interruption. But though desirable and essential for the
+_work_, it was a severe tax upon the powers of the _man_. There is, in
+fact, no doubt that Newton's brain suffered temporary aberration after
+this effort for a short time. The attack was slight, and it has been
+denied; but there are letters extant which are inexplicable otherwise,
+and moreover after a year or two he writes to his friends apologizing
+for strange and disjointed epistles, which he believed he had written
+without understanding clearly what he wrote. The derangement was,
+however, both slight and temporary: and it is only instructive to us as
+showing at what cost such a work as the _Principia_ must be produced,
+even by so mighty a mind as that of Newton.
+
+The first part of the work having been done, any ordinary mortal would
+have proceeded to publish it; but the fact is that after he had sent to
+the Royal Society his papers on optics, there had arisen controversies
+and objections; most of them rather paltry, to which he felt compelled
+to find answers. Many men would have enjoyed this part of the work, and
+taken it as evidence of interest and success. But to Newton's shy and
+retiring disposition these discussions were merely painful. He writes,
+indeed, his answers with great patience and ability, and ultimately
+converts the more reasonable of his opponents, but he relieves his mind
+in the following letter to the secretary of the Royal Society: "I see I
+have made myself a slave to philosophy, but if I get free of this
+present business I will resolutely bid adieu to it eternally, except
+what I do for my private satisfaction or leave to come out after me; for
+I see a man must either resolve to put out nothing new, or to become a
+slave to defend it." And again in a letter to Leibnitz: "I have been so
+persecuted with discussions arising out of my theory of light that I
+blamed my own imprudence for parting with so substantial a blessing as
+my quiet to run after a shadow." This shows how much he cared for
+contemporary fame.
+
+So he locked up the first part of the _Principia_ in his desk, doubtless
+intending it to be published after his death. But fortunately this was
+not so to be.
+
+In 1683, among the leading lights of the Royal Society, the same sort of
+notions about gravity and the solar system began independently to be
+bruited. The theory of gravitation seemed to be in the air, and Wren,
+Hooke, and Halley had many a talk about it.
+
+Hooke showed an experiment with a pendulum, which he likened to a planet
+going round the sun. The analogy is more superficial than real. It does
+not obey Kepler's laws; still it was a striking experiment. They had
+guessed at a law of inverse squares, and their difficulty was to prove
+what curve a body subject to it would describe. They knew it ought to be
+an ellipse if it was to serve to explain the planetary motion, and Hooke
+said he could prove that an ellipse it was; but he was nothing of a
+mathematician, and the others scarcely believed him. Undoubtedly he had
+shrewd inklings of the truth, though his guesses were based on little
+else than a most sagacious intuition. He surmised also that gravity was
+the force concerned, and asserted that the path of an ordinary
+projectile was an ellipse, like the path of a planet--which is quite
+right. In fact the beginnings of the discovery were beginning to dawn
+upon him in the well-known way in which things do dawn upon ordinary men
+of genius: and had Newton not lived we should doubtless, by the labours
+of a long chain of distinguished men, beginning with Hooke, Wren, and
+Halley, have been now in possession of all the truths revealed by the
+_Principia_. We should never have had them stated in the same form, nor
+proved with the same marvellous lucidity and simplicity, but the facts
+themselves we should by this time have arrived at. Their developments
+and completions, due to such men as Clairaut, Euler, D'Alembert,
+Lagrange, Laplace, Airy, Leverrier, Adams, we should of course not have
+had to the same extent; because the lives and energies of these great
+men would have been partially consumed in obtaining the main facts
+themselves.
+
+The youngest of the three questioners at the time we are speaking of was
+Edmund Halley, an able and remarkable man. He had been at Cambridge,
+doubtless had heard Newton lecture, and had acquired a great veneration
+for him.
+
+In January, 1684, we find Wren offering Hooke and Halley a prize, in the
+shape of a book worth forty shillings, if they would either of them
+bring him within two months a demonstration that the path of a planet
+subject to an inverse square law would be an ellipse. Not in two months,
+nor yet in seven, was there any proof forthcoming. So at last, in
+August, Halley went over to Cambridge to speak to Newton about the
+difficult problem and secure his aid. Arriving at his rooms he went
+straight to the point. He said, "What path will a body describe if it
+be attracted by a centre with a force varying as the inverse square of
+the distance." To which Newton at once replied, "An ellipse." "How on
+earth do you know?" said Halley in amazement. "Why, I have calculated
+it," and began hunting about for the paper. He actually couldn't find it
+just then, but sent it him shortly by post, and with it much more--in
+fact, what appeared to be a complete treatise on motion in general.
+
+With his valuable burden Halley hastened to the Royal Society and told
+them what he had discovered. The Society at his representation wrote to
+Mr. Newton asking leave that it might be printed. To this he consented;
+but the Royal Society wisely appointed Mr. Halley to see after him and
+jog his memory, in case he forgot about it. However, he set to work to
+polish it up and finish it, and added to it a great number of later
+developments and embellishments, especially the part concerning the
+lunar theory, which gave him a deal of trouble--and no wonder; for in
+the way he has put it there never was a man yet living who could have
+done the same thing. Mathematicians regard the achievement now as men
+might stare at the work of some demigod of a bygone age, wondering what
+manner of man this was, able to wield such ponderous implements with
+such apparent ease.
+
+To Halley the world owes a great debt of gratitude--first, for
+discovering the _Principia_; second, for seeing it through the press;
+and third, for defraying the cost of its publication out of his own
+scanty purse. For though he ultimately suffered no pecuniary loss,
+rather the contrary, yet there was considerable risk in bringing out a
+book which not a dozen men living could at the time comprehend. It is no
+small part of the merit of Halley that he recognized the transcendent
+value of the yet unfinished work, that he brought it to light, and
+assisted in its becoming understood to the best of his ability.
+
+Though Halley afterwards became Astronomer-Royal, lived to the ripe old
+age of eighty-six, and made many striking observations, yet he would be
+the first to admit that nothing he ever did was at all comparable in
+importance with his discovery of the _Principia_; and he always used to
+regard his part in it with peculiar pride and pleasure.
+
+And how was the _Principia_ received? Considering the abstruse nature of
+its subject, it was received with great interest and enthusiasm. In less
+than twenty years the edition was sold out, and copies fetched large
+sums. We hear of poor students copying out the whole in manuscript in
+order to possess a copy--not by any means a bad thing to do, however
+many copies one may possess. The only useful way really to read a book
+like that is to pore over every sentence: it is no book to be skimmed.
+
+While the _Principia_ was preparing for the press a curious incident of
+contact between English history and the University occurred. It seems
+that James II., in his policy of Catholicising the country, ordered both
+Universities to elect certain priests to degrees without the ordinary
+oaths. Oxford had given way, and the Dean of Christ Church was a
+creature of James's choosing. Cambridge rebelled, and sent eight of its
+members, among them Mr. Newton, to plead their cause before the Court of
+High Commission. Judge Jeffreys presided over the Court, and threatened
+and bullied with his usual insolence. The Vice-Chancellor of Cambridge
+was deprived of office, the other deputies were silenced and ordered
+away. From the precincts of this court of justice Newton returned to
+Trinity College to complete the _Principia_.
+
+By this time Newton was only forty-five years old, but his main work was
+done. His method of fluxions was still unpublished; his optics was
+published only imperfectly; a second edition of the _Principia_, with
+additions and improvements, had yet to appear; but fame had now come
+upon him, and with fame worries of all kinds.
+
+By some fatality, principally no doubt because of the interest they
+excited, every discovery he published was the signal for an outburst of
+criticism and sometimes of attack. I shall not go into these matters:
+they are now trivial enough, but it is necessary to mention them,
+because to Newton they evidently loomed large and terrible, and
+occasioned him acute torment.
+
+[Illustration: FIG. 68.--Newton when young. (_From an engraving by B.
+Reading after Sir Peter Lely._)]
+
+No sooner was the _Principia_ put than Hooke put in his claims for
+priority. And indeed his claims were not altogether negligible; for
+vague ideas of the same sort had been floating in his comprehensive
+mind, and he doubtless felt indistinctly conscious of a great deal more
+than he could really state or prove.
+
+By indiscreet friends these two great men were set somewhat at
+loggerheads, and worse might have happened had they not managed to come
+to close quarters, and correspond privately in a quite friendly manner,
+instead of acting through the mischievous medium of third parties. In
+the next edition Newton liberally recognizes the claims of both Hooke
+and Wren. However, he takes warning betimes of what he has to expect,
+and writes to Halley that he will only publish the first two books,
+those containing general theorems on motion. The third book--concerning
+the system of the world, _i.e._ the application to the solar system--he
+says "I now design to suppress. Philosophy is such an impertinently
+litigious lady that a man had as good be engaged in law-suits as have to
+do with her. I found it so formerly, and now I am no sooner come near
+her again but she gives me warning. The two books without the third will
+not so well bear the title 'Mathematical Principles of Natural
+Philosophy,' and therefore I had altered it to this, 'On the Free Motion
+of Two Bodies'; but on second thoughts I retain the former title: 'twill
+help the sale of the book--which I ought not to diminish now 'tis
+yours."
+
+However, fortunately, Halley was able to prevail upon him to publish the
+third book also. It is, indeed, the most interesting and popular of the
+three, as it contains all the direct applications to astronomy of the
+truths established in the other two.
+
+Some years later, when his method of fluxions was published, another and
+a worse controversy arose--this time with Leibnitz, who had also
+independently invented the differential calculus. It was not so well
+recognized then how frequently it happens that two men independently
+and unknowingly work at the very same thing at the same time. The
+history of science is now full of such instances; but then the friends
+of each accused the other of plagiarism.
+
+I will not go into the controversy: it is painful and useless. It only
+served to embitter the later years of two great men, and it continued
+long after Newton's death--long after both their deaths. It can hardly
+be called ancient history even now.
+
+But fame brought other and less unpleasant distractions than
+controversies. We are a curious, practical, and rather stupid people,
+and our one idea of honouring a man is to _vote_ for him in some way or
+other; so they sent Newton to Parliament. He went, I believe, as a Whig,
+but it is not recorded that he spoke. It is, in fact, recorded that he
+was once expected to speak when on a Royal Commission about some
+question of chronometers, but that he would not. However, I dare say he
+made a good average member.
+
+Then a little later it was realized that Newton was poor, that he still
+had to teach for his livelihood, and that though the Crown had continued
+his fellowship to him as Lucasian Professor without the necessity of
+taking orders, yet it was rather disgraceful that he should not be
+better off. So an appeal was made to the Government on his behalf, and
+Lord Halifax, who exerted himself strongly in the matter, succeeding to
+office on the accession of William III., was able to make him ultimately
+Master of the Mint, with a salary of some �1,200 a year. I believe he
+made rather a good Master, and turned out excellent coins: certainly he
+devoted his attention to his work there in a most exemplary manner.
+
+But what a pitiful business it all is! Here is a man sent by Heaven to
+do certain things which no man else could do, and so long as he is
+comparatively unknown he does them; but so soon as he is found out, he
+is clapped into a routine office with a big salary: and there is,
+comparatively speaking, an end of him. It is not to be supposed that he
+had lost his power, for he frequently solved problems very quickly which
+had been given out by great Continental mathematicians as a challenge to
+the world.
+
+We may ask why Newton allowed himself to be thus bandied about instead
+of settling himself down to the work in which he was so pre-eminently
+great. Well, I expect your truly great man never realizes how great he
+is, and seldom knows where his real strength lies. Certainly Newton did
+not know it. He several times talks of giving up philosophy altogether;
+and though he never really does it, and perhaps the feeling is one only
+born of some temporary overwork, yet he does not sacrifice everything
+else to it as he surely must had he been conscious of his own greatness.
+No; self-consciousness was the last thing that affected him. It is for a
+great man's contemporaries to discover him, to make much of him, and to
+put him in surroundings where he may flourish luxuriantly in his own
+heaven-intended way.
+
+However, it is difficult for us to judge of these things. Perhaps if he
+had been maintained at the national expense to do that for which he was
+preternaturally fitted, he might have worn himself out prematurely;
+whereas by giving him routine work the scientific world got the benefit
+of his matured wisdom and experience. It was no small matter to the
+young Royal Society to be able to have him as their President for
+twenty-four years. His portrait has hung over the President's chair ever
+since, and there I suppose it will continue to hang until the Royal
+Society becomes extinct.
+
+The events of his later life I shall pass over lightly. He lived a calm,
+benevolent life, universally respected and beloved. His silver-white
+hair when he removed his peruke was a venerable spectacle. A lock of it
+is still preserved, with many other relics, in the library of Trinity
+College. He died quietly, after a painful illness, at the ripe age of
+eighty-five. His body lay in state in the Jerusalem Chamber, and he was
+buried in Westminster Abbey, six peers bearing the pall. These things
+are to be mentioned to the credit of the time and the country; for
+after we have seen the calamitous spectacle of the way Tycho and Kepler
+and Galileo were treated by their ungrateful and unworthy countries, it
+is pleasant to reflect that England, with all its mistakes, yet
+recognized _her_ great man when she received him, and honoured him with
+the best she knew how to give.
+
+[Illustration: FIG. 69.--Sir Isaac Newton.]
+
+Concerning his character, one need only say that it was what one would
+expect and wish. It was characterized by a modest, calm, dignified
+simplicity. He lived frugally with his niece and her husband, Mr.
+Conduit, who succeeded him as Master of the Mint. He never married, nor
+apparently did he ever think of so doing. The idea, perhaps, did not
+naturally occur to him, any more than the idea of publishing his work
+did.
+
+He was always a deeply religious man and a sincere Christian, though
+somewhat of the Arian or Unitarian persuasion--so, at least, it is
+asserted by orthodox divines who understand these matters. He studied
+theology more or less all his life, and towards the end was greatly
+interested in questions of Biblical criticism and chronology. By some
+ancient eclipse or other he altered the recognized system of dates a few
+hundred years; and his book on the prophecies of Daniel and the
+Revelation of St. John, wherein he identifies the beast with the Church
+of Rome in quite the orthodox way, is still by some admired.
+
+But in all these matters it is probable that he was a merely ordinary
+man, with natural acumen and ability doubtless, but nothing in the least
+superhuman. In science, the impression he makes upon me is only
+expressible by the words inspired, superhuman.
+
+And yet if one realizes his method of work, and the calm, uninterrupted
+flow of all his earlier life, perhaps his achievements become more
+intelligible. When asked how he made his discoveries, he replied: "By
+always thinking unto them. I keep the subject constantly before me, and
+wait till the first dawnings open slowly by little and little into a
+full and clear light." That is the way--quiet, steady, continuous
+thinking, uninterrupted and unharassed brooding. Much may be done under
+those conditions. Much ought to be sacrificed to obtain those
+conditions. All the best thinking work of the world has been thus
+done.[18] Buffon said: "Genius is patience." So says Newton: "If I have
+done the public any service this way, it is due to nothing but industry
+and patient thought." Genius patience? No, it is not quite that, or,
+rather, it is much more than that; but genius without patience is like
+fire without fuel--it will soon burn itself out.
+
+
+
+
+NOTES FOR LECTURE IX
+
+  The _Principia_ published 1687.
+  Newton died               1727.
+
+
+THE LAW OF GRAVITATION.--Every particle of matter attracts every other
+particle of matter with a force proportional to the mass of each and to
+the inverse square of the distance between them.
+
+
+SOME OF NEWTON'S DEDUCTIONS.
+
+1. Kepler's second law (equable description of areas) proves that each
+planet is acted on by a force directed towards the sun as a centre of
+force.
+
+2. Kepler's first law proves that this central force diminishes in the
+same proportion as the square of the distance increases.
+
+3. Kepler's third law proves that all the planets are acted on by the
+same kind of force; of an intensity depending on the mass of the
+sun.[19]
+
+4. So by knowing the length of year and distance of any planet from the
+sun, the sun's mass can be calculated, in terms of that of the earth.
+
+5. For the satellites, the force acting depends on the mass of _their_
+central body, a planet. Hence the mass of any planet possessing a
+satellite becomes known.
+
+6. The force constraining the moon in her orbit is the same gravity as
+gives terrestrial bodies their weight and regulates the motion of
+projectiles. [Because, while a stone drops 16 feet in a second, the
+moon, which is 60 times as far from the centre of the earth, drops 16
+feet in a minute.]
+
+*      *      *      *      *
+
+7. The moon is attracted not only by the earth, but by the sun also;
+hence its orbit is perturbed, and Newton calculated out the chief of
+these perturbations, viz.:--
+
+     (The equation of the centre, discovered by Hipparchus.)
+
+     (_a_) The evection, discovered by Hipparchus and Ptolemy.
+
+     (_b_) The variation, discovered by Tycho Brah�.
+
+     (_c_) The annual equation, discovered by Tycho Brah�.
+
+     (_d_) The retrogression of the nodes, then being observed at
+     Greenwich by Flamsteed.
+
+     (_e_) The variation of inclination, then being observed at
+     Greenwich by Flamsteed.
+
+     (_f_) The progression of the apses (with an error of one-half).
+
+     (_g_) The inequality of apogee, previously unknown.
+
+     (_h_) The inequality of nodes, previously unknown.
+
+8. Each planet is attracted not only by the sun but by the other
+planets, hence their orbits are slightly affected by each other. Newton
+began the theory of planetary perturbations.
+
+9. He recognized the comets as members of the solar system, obedient to
+the same law of gravity and moving in very elongated ellipses; so their
+return could be predicted (_e.g._ Halley's comet).
+
+10. Applying the idea of centrifugal force to the earth considered as a
+rotating body, he perceived that it could not be a true sphere, and
+calculated its oblateness, obtaining 28 miles greater equatorial than
+polar diameter.
+
+11. Conversely, from the observed shape of Jupiter, or any planet, the
+length of its day could be estimated.
+
+12. The so-calculated shape of the earth, in combination with
+centrifugal force, causes the weight of bodies to vary with latitude;
+and Newton calculated the amount of this variation. 194 lbs. at pole
+balance 195 lbs. at equator.
+
+13. A homogeneous sphere attracts as if its mass were concentrated at
+its centre. For any other figure, such as an oblate spheroid, this is
+not exactly true. A hollow concentric spherical shell exerts no force on
+small bodies inside it.
+
+14. The earth's equatorial protuberance, being acted on by the
+attraction of the sun and moon, must disturb its axis of rotation in a
+calculated manner; and thus is produced the precession of the equinoxes.
+[The attraction of the planets on the same protuberance causes a smaller
+and rather different kind of precession.]
+
+15. The waters of the ocean are attracted towards the sun and moon on
+one side, and whirled a little further away than the solid earth on the
+other side: hence Newton explained all the main phenomena of the tides.
+
+16. The sun's mass being known, he calculated the height of the solar
+tide.
+
+17. From the observed heights of spring and neap tides he determined the
+lunar tide, and thence made an estimate of the mass of the moon.
+
+REFERENCE TABLE OF NUMERICAL DATA.
+
+ +---------+---------------+----------------------+-----------------+
+ |         |Masses in Solar| Height dropped by a  | Length of Day or|
+ |         |    System.    |stone in first second.|time of rotation.|
+ +---------+---------------+----------------------+-----------------+
+ |Mercury  |         �065  |       7�0 feet       |   24    hours   |
+ |Venus    |         �885  |      15�8  "         |   23-1/2  "     |
+ |Earth    |        1�000  |      16�1  "         |   24      "     |
+ |Mars     |         �108  |       6�2  "         |   24-1/2  "     |
+ |Jupiter  |      300�8    |      45�0  "         |   10      "     |
+ |Saturn   |       89�7    |      18�4  "         |   10-1/2  "     |
+ |The Sun  |   316000�     |     436�0  "         |   608     "     |
+ |The Moon |   about �012  |       3�7  "         |   702     "     |
+ +---------+---------------+----------------------+-----------------+
+
+The mass of the earth, taken above as unity, is 6,000 trillion tons.
+
+_Observatories._--Uraniburg flourished from 1576 to 1597; the
+Observatory of Paris was founded in 1667; Greenwich Observatory in 1675.
+
+_Astronomers-Royal._--Flamsteed, Halley, Bradley, Bliss, Maskelyne,
+Pond, Airy, Christie.
+
+
+
+
+LECTURE IX
+
+NEWTON'S "PRINCIPIA"
+
+
+The law of gravitation, above enunciated, in conjunction with the laws
+of motion rehearsed at the end of the preliminary notes of Lecture VII.,
+now supersedes the laws of Kepler and includes them as special cases.
+The more comprehensive law enables us to criticize Kepler's laws from a
+higher standpoint, to see how far they are exact and how far they are
+only approximations. They are, in fact, not precisely accurate, but the
+reason for every discrepancy now becomes abundantly clear, and can be
+worked out by the theory of gravitation.
+
+We may treat Kepler's laws either as immediate consequences of the law
+of gravitation, or as the known facts upon which that law was founded.
+Historically, the latter is the more natural plan, and it is thus that
+they are treated in the first three statements of the above notes; but
+each proposition may be worked inversely, and we might state them
+thus:--
+
+1. The fact that the force acting on each planet is directed to the sun,
+necessitates the equable description of areas.
+
+2. The fact that the force varies as the inverse square of the distance,
+necessitates motion in an ellipse, or some other conic section, with the
+sun in one focus.
+
+3. The fact that one attracting body acts on all the planets with an
+inverse square law, causes the cubes of their mean distances to be
+proportional to the squares of their periodic times.
+
+Not only these but a multitude of other deductions follow rigorously
+from the simple datum that every particle of matter attracts every other
+particle with a force directly proportional to the mass of each and to
+the inverse square of their mutual distance. Those dealt with in the
+_Principia_ are summarized above, and it will be convenient to run over
+them in order, with the object of giving some idea of the general
+meaning of each, without attempting anything too intricate to be readily
+intelligible.
+
+[Illustration: FIG. 70.]
+
+No. 1. Kepler's second law (equable description of areas) proves that
+each planet is acted on by a force directed towards the sun as a centre
+of force.
+
+The equable description of areas about a centre of force has already
+been fully, though briefly, established. (p. 175.) It is undoubtedly of
+fundamental importance, and is the earliest instance of the serious
+discussion of central forces, _i.e._ of forces directed always to a
+fixed centre.
+
+We may put it afresh thus:--OA has been the motion of a particle in a
+unit of time; at A it receives a knock towards C, whereby in the next
+unit it travels along AD instead of AB. Now the area of the triangle
+CAD, swept out by the radius vector in unit time, is 1/2_bh_; _h_ being
+the perpendicular height of the triangle from the base AC. (Fig. 70.)
+Now the blow at A, being along the base, has no effect upon _h_; and
+consequently the area remains just what it would have been without the
+blow. A blow directed to any point other than C would at once alter the
+area of the triangle.
+
+One interesting deduction may at once be drawn. If gravity were a
+radiant force emitted from the sun with a velocity like that of light,
+the moving planet would encounter it at a certain apparent angle
+(aberration), and the force experienced would come from a point a little
+in advance of the sun. The rate of description of areas would thus tend
+to increase; whereas in reality it is constant. Hence the force of
+gravity, if it travel at all, does so with a speed far greater than that
+of light. It appears to be practically instantaneous. (Cf. "Modern Views
+of Electricity," � 126, end of chap. xii.) Again, anything like a
+retarding effect of the medium through which the planets move would
+constitute a tangential force, entirely un-directed towards the sun.
+Hence no such frictional or retarding force can appreciably exist. It
+is, however, conceivable that both these effects might occur and just
+neutralize each other. The neutralization is unlikely to be exact for
+all the planets; and the fact is, that no trace of either effect has as
+yet been discovered. (See also p. 176.)
+
+The planets are, however, subject to forces not directed towards the
+sun, viz. their attractions for each other; and these perturbing forces
+do produce a slight discrepancy from Kepler's second law, but a
+discrepancy which is completely subject to calculation.
+
+No. 2. Kepler's first law proves that this central force diminishes in
+the same proportion as the square of the distance increases.
+
+To prove the connection between the inverse-square law of distance, and
+the travelling in a conic section with the centre of force in one focus
+(the other focus being empty), is not so simple. It obviously involves
+some geometry, and must therefore be left to properly armed students.
+But it may be useful to state that the inverse-square law of distance,
+although the simplest possible law for force emanating from a point or
+sphere, is not to be regarded as self-evident or as needing no
+demonstration. The force of a magnetic pole on a magnetized steel scrap,
+for instance, varies as the inverse cube of the distance; and the curve
+described by such a particle would be quite different from a conic
+section--it would be a definite class of spiral (called Cotes's spiral).
+Again, on an iron filing the force of a single pole might vary more
+nearly as the inverse fifth power; and so on. Even when the thing
+concerned is radiant in straight lines, like light, the law of inverse
+squares is not universally true. Its truth assumes, first, that the
+source is a point or sphere; next, that there is no reflection or
+refraction of any kind; and lastly, that the medium is perfectly
+transparent. The law of inverse squares by no means holds from a prairie
+fire for instance, or from a lighthouse, or from a street lamp in a fog.
+
+Mutual perturbations, especially the pull of Jupiter, prevent the path
+of a planet from being really and truly an ellipse, or indeed from being
+any simple re-entrant curve. Moreover, when a planet possesses a
+satellite, it is not the centre of the planet which ever attempts to
+describe the Keplerian ellipse, but it is the common centre of gravity
+of the two bodies. Thus, in the case of the earth and moon, the point
+which really does describe a close attempt at an ellipse is a point
+displaced about 3000 miles from the centre of the earth towards the
+moon, and is therefore only 1000 miles beneath the surface.
+
+No. 3. Kepler's third law proves that all the planets are acted on by
+the same kind of force; of an intensity depending on the mass of the
+sun.
+
+The third law of Kepler, although it requires geometry to state and
+establish it for elliptic motion (for which it holds just as well as it
+does for circular motion), is very easy to establish for circular
+motion, by any one who knows about centrifugal force. If _m_ is the mass
+of a planet, _v_ its velocity, _r_ the radius of its orbit, and _T_ the
+time of describing it; 2[pi]_r_ = _vT_, and the centripetal force
+needed to hold it in its orbit is
+
+       mv^2        4[pi]^2_mr_
+     --------  or  -----------
+       _r_             T^2
+
+Now the force of gravitative attraction between the planet and the sun
+is
+
+  _VmS_
+  -----,
+   r^2
+
+where _v_ is a fixed quantity called the gravitation-constant, to be
+determined if possible by experiment once for all. Now, expressing the
+fact that the force of gravitation _is_ the force holding the planet in,
+we write,
+
+  4[pi]^2_mr_    _VmS_
+  ----------- = ---------,
+     T^2          r^2
+
+whence, by the simplest algebra,
+
+   r^3      _VS_
+  ------ = ---------.
+   T^2      4[pi]^2
+
+The mass of the planet has been cancelled out; the mass of the sun
+remains, multiplied by the gravitation-constant, and is seen to be
+proportional to the cube of the distance divided by the square of the
+periodic time: a ratio, which is therefore the same for all planets
+controlled by the sun. Hence, knowing _r_ and _T_ for any single planet,
+the value of _VS_ is known.
+
+No. 4. So by knowing the length of year and distance of any planet from
+the sun, the sun's mass can be calculated, in terms of that of the
+earth.
+
+No. 5. For the satellites, the force acting depends on the mass of
+_their_ central body, a planet. Hence the mass of any planet possessing
+a satellite becomes known.
+
+The same argument holds for any other system controlled by a central
+body--for instance, for the satellites of Jupiter; only instead of _S_
+it will be natural to write _J_, as meaning the mass of Jupiter. Hence,
+knowing _r_ and _T_ for any one satellite of Jupiter, the value of _VJ_
+is known.
+
+Apply the argument also to the case of moon and earth. Knowing the
+distance and time of revolution of our moon, the value of _VE_ is at
+once determined; _E_ being the mass of the earth. Hence, _S_ and _J_,
+and in fact the mass of any central body possessing a visible satellite,
+are now known in terms of _E_, the mass of the earth (or, what is
+practically the same thing, in terms of _V_, the gravitation-constant).
+Observe that so far none of these quantities are known absolutely. Their
+relative values are known, and are tabulated at the end of the Notes
+above, but the finding of their absolute values is another matter, which
+we must defer.
+
+But, it may be asked, if Kepler's third law only gives us the mass of a
+_central_ body, how is the mass of a _satellite_ to be known? Well, it
+is not easy; the mass of no satellite is known with much accuracy. Their
+mutual perturbations give us some data in the case of the satellites of
+Jupiter; but to our own moon this method is of course inapplicable. Our
+moon perturbs at first sight nothing, and accordingly its mass is not
+even yet known with exactness. The mass of comets, again, is quite
+unknown. All that we can be sure of is that they are smaller than a
+certain limit, else they would perturb the planets they pass near.
+Nothing of this sort has ever been detected. They are themselves
+perturbed plentifully, but they perturb nothing; hence we learn that
+their mass is small. The mass of a comet may, indeed, be a few million
+or even billion tons; but that is quite small in astronomy.
+
+But now it may be asked, surely the moon perturbs the earth, swinging it
+round their common centre of gravity, and really describing its own
+orbit about this point instead of about the earth's centre? Yes, that is
+so; and a more precise consideration of Kepler's third law enables us to
+make a fair approximation to the position of this common centre of
+gravity, and thus practically to "weigh the moon," i.e. to compare its
+mass with that of the earth; for their masses will be inversely as their
+respective distances from the common centre of gravity or balancing
+point--on the simple steel-yard principle.
+
+Hitherto we have not troubled ourselves about the precise point about
+which the revolution occurs, but Kepler's third law is not precisely
+accurate unless it is attended to. The bigger the revolving body the
+greater is the discrepancy: and we see in the table preceding Lecture
+III., on page 57, that Jupiter exhibits an error which, though very
+slight, is greater than that of any of the other planets, when the sun
+is considered the fixed centre.
+
+     Let the common centre of gravity of earth and moon be displaced a
+     distance _x_ from the centre of the earth, then the moon's distance
+     from the real centre of revolution is not _r_, but _r-x_; and the
+     equation of centrifugal force to gravitative-attraction is strictly
+
+   4[pi]^2             _VE_
+  --------- (_r-x_) = ------,
+     T^2               r^2
+
+     instead of what is in the text above; and this gives a slightly
+     modified "third law." From this equation, if we have any distinct
+     method of determining _VE_ (and the next section gives such a
+     method), we can calculate _x_ and thus roughly weigh the moon,
+     since
+
+  _r-x_     E
+  ----- = -----,
+   _r_     E+M
+
+     but to get anything like a reasonable result the data must be very
+     precise.
+
+No. 6. The force constraining the moon in her orbit is the same gravity
+as gives terrestrial bodies their weight and regulates the motion of
+projectiles.
+
+Here we come to the Newtonian verification already several times
+mentioned; but because of its importance I will repeat it in other
+words. The hypothesis to be verified is that the force acting on the
+moon is the same kind of force as acts on bodies we can handle and
+weigh, and which gives them their weight. Now the weight of a mass _m_
+is commonly written _mg_, where _g_ is the intensity of terrestrial
+gravity, a thing easily measured; being, indeed, numerically equal to
+twice the distance a stone drops in the first second of free fall. [See
+table p. 205.] Hence, expressing that the weight of a body is due to
+gravity, and remembering that the centre of the earth's attraction is
+distant from us by one earth's radius (R), we can write
+
+         _Vm_E
+  _mg_ = ------,
+          R^2
+
+or
+
+_V_E = gR^2 = 95,522 cubic miles-per-second per second.
+
+But we already know _v_E, in terms of the moon's motion, as
+
+  4[pi]^2r^3
+  -----------
+      T^2
+
+approximately, [more accurately, see preceding note, this quantity is
+_V_(E + M)]; hence we can easily see if the two determinations of this
+quantity agree.[20]
+
+All these deductions are fundamental, and may be considered as the
+foundation of the _Principia_. It was these that flashed upon Newton
+during that moment of excitement when he learned the real size of the
+earth, and discovered his speculations to be true.
+
+The next are elaborations and amplifications of the theory, such as in
+ordinary times are left for subsequent generations of theorists to
+discover and work out.
+
+Newton did not work out these remoter consequences of his theory
+completely by any means: the astronomical and mathematical world has
+been working them out ever since; but he carried the theory a great way,
+and here it is that his marvellous power is most conspicuous.
+
+It is his treatment of No. 7, the perturbations of the moon, that
+perhaps most especially has struck all future mathematicians with
+amazement. No. 7, No. 14, No. 15, these are the most inspired of the
+whole.
+
+No. 7. The moon is attracted not only by the earth, but by the sun also;
+hence its orbit is perturbed, and Newton calculated out the chief of
+these perturbations.
+
+Now running through the perturbations (p. 203) in order:--The first is
+in parenthesis, because it is mere excentricity. It is not a true
+perturbation at all, and more properly belongs to Kepler.
+
+(_a_) The first true perturbation is what Ptolemy called "the evection,"
+the principal part of which is a periodic change in the ellipticity or
+excentricity of the moon's orbit, owing to the pull of the sun. It is a
+complicated matter, and Newton only partially solved it. I shall not
+attempt to give an account of it.
+
+(_b_) The next, "the variation," is a much simpler affair. It is caused
+by the fact that as the moon revolves round the earth it is half the
+time nearer to the sun than the earth is, and so gets pulled more than
+the average, while for the other fortnight it is further from the sun
+than the earth is, and so gets pulled less. For the week during which
+it is changing from a decreasing half to a new moon it is moving in the
+direction of the extra pull, and hence becomes new sooner than would
+have been expected. All next week it is moving against the same extra
+pull, and so arrives at quadrature (half moon) somewhat late. For the
+next fortnight it is in the region of too little pull, the earth gets
+pulled more than it does; the effect of this is to hurry it up for the
+third week, so that the full moon occurs a little early, and to retard
+it for the fourth week, so that the decreasing half moon like the
+increasing half occurs behind time again. Thus each syzygy (as new and
+full are technically called) is too early; each quadrature is too late;
+the maximum hurrying and slackening force being felt at the octants, or
+intermediate 45� points.
+
+(_c_) The "annual equation" is a fluctuation introduced into the other
+perturbations by reason of the varying distance of the disturbing body,
+the sun, at different seasons of the year. Its magnitude plainly depends
+simply on the excentricity of the earth's orbit.
+
+Both these perturbations, (_b_) and (_c_), Newton worked out completely.
+
+(_d_) and (_e_) Next come the retrogression of the nodes and the
+variation of the inclination, which at the time were being observed at
+Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged
+for data that he might complete their theory while he had his mind upon
+it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see
+list at end of notes above], and then Newton would have no difficulty in
+gaining such information as the national Observatory could give.
+
+The "inclination" meant is the angle between the plane of the moon's
+orbit and that of the earth. The plane of the earth's orbit round the
+sun is called the ecliptic; the plane of the moon's orbit round the
+earth is inclined to it at a certain angle, which is slowly changing,
+though in a periodic manner. Imagine a curtain ring bisected by a sheet
+of paper, and tilted to a certain angle; it may be likened to the moon's
+orbit, cutting the plane of the ecliptic. The two points at which the
+plane is cut by the ring are called "nodes"; and these nodes are not
+stationary, but are slowly regressing, _i.e._ travelling in a direction
+opposite to that of the moon itself. Also the angle of tilt is varying
+slowly, oscillating up and down in the course of centuries.
+
+(_f_) The two points in the moon's elliptic orbit where it comes nearest
+to or farthest from the earth, _i.e._ the points at the extremity of the
+long axis of the ellipse, are called separately perigee and apogee, or
+together "the apses." Now the pull of the sun causes the whole orbit to
+slowly revolve in its own plane, and consequently these apses
+"progress," so that the true path is not quite a closed curve, but a
+sort of spiral with elliptic loops.
+
+But here comes in a striking circumstance. Newton states with reference
+to this perturbation that theory only accounts for 1-1/2� per annum,
+whereas observation gives 3�, or just twice as much.
+
+This is published in the _Principia_ as a fact, without comment. It was
+for long regarded as a very curious thing, and many great mathematicians
+afterwards tried to find an error in the working. D'Alembert, Clairaut,
+and others attacked the problem, but were led to just the same result.
+It constituted the great outstanding difficulty in the way of accepting
+the theory of gravitation. It was suggested that perhaps the inverse
+square law was only a first approximation; that perhaps a more complete
+expression, such as
+
+   A       B
+  ---- + -----,
+  r^2     r^4
+
+must be given for it; and so on.
+
+Ultimately, Clairaut took into account a whole series of neglected
+terms, and it came out correct; thus verifying the theory.
+
+But the strangest part of this tale is to come. For only a few years
+ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was
+editing various old papers of Newton's, now in the possession of the
+Duke of Portland, and he found manuscripts bearing on this very point,
+and discovered that Newton had reworked out the calculations himself,
+had found the cause of the error, had taken into account the terms
+hitherto neglected, and so, fifty years before Clairaut, had completely,
+though not publicly, solved this long outstanding problem of the
+progression of the apses.
+
+(_g_) and (_h_) Two other inequalities he calculated out and predicted,
+viz. variation in the motions of the apses and the nodes. Neither of
+these had then been observed, but they were afterwards detected and
+verified.
+
+A good many other minor irregularities are now known--some thirty, I
+believe; and altogether the lunar theory, or problem of the moon's exact
+motion, is one of the most complicated and difficult in astronomy; the
+perturbations being so numerous and large, because of the enormous mass
+of the perturbing body.
+
+The disturbances experienced by the planets are much smaller, because
+they are controlled by the sun and perturbed by each other. The moon is
+controlled only by the earth, and perturbed by the sun. Planetary
+perturbations can be treated as a series of disturbances with some
+satisfaction: not so those of the moon. And yet it is the only way at
+present known of dealing with the lunar theory.
+
+To deal with it satisfactorily would demand the solution of such a
+problem as this:--Given three rigid spherical masses thrown into empty
+space with any initial motions whatever, and abandoned to gravity: to
+determine their subsequent motions. With two masses the problem is
+simple enough, being pretty well summed up in Kepler's laws; but with
+three masses, strange to say, it is so complicated as to be beyond the
+reach of even modern mathematics. It is a famous problem, known as that
+of "the three bodies," but it has not yet been solved. Even when it is
+solved it will be only a close approximation to the case of earth, moon,
+and sun, for these bodies are not spherical, and are not rigid. One may
+imagine how absurdly and hopelessly complicated a complete treatment of
+the motions of the entire solar system would be.
+
+No. 8. Each planet is attracted not only by the sun but by the other
+planets, hence their orbits are slightly affected by each other.
+
+The subject of planetary perturbation was only just begun by Newton.
+Gradually (by Laplace and others) the theory became highly developed;
+and, as everybody knows, in 1846 Neptune was discovered by means of it.
+
+No. 9. He recognized the comets as members of the solar system, obedient
+to the same law of gravity and moving in very elongated ellipses; so
+their return could be predicted.
+
+It was a long time before Newton recognized the comets as real members
+of the solar system, and subject to gravity like the rest. He at first
+thought they moved in straight lines. It was only in the second edition
+of the _Principia_ that the theory of comets was introduced.
+
+Halley observed a fine comet in 1682, and calculated its orbit on
+Newtonian principles. He also calculated when it ought to have been seen
+in past times; and he found the year 1607, when one was seen by Kepler;
+also the year 1531, when one was seen by Appian; again, he reckoned
+1456, 1380, 1305. All these appearances were the same comet, in all
+probability, returning every seventy-five or seventy-six years. The
+period was easily allowed to be not exact, because of perturbing
+planets. He then predicted its return for 1758, or perhaps 1759, a date
+he could not himself hope to see. He lived to a great age, but he died
+sixteen years before this date.
+
+As the time drew nigh, three-quarters of a century afterwards,
+astronomers were greatly interested in this first cometary prediction,
+and kept an eager look-out for "Halley's comet." Clairaut, a most
+eminent mathematician and student of Newton, proceeded to calculate out
+more exactly the perturbing influence of Jupiter, near which it had
+passed. After immense labour (for the difficulty of the calculation was
+extreme, and the mass of mere figures something portentous), he
+predicted its return on the 13th of April, 1759, but he considered that
+he might have made a possible error of a month. It returned on the 13th
+of March, 1759, and established beyond all doubt the rule of the
+Newtonian theory over comets.
+
+[Illustration: FIG. 71.--Well-known model exhibiting the oblate
+spheroidal form as a consequence of spinning about a central axis. The
+brass strip _a_ looks like a transparent globe when whirled, and bulges
+out equatorially.]
+
+No. 10. Applying the idea of centrifugal force to the earth considered
+as a rotating body, he perceived that it could not be a true sphere, and
+calculated its oblateness, obtaining 28 miles greater equatorial than
+polar diameter.
+
+Here we return to one of the more simple deductions. A spinning body of
+any kind tends to swell at its circumference (or equator), and shrink
+along its axis (or poles). If the body is of yielding material, its
+shape must alter under the influence of centrifugal force; and if a
+globe of yielding substance subject to known forces rotates at a
+definite pace, its shape can be calculated. Thus a plastic sphere the
+size of the earth, held together by its own gravity, and rotating once a
+day, can be shown to have its equatorial diameter twenty-eight miles
+greater than its polar diameter: the two diameters being 8,000 and 8,028
+respectively. Now we have no guarantee that the earth is of yielding
+material: for all Newton could tell it might be extremely rigid. As a
+matter of fact it is now very nearly rigid. But he argued thus. The
+water on it is certainly yielding, and although the solid earth might
+decline to bulge at the equator in deference to the diurnal rotation,
+that would not prevent the ocean from flowing from the poles to the
+equator and piling itself up as an equatorial ocean fourteen miles deep,
+leaving dry land everywhere near either pole. Nothing of this sort is
+observed: the distribution of land and water is not thus regulated.
+Hence, whatever the earth may be now, it must once have been plastic
+enough to accommodate itself perfectly to the centrifugal forces, and to
+take the shape appropriate to a perfectly plastic body. In all
+probability it was once molten, and for long afterwards pasty.
+
+Thus, then, the shape of the earth can be calculated from the length of
+its day and the intensity of its gravity. The calculation is not
+difficult: it consists in imagining a couple of holes bored to the
+centre of the earth, one from a pole and one from the equator; filling
+these both with water, and calculating how much higher the water will
+stand in one leg of the gigantic V tube so formed than in the other. The
+answer comes out about fourteen miles.
+
+The shape of the earth can now be observed geodetically, and it accords
+with calculation, but the observations are extremely delicate; in
+Newton's time the _size_ was only barely known, the _shape_ was not
+observed till long after; but on the principles of mechanics, combined
+with a little common-sense reasoning, it could be calculated with
+certainty and accuracy.
+
+No. 11. From the observed shape of Jupiter or any planet the length of
+its day could be estimated.
+
+Jupiter is much more oblate than the earth. Its two diameters are to one
+another as 17 is to 16; the ellipticity of its disk is manifest to
+simple inspection. Hence we perceive that its whirling action must be
+more violent--it must rotate quicker. As a matter of fact its day is ten
+
+[Illustration: FIG. 72.--Jupiter.]
+
+hours long--five hours daylight and five hours night. The times of
+rotation of other bodies in the solar system are recorded in a table
+above.
+
+No. 12. The so-calculated shape of the earth, in combination with
+centrifugal force, causes the weight of bodies to vary with latitude;
+and Newton calculated the amount of this variation. 194 lbs. at pole
+balance 195 lbs. at equator.
+
+But following from the calculated shape of the earth follow several
+interesting consequences. First of all, the intensity of gravity will
+not be the same everywhere; for at the equator a stone is further from
+the average bulk of the earth (say the centre) than it is at the poles,
+and owing to this fact a mass of 590 pounds at the pole; would suffice
+to balance 591 pounds at the equator, if the two could be placed in the
+pans of a gigantic balance whose beam straddled along an earth's
+quadrant. This is a _true_ variation of gravity due to the shape of the
+earth. But besides this there is a still larger _apparent_ variation due
+to centrifugal force, which affects all bodies at the equator but not
+those at the poles. From this cause, even if the earth were a true
+sphere, yet if it were spinning at its actual pace, 288 pounds at the
+pole could balance 289 pounds at the equator; because at the equator the
+true weight of the mass would not be fully appreciated, centrifugal
+force would virtually diminish it by 1/289th of its amount.
+
+In actual fact both causes co-exist, and accordingly the total variation
+of gravity observed is compounded of the real and the apparent effects;
+the result is that 194 pounds at a pole weighs as much as 195 pounds at
+the equator.
+
+No. 13. A homogeneous sphere attracts as if its mass were concentrated
+at its centre. For any other figure, such as an oblate spheroid, this is
+not exactly true. A hollow concentric spherical shell exerts no force on
+small bodies inside it.
+
+A sphere composed of uniform material, or of materials arranged in
+concentric strata, can be shown to attract external bodies as if its
+mass were concentrated at its centre. A hollow sphere, similarly
+composed, does the same, but on internal bodies it exerts no force at
+all.
+
+Hence, at all distances above the surface of the earth, gravity
+decreases in inverse proportion as the square of the distance from the
+centre of the earth increases; but, if you descend a mine, gravity
+decreases in this case also as you leave the surface, though not at the
+same rate as when you went up. For as you penetrate the crust you get
+inside a concentric shell, which is thus powerless to act upon you, and
+the earth you are now outside is a smaller one. At what rate the force
+decreases depends on the distribution of density; if the density were
+uniform all through, the law of variation would be the direct distance,
+otherwise it would be more complicated. Anyhow, the intensity of gravity
+is a maximum at the surface of the earth, and decreases as you travel
+from the surface either up or down.
+
+No. 14. The earth's equatorial protuberance, being acted on by the
+attraction of the sun and moon, must disturb its axis of rotation in a
+calculated manner; and thus is produced the precession of the equinoxes.
+
+Here we come to a truly awful piece of reasoning. A sphere attracts as
+if its mass were concentrated at its centre (No. 12), but a spheroid
+does not. The earth is a spheroid, and hence it pulls and is pulled by
+the moon with a slightly uncentric attraction. In other words, the line
+of pull does not pass through its precise centre. Now when we have a
+spinning body, say a top, overloaded on one side so that gravity acts on
+it unsymmetrically, what happens? The axis of rotation begins to rotate
+cone-wise, at a pace which depends on the rate of spin, and on the shape
+and mass of the top, as well as on the amount and leverage of the
+overloading.
+
+Newton calculated out the rapidity of this conical motion of the axis of
+the earth, produced by the slightly unsymmetrical pull of the moon, and
+found that it would complete a revolution in 26,000 years--precisely
+what was wanted to explain the precession of the equinoxes. In fact he
+had discovered the physical cause of that precession.
+
+Observe that there were three stages in this discovery of precession:--
+
+First, the observation by Hipparchus, that the nodes, or intersections
+of the earth's orbit (the sun's apparent orbit) with the plane of the
+equator, were not stationary, but slowly moved.
+
+Second, the description of this motion by Copernicus, by the statement
+that it was due to a conical motion of the earth's axis of rotation
+about its centre as a fixed point.
+
+Third, the explanation of this motion by Newton as due to the pull of
+the moon on the equatorial protuberance of the earth.
+
+The explanation _could_ not have been previously suspected, for the
+shape of the earth, on which the whole theory depends, was entirely
+unknown till Newton calculated it.
+
+Another and smaller motion of a somewhat similar kind has been worked
+out since: it is due to the unsymmetrical attraction of the other
+planets for this same equatorial protuberance. It shows itself as a
+periodic change in the obliquity of the ecliptic, or so-called recession
+of the apses, rather than as a motion of the nodes.[21]
+
+No. 15. The waters of the ocean are attracted towards the sun and moon
+on one side, and whirled a little farther away than the solid earth on
+the other side: hence Newton explained all the main phenomena of the
+tides.
+
+And now comes another tremendous generalization. The tides had long been
+an utter mystery. Kepler likens the earth to an animal, and the tides to
+his breathings and inbreathings, and says they follow the moon.
+
+Galileo chaffs him for this, and says that it is mere superstition to
+connect the moon with the tides.
+
+Descartes said the moon pressed down upon the waters by the centrifugal
+force of its vortex, and so produced a low tide under it.
+
+Everything was fog and darkness on the subject. The legend goes that an
+astronomer threw himself into the sea in despair of ever being able to
+explain the flux and reflux of its waters.
+
+Newton now with consummate skill applied his theory to the effect of
+the moon upon the ocean, and all the main details of tidal action
+gradually revealed themselves to him.
+
+He treated the water, rotating with the earth once a day, somewhat as if
+it were a satellite acted on by perturbing forces. The moon as it
+revolves round the earth is perturbed by the sun. The ocean as it
+revolves round the earth (being held on by gravitation just as the moon
+is) is perturbed by both sun and moon.
+
+The perturbing effect of a body varies directly as its mass, and
+inversely as the cube of its distance. (The simple law of inverse square
+does not apply, because a perturbation is a differential effect: the
+satellite or ocean when nearer to the perturbing body than the rest of
+the earth, is attracted more, and when further off it is attracted less
+than is the main body of the earth; and it is these differences alone
+which constitute the perturbation.) The moon is the more powerful of the
+two perturbing bodies, hence the main tides are due to the moon; and its
+chief action is to cause a pair of low waves or oceanic humps, of
+gigantic area, to travel round the earth once in a lunar day, _i.e._ in
+about 24 hours and 50 minutes. The sun makes a similar but still lower
+pair of low elevations to travel round once in a solar day of 24 hours.
+And the combination of the two pairs of humps, thus periodically
+overtaking each other, accounts for the well-known spring and neap
+tides,--spring tides when their maxima agree, neap tides when the
+maximum of one coincides with the minimum of the other: each of which
+events happens regularly once a fortnight.
+
+These are the main effects, but besides these there are the effects of
+varying distances and obliquity to be taken into account; and so we have
+a whole series of minor disturbances, very like those discussed in No.
+7, under the lunar theory, but more complex still, because there are two
+perturbing bodies instead of only one.
+
+The subject of the tides is, therefore, very recondite; and though one
+may give some elementary account of its main features, it will be best
+to defer this to a separate lecture (Lecture XVII).
+
+I had better, however, here say that Newton did not limit himself to the
+consideration of the primary oceanic humps: he pursued the subject into
+geographical detail. He pointed out that, although the rise and fall of
+the tide at mid-ocean islands would be but small, yet on stretches of
+coast the wave would fling itself, and by its momentum would propel the
+waters, to a much greater height--for instance, 20 or 30 feet;
+especially in some funnel-shaped openings like the Bristol Channel and
+the Bay of Fundy, where the concentrated impetus of the water is
+enormous.
+
+He also showed how the tidal waves reached different stations in
+successive regular order each day; and how some places might be fed with
+tide by two distinct channels; and that if the time of these channels
+happened to differ by six hours, a high tide might be arriving by one
+channel and a low tide by the other, so that the place would only feel
+the difference, and so have a very small observed rise and fall;
+instancing a port in China (in the Gulf of Tonquin) where that
+approximately occurs.
+
+In fact, although his theory was not, as we now know, complete or final,
+yet it satisfactorily explained a mass of intricate detail as well as
+the main features of the tides.
+
+No. 16. The sun's mass being known, he calculated the height of the
+solar tide.
+
+No. 17. From the observed heights of spring and neap tides he determined
+the lunar tide, and thence made an estimate of the mass of the moon.
+
+Knowing the sun's mass and distance, it was not difficult for Newton to
+calculate the height of the protuberance caused by it in a pasty ocean
+covering the whole earth. I say pasty, because, if there was any
+tendency for impulses to accumulate, as timely pushes given to a
+pendulum accumulate, the amount of disturbance might become excessive,
+and its calculation would involve a multitude of data. The Newtonian
+tide ignored this, thus practically treating the motion as either
+dead-beat, or else the impulses as very inadequately timed. With this
+reservation the mid-ocean tide due to the action of the sun alone comes
+out about one foot, or let us say one foot for simplicity. Now the
+actual tide observed in mid-Atlantic is at the springs about four feet,
+at the neaps about two. The spring tide is lunar plus solar; the neap
+tide is lunar minus solar. Hence it appears that the tide caused by the
+moon alone must be about three feet, when unaffected by momentum. From
+this datum Newton made the first attempt to approximately estimate the
+mass of the moon. I said that the masses of satellites must be
+estimated, if at all, by the perturbation they are able to cause. The
+lunar tide is a perturbation in the diurnal motion of the sea, and its
+amount is therefore a legitimate mode of calculating the moon's mass.
+The available data were not at all good, however; nor are they even now
+very perfect; and so the estimate was a good way out. It is now
+considered that the mass of the moon is about one-eightieth that of the
+earth.
+
+*      *      *      *      *
+
+Such are some of the gems extracted from their setting in the
+_Principia_, and presented as clearly as I am able before you.
+
+Do you realize the tremendous stride in knowledge--not a stride, as
+Whewell says, nor yet a leap, but a flight--which has occurred between
+the dim gropings of Kepler, the elementary truths of Galileo, the
+fascinating but wild speculations of Descartes, and this magnificent and
+comprehensive system of ordered knowledge. To some his genius seemed
+almost divine. "Does Mr. Newton eat, drink, sleep, like other men?" said
+the Marquis de l'H�pital, a French mathematician of no mean eminence; "I
+picture him to myself as a celestial genius, entirely removed from the
+restrictions of ordinary matter." To many it seemed as if there was
+nothing more to be discovered, as if the universe were now explored, and
+only a few fragments of truth remained for the gleaner. This is the
+attitude of mind expressed in Pope's famous epigram:--
+
+  "Nature and Nature's laws lay hid in Night,
+  God said, Let Newton be, and all was light."
+
+This feeling of hopelessness and impotence was very natural after the
+advent of so overpowering a genius, and it prevailed in England for
+fully a century. It was very natural, but it was very mischievous; for,
+as a consequence, nothing of great moment was done by England in
+science, and no Englishman of the first magnitude appeared, till some
+who are either living now or who have lived within the present century.
+
+It appeared to his contemporaries as if he had almost exhausted the
+possibility of discovery; but did it so appear to Newton? Did it seem to
+him as if he had seen far and deep into the truths of this great and
+infinite universe? It did not. When quite an old man, full of honour and
+renown, venerated, almost worshipped, by his contemporaries, these were
+his words:--
+
+"I know not what the world will think of my labours, but to myself it
+seems that I have been but as a child playing on the sea-shore; now
+finding some pebble rather more polished, and now some shell rather more
+agreeably variegated than another, while the immense ocean of truth
+extended itself unexplored before me."
+
+And so it must ever seem to the wisest and greatest of men when brought
+into contact with the great things of God--that which they know is as
+nothing, and less than nothing, to the infinitude of which they are
+ignorant.
+
+Newton's words sound like a simple and pleasing echo of the words of
+that great unknown poet, the writer of the book of Job:--
+
+  "Lo, these are parts of His ways,
+  But how little a portion is heard of Him;
+  The thunder of His power, who can understand?"
+
+END OF PART I.
+
+
+
+
+PART II
+
+_A COUPLE OF CENTURIES' PROGRESS._
+
+
+
+
+NOTES TO LECTURE X
+
+_Science during the century after Newton_
+
+The _Principia_ published, 1687
+
+  Roemer                1644-1710
+  James Bradley         1692-1762
+  Clairaut              1713-1765
+  Euler                 1707-1783
+  D'Alembert            1717-1783
+  Lagrange              1736-1813
+  Laplace               1749-1827
+  William Herschel      1738-1822
+
+
+_Olaus Roemer_ was born in Jutland, and studied at Copenhagen. Assisted
+Picard in 1671 to determine the exact position of Tycho's observatory on
+Huen. Accompanied Picard to Paris, and in 1675 read before the Academy
+his paper "On Successive Propagation of Light as revealed by a certain
+inequality in the motion of Jupiter's First Satellite." In 1681 he
+returned to Copenhagen as Professor of Mathematics and Astronomy, and
+died in 1710. He invented the transit instrument, mural circle,
+equatorial mounting for telescopes, and most of the other principal
+instruments now in use in observatories. He made as many observations as
+Tycho Brah�, but the records of all but the work of three days were
+destroyed by a great fire in 1728.
+
+_Bradley_, Professor of Astronomy at Oxford, discovered the aberration
+of light in 1729, while examining stars for parallax, and the nutation
+of the earth's axis in 1748. Was appointed Astronomer-Royal in 1742.
+
+
+
+
+LECTURE X
+
+ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT
+
+
+At Newton's death England stood pre-eminent among the nations of Europe
+in the sphere of science. But the pre-eminence did not last long. Two
+great discoveries were made very soon after his decease, both by
+Professor Bradley, of Oxford, and then there came a gap. A moderately
+great man often leaves behind him a school of disciples able to work
+according to their master's methods, and with a healthy spirit of
+rivalry which stimulates and encourages them. Newton left, indeed, a
+school of disciples, but his methods of work were largely unknown to
+them, and such as were known were too ponderous to be used by ordinary
+men. Only one fresh result, and that a small one, has ever been attained
+by other men working according to the methods of the _Principia_. The
+methods were studied and commented on in England to the exclusion of all
+others for nigh a century, and as a consequence no really important work
+was done.
+
+On the Continent, however, no such system of slavish imitation
+prevailed. Those methods of Newton's which had been simultaneously
+discovered by Leibnitz were more thoroughly grasped, modified, extended,
+and improved. There arose a great school of French and German
+mathematicians, and the laurels of scientific discovery passed to France
+and Germany--more especially, perhaps, at this time to France. England
+has never wholly recovered them. During the present century this country
+has been favoured with some giants who, as they become distant enough
+for their true magnitude to be perceived, may possibly stand out as
+great as any who have ever lived; but for the mass and bulk of
+scientific work at the present day we have to look to Germany, with its
+enlightened Government and extensive intellectual development. England,
+however, is waking up, and what its Government does not do, private
+enterprise is beginning to accomplish. The establishment of centres of
+scientific and literary activity in the great towns of England, though
+at present they are partially encumbered with the supply of education of
+an exceedingly rudimentary type, is a movement that in the course of
+another century or so will be seen to be one of the most important and
+fruitful steps ever taken by this country. On the Continent such centres
+have long existed; almost every large town is the seat of a University,
+and they are now liberally endowed. The University of Bologna (where,
+you may remember, Copernicus learnt mathematics) has recently celebrated
+its 800th anniversary.
+
+The scientific history of the century after Newton, summarized in the
+above table of dates, embraces the labours of the great mathematicians
+Clairaut, Euler, D'Alembert, and especially of Lagrange and Laplace.
+
+But the main work of all these men was hardly pioneering work. It was
+rather the surveying, and mapping out, and bringing into cultivation, of
+lands already discovered. Probably Herschel may be justly regarded as
+the next true pioneer. We shall not, however, properly appreciate the
+stages through which astronomy has passed, nor shall we be prepared
+adequately to welcome the discoveries of modern times unless we pay some
+attention to the intervening age. Moreover, during this era several
+facts of great moment gradually came into recognition; and the
+importance of the discovery we have now to speak of can hardly be
+over-estimated.
+
+Our whole direct knowledge of the planetary and stellar universe, from
+the early observations of the ancients down to the magnificent
+discoveries of a Herschel, depends entirely upon our happening to
+possess a sense of sight. To no other of our senses do any other worlds
+than our own in the slightest degree appeal. We touch them or hear them
+never. Consequently, if the human race had happened to be blind, no
+other world but the one it groped its way upon could ever have been
+known or imagined by it. The outside universe would have existed, but
+man would have been entirely and hopelessly ignorant of it. The bare
+idea of an outside universe beyond the world would have been
+inconceivable, and might have been scouted as absurd. We do possess the
+sense of sight; but is it to be supposed that we possess every sense
+that can be possessed by finite beings? There is not the least ground
+for such an assumption. It is easy to imagine a deaf race or a blind
+race: it is not so easy to imagine a race more highly endowed with
+senses than our own; and yet the sense of smell in animals may give us
+some aid in thinking of powers of perception which transcend our own in
+particular directions. If there were a race with higher or other senses
+than our own, or if the human race should ever in the process of
+development acquire such extra sense-organs, a whole universe of
+existent fact might become for the first time perceived by us, and we
+should look back upon our past state as upon a blind chrysalid form of
+existence in which we had been unconscious of all this new wealth of
+perception.
+
+It cannot be too clearly and strongly insisted on and brought home to
+every mind, that the mode in which the universe strikes us, our view of
+the universe, our whole idea of matter, and force, and other worlds, and
+even of consciousness, depends upon the particular set of sense-organs
+with which we, as men, happen to be endowed. The senses of force, of
+motion, of sound, of light, of touch, of heat, of taste, and of
+smell--these we have, and these are the things we primarily know. All
+else is inference founded upon these sensations. So the world appears to
+us. But given other sense-organs, and it might appear quite otherwise.
+What it is actually and truly like, therefore, is quite and for ever
+beyond us--so long as we are finite beings.
+
+Without eyes, astronomy would be non-existent. Light it is which conveys
+all the information we possess, or, as it would seem, ever can possess,
+concerning the outer and greater universe in which this small world
+forms a speck. Light is the channel, the messenger of information; our
+eyes, aided by telescopes, spectroscopes, and many other "scopes" that
+may yet be invented, are the means by which we read the information that
+light brings.
+
+Light travels from the stars to our eyes: does it come instantaneously?
+or does it loiter by the way? for if it lingers it is not bringing us
+information properly up to date--it is only telling us what the state of
+affairs was when it started on its long journey.
+
+Now, it is evidently a matter of interest to us whether we see the sun
+as he is now, or only as he was some three hundred years ago. If the
+information came by express train it would be three hundred years behind
+date, and the sun might have gone out in the reign of Queen Anne without
+our being as yet any the wiser. The question, therefore, "At what rate
+does our messenger travel?" is evidently one of great interest for
+astronomers, and many have been the attempts made to solve it. Very
+likely the ancient Greeks pondered over this question, but the earliest
+writer known to me who seriously discussed the question is Galileo. He
+suggests a rough experimental means of attacking it. First of all, it
+plainly comes quicker than sound. This can be perceived by merely
+watching distant hammering, or by noticing that the flash of a pistol is
+seen before its report is heard, or by listening to the noise of a
+flash of lightning. Sound takes five seconds to travel a mile--it has
+about the same speed as a rifle bullet; but light is much quicker than
+that.
+
+The rude experiment suggested by Galileo was to send two men with
+lanterns and screens to two distant watch-towers or neighbouring
+mountain tops, and to arrange that each was to watch alternate displays
+and obscurations of the light made by the other, and to imitate them as
+promptly as possible. Either man, therefore, on obscuring or showing his
+own light would see the distant glimmer do the same, and would be able
+to judge if there was any appreciable interval between his own action
+and the response of the distant light. The experiment was actually tried
+by the Florentine Academicians,[22] with the result that, as practice
+improved, the interval became shorter and shorter, so that there was no
+reason to suppose that there was any real interval at all. Light, in
+fact, seemed to travel instantaneously.
+
+Well might they have arrived at this result. Even if they had made far
+more perfect arrangements--for instance, by arranging a looking-glass at
+one of the stations in which a distant observer might see the reflection
+of his own lantern, and watch the obscurations and flashings made by
+himself, without having to depend on the response of human
+mechanism--even then no interval whatever could have been detected.
+
+If, by some impossibly perfect optical arrangement, a lighthouse here
+were made visible to us after reflection in a mirror erected at New
+York, so that the light would have to travel across the Atlantic and
+back before it could be seen, even then the appearance of the light on
+removing a shutter, or the eclipse on interposing it, would seem to
+happen quite instantaneously. There would certainly be an interval: the
+interval would be the fiftieth part of a second (the time a stone takes
+to drop 1/13th of an inch), but that is too short to be securely
+detected without mechanism. With mechanism the thing might be managed,
+for a series of shutters might be arranged like the teeth of a large
+wheel; so that, when the wheel rotates, eclipses follow one another very
+rapidly; if then an eye looked through the same opening as that by which
+the light goes on its way to the distant mirror, a tooth might have
+moved sufficiently to cover up this space by the time the light
+returned; in which case the whole would appear dark, for the light would
+be stopped by a tooth, either at starting or at returning, continually.
+At higher speeds of rotation some light would reappear, and at lower
+speeds it would also reappear; by noticing, therefore, the precise speed
+at which there was constant eclipse the velocity of light could be
+determined.
+
+[Illustration: FIG. 73.--Diagram of eye looking at a light reflected in
+a distant mirror through the teeth of a revolving wheel.]
+
+This experiment has now been made in a highly refined form by Fizeau,
+and repeated by M. Cornu with prodigious care and accuracy. But with
+these recent matters we have no concern at present. It may be
+instructive to say, however, that if the light had to travel two miles
+altogether, the wheel would have to possess 450 teeth and to spin 100
+times a second (at the risk of flying to pieces) in order that the ray
+starting through any one of the gaps might be stopped on returning by
+the adjacent tooth.
+
+Well might the velocity of light be called instantaneous by the early
+observers. An ordinary experiment seemed (and was) hopeless, and light
+was supposed to travel at an infinite speed. But a phenomenon was
+noticed in the heavens by a quick-witted and ingenious Danish
+astronomer, which was not susceptible of any ordinary explanation, and
+which he perceived could at once be explained if light had a certain
+rate of travel--great, indeed, but something short of infinite. This
+phenomenon was connected with the satellites of Jupiter, and the
+astronomer's name was Roemer. I will speak first of the observation and
+then of the man.
+
+[Illustration: FIG. 74.--Fizeau's wheel, shewing the appearance of
+distant image seen through its teeth. 1st, when stationary, next when
+revolving at a moderate speed, last when revolving at the high speed
+just sufficient to cause eclipse.]
+
+Jupiter's satellites are visible, precisely as our own moon is, by
+reason of the shimmer of sunlight which they reflect. But as they
+revolve round their great planet they plunge into his shadow at one part
+of their course, and so become eclipsed from sunshine and invisible to
+us. The moment of disappearance can be sharply observed.
+
+Take the first satellite as an example. The interval between successive
+eclipses ought to be its period of revolution round Jupiter. Observe
+this period. It was not uniform. On the average it was 42 hours 47
+minutes, but it seemed to depend on the time of year. When Roemer
+observed in spring it was less, and in autumn it was more than usual.
+This was evidently a puzzling fact: what on earth can our year have to
+do with the motion of a moon of Jupiter's? It was probably, therefore,
+only an apparent change, caused either by our greater or less distance
+from Jupiter, or else by our greater or less speed of travelling to or
+from him. Considering it thus, he was led to see that, when the time of
+revolution seemed longest, we were receding fastest from Jupiter, and
+when shortest, approaching fastest.
+
+_If_, then, light took time on its journey, _if_ it travelled
+progressively, the whole anomaly would be explained.
+
+In a second the earth goes nineteen miles; therefore in 42-3/4 hours
+(the time of revolution of Jupiter's first satellite) it goes 2�9
+million (say three million) miles. The eclipse happens punctually, but
+we do not see it till the light conveying the information has travelled
+the extra three million miles and caught up the earth. Evidently,
+therefore, by observing how much the apparent time of revolution is
+lengthened in one part of the earth's orbit and shortened in another,
+getting all the data accurately, and assuming the truth of our
+hypothetical explanation, we can calculate the velocity of light. This
+is what Roemer did.
+
+Now the maximum amount of retardation is just about fifteen seconds.
+Hence light takes this time to travel three million miles; therefore its
+velocity is three million divided by fifteen, say 200,000, or, as we now
+know more exactly, 186,000 miles every second. Note that the delay does
+not depend on our _distance_, but on our _speed_. One can tell this by
+common-sense as soon as we grasp the general idea of the explanation. A
+velocity cannot possibly depend on a distance only.
+
+[Illustration: FIG. 75.--Eclipses of one of Jupiter's satellites. A
+diagram intended to illustrate the dependence of its apparent time of
+revolution (from eclipse to eclipse) on the motion of the earth; but not
+illustrating the matter at all well. TT' T'' are successive positions of
+the earth, while JJ' J'' are corresponding positions of Jupiter.]
+
+Roemer's explanation of the anomaly was not accepted by astronomers. It
+excited some attention, and was discussed, but it was found not
+obviously applicable to any of the satellites except the first, and not
+very simply and satisfactorily even to that. I have, of course, given
+you the theory in its most elementary and simple form. In actual fact a
+host of disturbing and complicated considerations come in--not so
+violently disturbing for the first satellite as for the others, because
+it moves so quickly, but still complicated enough.
+
+The fact is, the real motion of Jupiter's satellites is a most difficult
+problem. The motion even of our own moon (the lunar theory) is difficult
+enough: perturbed as its motion is by the sun. You know that Newton said
+it cost him more labour than all the rest of the _Principia_. But the
+motion of Jupiter's satellites is far worse. No one, in fact, has yet
+worked their theory completely out. They are perturbed by the sun, of
+course, but they also perturb each other, and Jupiter is far from
+spherical. The shape of Jupiter, and their mutual attractions, combine
+to make their motions most peculiar and distracting.
+
+Hence an error in the time of revolution of a satellite was not
+_certainly_ due to the cause Roemer suggested, unless one could be sure
+that the inequality was not a real one, unless it could be shown that
+the theory of gravitation was insufficient to account for it. This had
+not then been done; so the half-made discovery was shelved, and properly
+shelved, as a brilliant but unverified speculation. It remained on the
+shelf for half a century, and was no doubt almost forgotten.
+
+[Illustration: FIG. 76.--A Transit-instrument for the British
+astronomical expedition, 1874. Shewing in its essential features the
+simplest form of such an instrument.]
+
+Now a word or two about the man. He was a Dane, educated at Copenhagen,
+and learned in the mathematics. We first hear of him as appointed to
+assist Picard, the eminent French geodetic surveyor (whose admirable
+work in determining the length of a degree you remember in connection
+with Newton), who had come over to Denmark with the object of fixing the
+exact site of the old and extinct Tychonic observatory in the island of
+Huen. For of course the knowledge of the exact latitude and longitude of
+every place whence numerous observations have been taken must be an
+essential to the full interpretation of those observations. The
+measurements being finished, young Roemer accompanied Picard to Paris,
+and here it was, a few years after, that he read his famous paper
+concerning "An Inequality in the Motion of Jupiter's First Satellite,"
+and its explanation by means of an hypothesis of "the successive
+propagation of light."
+
+The later years of his life he spent in Copenhagen as a professor in the
+University and an enthusiastic observer of the heavens,--not a
+descriptive observer like Herschel, but a measuring observer like Sir
+George Airy or Tycho Brah�. He was, in fact, a worthy follower of Tycho,
+and the main work of his life is the development and devising of new and
+more accurate astronomical instruments. Many of the large and accurate
+instruments with which a modern observatory is furnished are the
+invention of this Dane. One of the finest observatories in the world is
+the Russian one at Pulkowa, and a list of the instruments there reads
+like an extended catalogue of Roemer's inventions.
+
+He not only _invented_ the instruments, he had them made, being allowed
+money for the purpose; and he used them vigorously, so that at his death
+he left great piles of manuscript stored in the national observatory.
+
+Unfortunately this observatory was in the heart of the city, and was
+thus exposed to a danger from which such places ought to be as far as
+possible exempt.
+
+Some eighteen years after Roemer's death a great conflagration broke out
+in Copenhagen, and ruined large portions of the city. The successor to
+Roemer, Horrebow by name, fled from his house, with such valuables as he
+possessed, to the observatory, and there went on with his work. But
+before long the wind shifted, and to his horror he saw the flames
+coming his way. He packed up his own and his predecessor's manuscript
+observations in two cases, and prepared to escape with them, but the
+neighbours had resorted to the observatory as a place of safety, and so
+choked up the staircase with their property that he was barely able to
+escape himself, let alone the luggage, and everything was lost.
+
+[Illustration: FIG. 77.--Diagram of equatorially mounted telescope; CE
+is the polar axis parallel to the axis of the earth; AB the declination
+axis. The diurnal motion is compensated by motion about the polar axis
+only, the other being clamped.]
+
+Of all the observations, only three days' work remains, and these were
+carefully discussed by Dr. Galle, of Berlin, in 1845, and their
+nutriment extracted. These ancient observations are of great use for
+purposes of comparison with the present state of the heavens, and throw
+light upon possible changes that are going on. Of course nowadays such a
+series of observations would be printed and distributed in many
+libraries, and so made practically indestructible.
+
+Sad as the disaster was to the posthumous fame of the great observer, a
+considerable compensation was preparing. The very year that the fire
+occurred in Denmark a quiet philosopher in England was speculating and
+brooding on a remarkable observation that he had made concerning the
+apparent motion of certain stars, and he was led thereby to a discovery
+of the first magnitude concerning the speed of light--a discovery which
+resuscitated the old theory of Roemer about Jupiter's satellites, and
+made both it and him immortal.
+
+James Bradley lived a quiet, uneventful, studious life, mainly at Oxford
+but afterwards at the National Observatory at Greenwich, of which he was
+third Astronomer-Royal, Flamsteed and Halley having preceded him in that
+office. He had taken orders, and lectured at Oxford as Savilian
+Professor. It is said that he pondered his great discovery while pacing
+the Long Walk at Magdalen College--and a beautiful place it is to
+meditate in.
+
+Bradley was engaged in making observations to determine if possible the
+parallax of some of the fixed stars. Parallax means the apparent
+relative shift of bodies due to a change in the observer's position. It
+is parallax which we observe when travelling by rail and looking out of
+window at the distant landscape. Things at different distances are left
+behind at different apparent rates, and accordingly they seem to move
+relatively to each other. The most distant objects are least affected;
+and anything enormously distant, like the moon, is not subject to this
+effect, but would retain its position however far we travelled, unless
+we had some extraordinarily precise means of observation.
+
+So with the fixed stars: they were being observed from a moving
+carriage--viz. the earth--and one moving at the rate of nineteen miles a
+second. Unless they were infinitely distant, or unless they were all at
+the same distance, they must show relative apparent motions among
+themselves. Seen from one point of the earth's orbit, and then in six
+months from an opposite point, nearly 184 million miles away, surely
+they must show some difference of aspect.
+
+Remember that the old Copernican difficulty had never been removed. If
+the earth revolved round the sun, how came it that the fixed stars
+showed no parallax? The fact still remained a surprise, and the question
+a challenge. Picard, like other astronomers, supposed that it was only
+because the methods of observation had not been delicate enough; but now
+that, since the invention of the telescope and the founding of National
+Observatories, accuracy hitherto undreamt of was possible, why not
+attack the problem anew? This, then, he did, watching the stars with
+great care to see if in six months they showed any change in absolute
+position with reference to the pole of the heavens; any known secular
+motion of the pole, such as precession, being allowed for. Already he
+thought he detected a slight parallax for several stars near the pole,
+and the subject was exciting much interest.
+
+Bradley determined to attempt the same investigation. He was not
+destined to succeed in it. Not till the present century was success in
+that most difficult observation achieved; and even now it cannot be done
+by the absolute methods then attempted; but, as so often happens,
+Bradley, in attempting one thing, hit upon another, and, as it happened,
+one of still greater brilliance and importance. Let us trace the stages
+of his discovery.
+
+Atmospheric refraction made horizon observations useless for the
+delicacy of his purpose, so he chose stars near the zenith, particularly
+one--[gamma] Draconis. This he observed very carefully at different
+seasons of the year by means of an instrument specially adapted for
+zenith observations, viz. a zenith sector. The observations were made in
+conjunction with a friend of his, an amateur astronomer named Molyneux,
+and they were made at Kew. Molyneux was shortly made First Lord of the
+Admiralty, or something important of that sort, and gave up frivolous
+pursuits. So Bradley observed alone. They observed the star accurately
+early in the month of December, and then intended to wait six months.
+But from curiosity Bradley observed it again only about a week later. To
+his surprise, he found that it had already changed its position. He
+recorded his observation on the back of an old envelope: it was his wont
+thus to use up odd scraps of paper--he was not, I regret to say, a tidy
+or methodical person--and this odd piece of paper turned up long
+afterwards among his manuscripts. It has been photographed and preserved
+as an historical relic.
+
+Again and again he repeated the observation of the star, and continually
+found it moving still a little further and further south, an excessively
+small motion, but still an appreciable one--not to be set down to errors
+of observation. So it went on till March. It then waited, and after a
+bit longer began to return, until June. By September it was displaced as
+much to the north as it had been to the south, and by December it had
+got back to its original position. It had described, in fact, a small
+oscillation in the course of the year. The motion affected neighbouring
+stars in a similar way, and was called an "aberration," or wandering
+from their true place.
+
+For a long time Bradley pondered over this observation, and over others
+like them which he also made. He found one group of stars describing
+small circles, while others at a distance from them were oscillating in
+straight lines, and all the others were describing ellipses. Unless this
+state of things were cleared up, accurate astronomy was impossible. The
+fixed stars!--they were not fixed a bit. To refined and accurate
+observation, such as was now possible, they were all careering about in
+little orbits having a reference to the earth's year, besides any proper
+motion which they might really have of their own, though no such motion
+was at present known. Not till Herschel was that discovered; not till
+this extraordinary aberration was allowed for could it be discovered.
+The effect observed by Bradley and Molyneux must manifestly be only an
+apparent motion: it was absurd to suppose a real stellar motion
+regulating itself according to the position of the earth. Parallax could
+not do it, for that would displace stars relatively among each other--it
+would not move similarly a set of neighbouring stars.
+
+At length, four years after the observation, the explanation struck him,
+while in a boat upon the Thames. He noticed the apparent direction of
+the wind changed whenever the boat started. The wind veered when the
+boat's motion changed. Of course the cause of this was obvious
+enough--the speed of the wind and the speed of the boat were compounded,
+and gave an apparent direction of the wind other than the true
+direction. But this immediately suggested a cause for what he had
+observed in the heavens. He had been observing an apparent direction of
+the stars other than the true direction, because he was observing from a
+moving vehicle. The real direction was doubtless fixed: the apparent
+direction veered about with the motion of the earth. It must be that
+light did not travel instantaneously, but gradually, as Roemer had
+surmised fifty years ago; and that the motion of the light was
+compounded with the motion of the earth.
+
+Think of a stream of light or anything else falling on a moving
+carriage. The carriage will run athwart the stream, the occupants of the
+carriage will mistake its true direction. A rifle fired through the
+windows of a railway carriage by a man at rest outside would make its
+perforations not in the true line of fire unless the train is
+stationary. If the train is moving, the line joining the holes will
+point to a place in advance of where the rifle is really located.
+
+So it is with the two glasses of a telescope, the object-glass and
+eye-piece, which are pierced by the light; an astronomer, applying his
+eye to the tube and looking for the origin of the disturbance, sees it
+apparently, but not in its real position--its apparent direction is
+displaced in the direction of the telescope's motion; by an amount
+depending on the ratio of the velocity of the earth to the velocity of
+light, and on the angle between those two directions.
+
+[Illustration: FIG. 78.--Aberration diagram. The light-ray L penetrates
+the object-glass of the moving telescope at O, but does not reach the
+eye-piece until the telescope has travelled to the second position.
+Consequently a moving telescope does not point out the true direction of
+the light, but aims at a point a little in advance.]
+
+But how minute is the displacement! The greatest effect is obtained when
+the two motions are at right angles to each other, _i.e._ when the star
+seen is at right angles to the direction of the earth's motion, but even
+then it is only 20", or 1/180th part of a degree; one-ninetieth of the
+moon's apparent diameter. It could not be detected without a cross-wire
+in the telescope, and would only appear as a slight displacement from
+the centre of the field, supposing the telescope accurately pointed to
+the true direction.
+
+But if this explanation be true, it at once gives a method of
+determining the velocity of light. The maximum angle of deviation,
+represented as a ratio of arc � radius, amounts to
+
+       1                     1
+  ------------  -  �0001 = ------
+  180 � 57-1/3             10,000
+
+(a gradient of 1 foot in two miles). In other words, the velocity of
+light must be 10,000 times as great as the velocity of the earth in its
+orbit. This amounts to a speed of 190,000 miles a second--not so very
+different from what Roemer had reckoned it in order to explain the
+anomalies of Jupiter's first satellite.
+
+Stars in the direction in which the earth was moving would not be thus
+affected; there would be nothing in mere approach or recession to alter
+direction or to make itself in any way visible. Stars at right angles to
+the earth's line of motion would be most affected, and these would be
+all displaced by the full amount of 20 seconds of arc. Stars in
+intermediate directions would be displaced by intermediate amounts.
+
+But the line of the earth's motion is approximately a circle round the
+sun, hence the direction of its advance is constantly though slowly
+changing, and in one year it goes through all the points of the compass.
+The stars, being displaced always in the line of advance, must similarly
+appear to describe little closed curves, always a quadrant in advance of
+the earth, completing their orbits once a year. Those near the pole of
+the ecliptic will describe circles, being always at right angles to the
+motion. Those in the plane of the ecliptic (near the zodiac) will be
+sometimes at right angles to the motion, but at other times will be
+approached or receded from; hence these will oscillate like pendulums
+once a year; and intermediate stars will have intermediate motions--that
+is to say, will describe ellipses of varying excentricity, but all
+completed in a year, and all with the major axis 20". This agreed very
+closely with what was observed.
+
+The main details were thus clearly and simply explained by the
+hypothesis of a finite velocity for light, "the successive propagation
+of light in time." This time there was no room for hesitation, and
+astronomers hailed the discovery with enthusiasm.
+
+Not yet, however, did Bradley rest. The finite velocity of light
+explained the major part of the irregularities he had observed, but not
+the whole. The more carefully he measured the amount of the deviation,
+the less completely accurate became its explanation.
+
+There clearly was a small outstanding error or discrepancy; the stars
+were still subject to an unexplained displacement--not, indeed, a
+displacement that repeated itself every year, but one that went through
+a cycle of changes in a longer period.
+
+The displacement was only about half that of aberration, and having a
+longer period was rather more difficult to detect securely. But the
+major difficulty was the fact that the two sorts of disturbances were
+co-existent, and the skill of disentangling them, and exhibiting the
+true and complete cause of each inequality, was very brilliant.
+
+For nineteen years did Bradley observe this minor displacement, and in
+that time he saw it go through a complete cycle. Its cause was now clear
+to him; the nineteen-year period suggested the explanation. It is the
+period in which the moon goes through all her changes--a period known to
+the ancients as the lunar cycle, or Metonic cycle, and used by them to
+predict eclipses. It is still used for the first rough approximation to
+the prediction of eclipses, and to calculate Easter. The "Golden Number"
+of the Prayer-book is the number of the year in this cycle.
+
+The cause of the second inequality, or apparent periodic motion of the
+stars, Bradley made out to be a nodding motion of the earth's axis.
+
+The axis of the earth describes its precessional orbit or conical
+motion every 26,000 years, as had long been known; but superposed upon
+this great movement have now been detected minute nods, each with a
+period of nineteen years.
+
+The cause of the nodding is completely accounted for by the theory of
+gravitation, just as the precession of the equinoxes was. Both
+disturbances result from the attraction of the moon on the non-spherical
+earth--on its protuberant equator.
+
+"Nutation" is, in fact, a small perturbation of precession. The motion
+may be observed in a non-sleeping top. The slow conical motion of the
+top's slanting axis represents the course of precession. Sometimes this
+path is loopy, and its little nods correspond to nutation.
+
+The probable existence of some such perturbation had not escaped the
+sagacity of Newton, and he mentions something about it in the
+_Principia_, but thinks it too small to be detected by observation. He
+was thinking, however, of a solar disturbance rather than a lunar one,
+and this is certainly very small, though it, too, has now been observed.
+
+Newton was dead before Bradley made these great discoveries, else he
+would have been greatly pleased to hear of them.
+
+These discoveries of aberration and nutation, says Delambre, the great
+French historian of science, secure to their author a distinguished
+place after Hipparchus and Kepler among the astronomers of all ages and
+all countries.
+
+
+
+
+NOTES TO LECTURE XI
+
+
+_Lagrange_ and _Laplace_, both tremendous mathematicians, worked very
+much in alliance, and completed Newton's work. The _M�canique C�leste_
+contains the higher intricacies of astronomy mathematically worked out
+according to the theory of gravitation. They proved the solar system to
+be stable; all its inequalities being periodic, not cumulative. And
+Laplace suggested the "nebular hypothesis" concerning the origin of sun
+and planets: a hypothesis previously suggested, and to some extent,
+elaborated, by Kant.
+
+A list of some of the principal astronomical researches of Lagrange and
+Laplace:--Libration of the moon. Long inequality of Jupiter and Saturn.
+Perturbations of Jupiter's satellites. Perturbations of comets.
+Acceleration of the moon's mean motion. Improved lunar theory.
+Improvements in the theory of the tides. Periodic changes in the form
+and obliquity of the earth's orbit. Stability of the solar system
+considered as an assemblage of rigid bodies subject to gravity.
+
+The two equations which establish the stability of the solar system
+are:--
+
+  _Sum (me^2[square root]d) = constant,_
+
+  and
+
+  _Sum (m tan^2[theta][square root]d) = constant;_
+
+where _m_ is the mass of each planet, _d_ its mean distance from the
+sun, _e_ the excentricity of its orbit, and [theta] the inclination
+of its plane. However the expressions above formulated may change for
+individual planets, the sum of them for all the planets remains
+invariable.
+
+The period of the variations in excentricity of the earth's orbit is
+86,000 years; the period of conical revolution of the earth's axis is
+25,800 years. About 18,000 years ago the excentricity was at a maximum.
+
+
+
+
+LECTURE XI
+
+LAGRANGE AND LAPLACE--THE STABILITY OF THE SOLAR SYSTEM, AND THE NEBULAR
+HYPOTHESIS
+
+
+Laplace was the son of a small farmer or peasant of Normandy. His
+extraordinary ability was noticed by some wealthy neighbours, and by
+them he was sent to a good school. From that time his career was one
+brilliant success, until in the later years of his life his prominence
+brought him tangibly into contact with the deteriorating influence of
+politics. Perhaps one ought rather to say trying than deteriorating; for
+they seem trying to a strong character, deteriorating to a weak one--and
+unfortunately, Laplace must be classed in this latter category.
+
+It has always been the custom in France for its high scientific men to
+be conspicuous also in politics. It seems to be now becoming the fashion
+in this country also, I regret to say.
+
+The _life_ of Laplace is not specially interesting, and I shall not go
+into it. His brilliant mathematical genius is unquestionable, and almost
+unrivalled. He is, in fact, generally considered to come in this respect
+next after Newton. His talents were of a more popular order than those
+of Lagrange, and accordingly he acquired fame and rank, and rose to the
+highest dignities. Nevertheless, as a man and a politician he hardly
+commands our respect, and in time-serving adjustability he is comparable
+to the redoubtable Vicar of Bray. His scientific insight and genius
+were however unquestionably of the very highest order, and his work has
+been invaluable to astronomy.
+
+I will give a short sketch of some of his investigations, so far as they
+can be made intelligible without overmuch labour. He worked very much in
+conjunction with Lagrange, a more solid though a less brilliant man, and
+it is both impossible and unnecessary for us to attempt to apportion
+respective shares of credit between these two scientific giants, the
+greatest scientific men that France ever produced.
+
+First comes a research into the libration of the moon. This was
+discovered by Galileo in his old age at Arcetri, just before his
+blindness. The moon, as every one knows, keeps the same face to the
+earth as it revolves round it. In other words, it does not rotate with
+reference to the earth, though it does rotate with respect to outside
+bodies. Its libration consists in a sort of oscillation, whereby it
+shows us now a little more on one side, now a little more on the other,
+so that altogether we are cognizant of more than one-half of its
+surface--in fact, altogether of about three-fifths. It is a simple and
+unimportant matter, easily explained.
+
+     The motion of the moon may be analyzed into a rotation about its
+     own axis combined with a revolution about the earth. The speed of
+     the rotation is quite uniform, the speed of the revolution is not
+     quite uniform, because the orbit is not circular but elliptical,
+     and the moon has to travel faster in perigee than in apogee (in
+     accordance with Kepler's second law). The consequence of this is
+     that we see a little too far round the body of the moon, first on
+     one side, then on the other. Hence it _appears_ to oscillate
+     slightly, like a lop-sided fly-wheel whose revolutions have been
+     allowed to die away so that they end in oscillations of small
+     amplitude.[23] Its axis of rotation, too, is not precisely
+     perpendicular to its plane of revolution, and therefore we
+     sometimes see a few hundred miles beyond its north pole, sometimes
+     a similar amount beyond its south. Lastly, there is a sort of
+     parallax effect, owing to the fact that we see the rising moon from
+     one point of view, and the setting moon from a point 8,000 miles
+     distant; and this base-line of the earth's diameter gives us again
+     some extra glimpses. This diurnal or parallactic libration is
+     really more effective than the other two in extending our vision
+     into the space-facing hemisphere of the moon.
+
+     These simple matters may as well be understood, but there is
+     nothing in them to dwell upon. The far side of the moon is probably
+     but little worth seeing. Its features are likely to be more blurred
+     with accumulations of meteoric dust than are those of our side, but
+     otherwise they are likely to be of the same general character.
+
+The thing of real interest is the fact that the moon does turn the same
+face towards us; _i.e._ has ceased to rotate with respect to the earth
+(if ever it did so). The stability of this state of things was shown by
+Lagrange to depend on the shape of the moon. It must be slightly
+egg-shape, or prolate--extended in the direction of the earth; its
+earth-pointing diameter being a few hundred feet longer than its visible
+diameter; a cause slight enough, but nevertheless sufficient to maintain
+stability, except under the action of a distinct disturbing cause. The
+prolate or lemon-like shape is caused by the gravitative pull of the
+earth, balanced by the centrifugal whirl. The two forces balance each
+other as regards motion, but between them they have strained the moon a
+trifle out of shape. The moon has yielded as if it were perfectly
+plastic; in all probability it once was so.
+
+It may be interesting to note for a moment the correlative effect of
+this aspect of the moon, if we transfer ourselves to its surface in
+imagination, and look at the earth (cf. Fig. 41). The earth would be
+like a gigantic moon of four times our moon's diameter, and would go
+through its phases in regular order. But it would not rise or set: it
+would be fixed in the sky, and subject only to a minute oscillation to
+and fro once a month, by reason of the "libration" we have been speaking
+of. Its aspect, as seen by markings on its surface, would rapidly
+change, going through a cycle in twenty-four hours; but its permanent
+features would be usually masked by lawless accumulations of cloud,
+mainly aggregated in rude belts parallel to the equator. And these
+cloudy patches would be the most luminous, the whitest portions; for of
+course it would be their silver lining that we would then be looking
+on.[24]
+
+Next among the investigations of Lagrange and Laplace we will mention
+the long inequality of Jupiter and Saturn. Halley had found that Jupiter
+was continually lagging behind its true place as given by the theory of
+gravitation; and, on the other hand, that Saturn was being accelerated.
+The lag on the part of Jupiter amounted to about 34-1/2 minutes in a
+century. Overhauling ancient observations, however, Halley found signs
+of the opposite state of things, for when he got far enough back Jupiter
+was accelerated and Saturn was being retarded.
+
+Here was evidently a case of planetary perturbation, and Laplace and
+Lagrange undertook the working of it out. They attacked it as a case of
+the problem of three bodies, viz. the sun, Jupiter, and Saturn; which
+are so enormously the biggest of the known bodies in the system that
+insignificant masses like the Earth, Mars, and the rest, may be wholly
+neglected. They succeeded brilliantly, after a long and complex
+investigation: succeeded, not in solving the problem of the three
+bodies, but, by considering their mutual action as perturbations
+superposed on each other, in explaining the most conspicuous of the
+observed anomalies of their motion, and in laying the foundation of a
+general planetary theory.
+
+[Illustration: FIG. 79.--Shewing the three conjunction places in the
+orbits of Jupiter and Saturn. The two planets are represented as leaving
+one of the conjunctions where Jupiter was being pulled back and Saturn
+being pulled forward by their mutual attraction.]
+
+     One of the facts that plays a large part in the result was known to
+     the old astrologers, viz. that Jupiter and Saturn come into
+     conjunction with a certain triangular symmetry; the whole scheme
+     being called a trigon, and being mentioned several times by Kepler.
+     It happens that five of Jupiter's years very nearly equal two of
+     Saturn's,[25] so that they get very nearly into conjunction three
+     times in every five Jupiter years, but not exactly. The result of
+     this close approach is that periodically one pulls the other on and
+     is itself pulled back; but since the three points progress, it is
+     not always the same planet which gets pulled back. The complete
+     theory shows that in the year 1560 there was no marked
+     perturbation: before that it was in one direction, while afterwards
+     it was in the other direction, and the period of the whole cycle of
+     disturbances is 929 of our years. The solution of this long
+     outstanding puzzle by the theory of gravitation was hailed with the
+     greatest enthusiasm by astronomers, and it established the fame of
+     the two French mathematicians.
+
+Next they attacked the complicated problem of the motions of Jupiter's
+satellites. They succeeded in obtaining a theory of their motions which
+represented fact very nearly indeed, and they detected the following
+curious relationship between the satellites:--The speed of the first
+satellite + twice the speed of the second is equal to the speed of the
+third.
+
+They found this, not empirically, after the manner of Kepler, but as a
+deduction from the law of gravitation; for they go on to show that even
+if the satellites had not started with this relation they would sooner
+or later, by mutual perturbation, get themselves into it. One singular
+consequence of this, and of another quite similar connection between
+their positions, is that all three satellites can never be eclipsed at
+once.
+
+The motion of the fourth satellite is less tractable; it does not so
+readily form an easy system with the others.
+
+After these great successes the two astronomers naturally proceeded to
+study the mutual perturbations of all other bodies in the solar system.
+And one very remarkable discovery they made concerning the earth and
+moon, an account of which will be interesting, though the details and
+processes of calculation are quite beyond us in a course like this.
+
+Astronomical theory had become so nearly perfect by this time, and
+observations so accurate, that it was possible to calculate many
+astronomical events forwards or backwards, over even a thousand years or
+more, with admirable precision.
+
+Now, Halley had studied some records of ancient eclipses, and had
+calculated back by means of the lunar theory to see whether the
+calculation of the time they ought to occur would agree with the record
+of the time they did occur. To his surprise he found a discrepancy, not
+a large one, but still one quite noticeable. To state it as we know it
+now:--An eclipse a century ago happened twelve seconds later than it
+ought to have happened by theory; two centuries back the error amounted
+to forty-eight seconds, in three centuries it would be 108 seconds, and
+so on; the lag depending on the square of the time. By research, and
+help from scholars, he succeeded in obtaining the records of some very
+ancient eclipses indeed. One in Egypt towards the end of the tenth
+century A.D.; another in 201 A.D.; another a little before Christ; and
+one, the oldest of all of which any authentic record has been preserved,
+observed by the Chald�an astronomers in Babylon in the reign of
+Hezekiah.
+
+Calculating back to this splendid old record of a solar eclipse, over
+the intervening 2,400 years, the calculated and the observed times were
+found to disagree by nearly two hours. Pondering over an explanation of
+the discrepancy, Halley guessed that it must be because the moon's
+motion was not uniform, it must be going quicker and quicker, gaining
+twelve seconds each century on its previous gain--a discovery announced
+by him as "the acceleration of the moon's mean motion." The month was
+constantly getting shorter.
+
+What was the physical cause of this acceleration according to the theory
+of gravitation? Many attacked the question, but all failed. This was the
+problem Laplace set himself to work out. A singular and beautiful result
+rewarded his efforts.
+
+You know that the earth describes an elliptic orbit round the sun: and
+that an ellipse is a circle with a certain amount of flattening or
+"excentricity."[26] Well, Laplace found that the excentricity of the
+earth's orbit must be changing, getting slightly less; and that this
+change of excentricity would have an effect upon the length of the
+month. It would make the moon go quicker.
+
+One can almost see how it comes about. A decrease in excentricity means
+an increase in mean distance of the earth from the sun. This means to
+the moon a less solar perturbation. Now one effect of the solar
+perturbation is to keep the moon's orbit extra large: if the size of its
+orbit diminishes, its velocity must increase, according to Kepler's
+third law.
+
+Laplace calculated the amount of acceleration so resulting, and found it
+ten seconds a century; very nearly what observation required; for,
+though I have quoted observation as demanding twelve seconds per
+century, the facts were not then so distinctly and definitely
+ascertained.
+
+This calculation for a long time seemed thoroughly satisfactory, but it
+is not the last word on the subject. Quite lately an error has been
+found in the working, which diminishes the theoretical
+gravitation-acceleration to six seconds a century instead of ten, thus
+making it insufficient to agree exactly with fact. The theory of
+gravitation leaves an outstanding error. (The point is now almost
+thoroughly understood, and we shall return to it in Lecture XVIII).
+
+But another question arises out of this discussion. I have spoken of the
+excentricity of the earth's orbit as decreasing. Was it always
+decreasing? and if so, how far back was it so excentric that at
+perihelion the earth passed quite near the sun? If it ever did thus pass
+near the sun, the inference is manifest--the earth must at one time have
+been thrown off, or been separated off, from the sun.
+
+If a projectile could be fired so fast that it described an orbit round
+the earth--and the speed of fire to attain this lies between five and
+seven miles a second (not less than the one, nor more than the
+other)--it would ever afterwards pass through its point of projection
+as one point of its elliptic orbit; and its periodic return through that
+point would be the sign of its origin. Similarly, if a satellite does
+_not_ come near its central orb, and can be shown never to have been
+near it, the natural inference is that it has _not_ been born from it,
+but has originated in some other way.
+
+The question which presented itself in connexion with the variable
+ellipticity of the earth's orbit was the following:--Had it always been
+decreasing, so that once it was excentric enough just to graze the sun
+at perihelion as a projected body would do?
+
+Into the problem thus presented Lagrange threw himself, and he succeeded
+in showing that no such explanation of the origin of the earth is
+possible. The excentricity of the orbit, though now decreasing, was not
+always decreasing; ages ago it was increasing: it passes through
+periodic changes. Eighteen thousand years ago its excentricity was a
+maximum; since then it has been diminishing, and will continue to
+diminish for 25,000 years more, when it will be an almost perfect
+circle; it will then begin to increase again, and so on. The obliquity
+of the ecliptic is also changing periodically, but not greatly: the
+change is less than three degrees.
+
+This research has, or ought to have, the most transcendent interest for
+geologists and geographers. You know that geologists find traces of
+extraordinary variations of temperature on the surface of the earth.
+England was at one time tropical, at another time glacial. Far away
+north, in Spitzbergen, evidence of the luxuriant vegetation of past ages
+has been found; and the explanation of these great climatic changes has
+long been a puzzle. Does not the secular variation in excentricity of
+the earth's orbit, combined with the precession of the equinoxes, afford
+a key? And if a key at all, it will be an accurate key, and enable us to
+calculate back with some precision to the date of the glacial epoch;
+and again to the time when a tropical flora flourished in what is now
+northern Europe, _i.e._ to the date of the Carboniferous era.
+
+This aspect of the subject has recently been taught with vigour and
+success by Dr. Croll in his book "Climate and Time."
+
+     A brief and partial explanation of the matter may be given, because
+     it is a point of some interest and is also one of fair simplicity.
+
+     Every one knows that the climatic conditions of winter and summer
+     are inverted in the two hemispheres, and that at present the sun is
+     nearest to us in our (northern) winter. In other words, the earth's
+     axis is inclined so as to tilt its north pole away from the sun at
+     perihelion, or when the earth is at the part of its elliptic orbit
+     nearest the sun's focus; and to tilt it towards the sun at
+     aphelion. The result of this present state of things is to diminish
+     the intensity of the average northern winter and of the average
+     northern summer, and on the other hand to aggravate the extremes of
+     temperature in the southern hemisphere; all other things being
+     equal. Of course other things are not equal, and the distribution
+     of land and sea is a still more powerful climatic agent than is the
+     three million miles or so extra nearness of the sun. But it is
+     supposed that the Antarctic ice-cap is larger than the northern,
+     and increased summer radiation with increased winter cold would
+     account for this.
+
+     But the present state of things did not always obtain. The conical
+     movement of the earth's axis (now known by a curious perversion of
+     phrase as "precession") will in the course of 13,000 years or so
+     cause the tilt to be precisely opposite, and then we shall have the
+     more extreme winters and summers instead of the southern
+     hemisphere.
+
+     If the change were to occur now, it might not be overpowering,
+     because now the excentricity is moderate. But if it happened some
+     time back, when the excentricity was much greater, a decidedly
+     different arrangement of climate may have resulted. There is no
+     need to say _if_ it happened some time back: it did happen, and
+     accordingly an agent for affecting the distribution of mean
+     temperature on the earth is to hand; though whether it is
+     sufficient to achieve all that has been observed by geologists is a
+     matter of opinion.
+
+     Once more, the whole diversity of the seasons depends on the tilt
+     of the earth's axis, the 23� by which it is inclined to a
+     perpendicular to the orbital plane; and this obliquity or tilt is
+     subject to slow fluctuations. Hence there will come eras when all
+     causes combine to produce a maximum extremity of seasons in the
+     northern hemisphere, and other eras when it is the southern
+     hemisphere which is subject to extremes.
+
+But a grander problem still awaited solution--nothing less than the fate
+of the whole solar system. Here are a number of bodies of various sizes
+circulating at various rates round one central body, all attracted by
+it, and all attracting each other, the whole abandoned to the free play
+of the force of gravitation: what will be the end of it all? Will they
+ultimately approach and fall into the sun, or will they recede further
+and further from him, into the cold of space? There is a third possible
+alternative: may they not alternately approach and recede from him, so
+as on the whole to maintain a fair approximation to their present
+distances, without great and violent extremes of temperature either way?
+
+If any one planet of the system were to fall into the sun, more
+especially if it were a big one like Jupiter or Saturn, the heat
+produced would be so terrific that life on this earth would be
+destroyed, even at its present distance; so that we are personally
+interested in the behaviour of the other planets as well as in the
+behaviour of our own.
+
+The result of the portentously difficult and profoundly interesting
+investigation, here sketched in barest outline, is that the solar system
+is stable: that is to say, that if disturbed a little it will oscillate
+and return to its old state; whereas if it were unstable the slightest
+disturbance would tend to accumulate, and would sooner or later bring
+about a catastrophe. A hanging pendulum is stable, and oscillates about
+a mean position; its motion is periodic. A top-heavy load balanced on a
+point is unstable. All the changes of the solar system are periodic,
+_i.e._ they repeat themselves at regular intervals, and they never
+exceed a certain moderate amount.
+
+The period is something enormous. They will not have gone through all
+their changes until a period of 2,000,000 years has elapsed. This is
+the period of the planetary oscillation: "a great pendulum of eternity
+which beats ages as our pendulums beat seconds." Enormous it seems; and
+yet we have reason to believe that the earth has existed through many
+such periods.
+
+     The two laws of stability discovered and stated by Lagrange and
+     Laplace I can state, though they may be difficult to understand:--
+
+     Represent the masses of the several planets by m_1, m_2, &c.; their
+     mean distances from the sun (or radii vectores) by r_1, r_2, &c.;
+     the excentricities of their orbits by e_1, e_2, &c.; and the
+     obliquity of the planes of these orbits, reckoned from a single
+     plane of reference or "invariable plane," by [theta]_1, [theta]_2,
+     &c.; then all these quantities (except m) are liable to
+     fluctuate; but, however much they change, an increase for one
+     planet will be accompanied by a decrease for some others; so that,
+     taking all the planets into account, the sum of a set of terms like
+     these, m_1e_1^2 [square root]r_1 + m_2e_2^2 [square root]r_2
+     + &c., will remain always the same. This is summed up briefly in
+     the following statement:
+
+  [Sigma](me^2 [square root]r) = constant.
+
+     That is one law, and the other is like it, but with inclination of
+     orbit instead of excentricity, viz.:
+
+  [Sigma](m[theta]^2 [square root]r) = constant.
+
+     The value of each of these two constants can at any time be
+     calculated. At present their values are small. Hence they always
+     were and always will be small; being, in fact, invariable. Hence
+     neither _e_ nor _r_ nor [theta] can ever become infinite, nor can
+     their average value for the system ever become zero.
+
+The planets may share the given amount of total excentricity and
+obliquity in various proportions between themselves; but even if it were
+all piled on to one planet it would not be very excessive, unless the
+planet were so small a one as Mercury; and it would be most improbable
+that one planet should ever have all the excentricity of the solar
+system heaped upon itself. The earth, therefore, never has been, nor
+ever will be, enormously nearer the sun than it is at present: nor can
+it ever get very much further off. Its changes are small and are
+periodic--an increase is followed by a decrease, like the swing of a
+pendulum.
+
+The above two laws have been called the Magna Charta of the solar
+system, and were long supposed to guarantee its absolute permanence. So
+far as the theory of gravitation carries us, they do guarantee its
+permanence; but something more remains to be said on the subject in a
+future lecture (XVIII).
+
+And now, finally, we come to a sublime speculation, thrown out by
+Laplace, not as the result of profound calculation, like the results
+hitherto mentioned, not following certainly from the theory of
+gravitation, or from any other known theory, and therefore not to be
+accepted as more than a brilliant hypothesis, to be confirmed or
+rejected as our knowledge extends. This speculation is the "Nebular
+hypothesis." Since the time of Laplace the nebular hypothesis has had
+ups and downs of credence, sometimes being largely believed in,
+sometimes being almost ignored. At the present time it holds the field
+with perhaps greater probability of ultimate triumph than has ever
+before seemed to belong to it--far greater than belonged to it when
+first propounded.
+
+It had been previously stated clearly and well by the philosopher Kant,
+who was intensely interested in "the starry heavens" as well as in the
+"mind of man," and who shewed in connexion with astronomy also a most
+surprising genius. The hypothesis ought by rights perhaps to be known
+rather by his name than by that of Laplace.
+
+The data on which it was founded are these:--Every motion in the solar
+system known at that time took place in one direction, and in one
+direction only. Thus the planets revolve round the sun, all going the
+same way round; moons revolve round the planets, still maintaining the
+same direction of rotation, and all the bodies that were known to rotate
+on their own axis did so with still the same kind of spin. Moreover,
+all these motions take place in or near a single plane. The ancients
+knew that sun moon and planets all keep near to the ecliptic, within a
+belt known as the zodiac: none strays away into other parts of the sky.
+Satellites also, and rings, are arranged in or near the same plane; and
+the plane of diurnal spin, or equator of the different bodies, is but
+slightly tilted.
+
+Now all this could not be the result of chance. What could have caused
+it? Is there any connection or common ancestry possible, to account for
+this strange family likeness? There is no connection now, but there may
+have been once. Must have been, we may almost say. It is as though they
+had once been parts of one great mass rotating as a whole; for if such a
+rotating mass broke up, its parts would retain its direction of
+rotation. But such a mass, filling all space as far as or beyond Saturn,
+although containing the materials of the whole solar system in itself,
+must have been of very rare consistency. Occupying so much bulk it could
+not have been solid, nor yet liquid, but it might have been gaseous.
+
+Are there any such gigantic rotating masses of gas in the heaven now?
+Certainly there are; there are the nebul�. Some of the nebul� are now
+known to be gaseous, and some of them at least are in a state of
+rotation. Laplace could not have known this for certain, but he
+suspected it. The first distinctly spiral nebula was discovered by the
+telescope of Lord Rosse; and quite recently a splendid photograph of the
+great Andromeda nebula, by our townsman, Mr. Isaac Roberts, reveals what
+was quite unsuspected--and makes it clear that this prodigious mass also
+is in a state of extensive and majestic whirl.
+
+Very well, then, put this problem:--A vast mass of rotating gas is left
+to itself to cool for ages and to condense as it cools: how will it
+behave? A difficult mathematical problem, worthy of being attacked
+to-day; not yet at all adequately treated. There are those who believe
+that by the complete treatment of such a problem all the history of the
+solar system could be evolved.
+
+[Illustration: FIG. 80.--Lord Rosse's drawing of the spiral nebula in
+Canes Venatici, with the stub marks of the draughtsman unduly emphasised
+into features by the engraver.]
+
+Laplace pictured to himself this mass shrinking and thereby whirling
+more and more rapidly. A spinning body shrinking in size and retaining
+its original amount of rotation, as it will unless a brake is applied,
+must spin more and more rapidly as it shrinks. It has what
+mathematicians call a constant moment of momentum; and what it loses in
+leverage, as it shrinks, it gains in speed. The mass is held together by
+gravitation, every particle attracting every other particle; but since
+all the particles are describing curved paths, they will tend to fly off
+tangentially, and only a small excess of the gravitation force over the
+centrifugal is left to pull the particles in, and slowly to concentrate
+the nebula. The mutual gravitation of the parts is opposed by the
+centrifugal force of the whirl. At length a point is reached where the
+two forces balance. A portion outside a certain line will be in
+equilibrium; it will be left behind, and the rest must contract without
+it. A ring is formed, and away goes the inner nucleus contracting
+further and further towards a centre. After a time another ring will be
+left behind in the same way, and so on. What happens to these rings?
+They rotate with the motion they possess when thrown or shrunk off; but
+will they remain rings? If perfectly regular they may; if there be any
+irregularity they are liable to break up. They will break into one or
+two or more large masses, which are ultimately very likely to collide
+and become one. The revolving body so formed is still a rotating gaseous
+mass; and it will go on shrinking and cooling and throwing off rings,
+like the larger nucleus by which it has been abandoned. As any nucleus
+gets smaller, its rate of rotation increases, and so the rings last
+thrown off will be spinning faster than those thrown off earliest. The
+final nucleus or residual central body will be rotating fastest of all.
+
+The nucleus of the whole original mass we now see shrunk up into what we
+call the sun, which is spinning on its axis once every twenty-five days.
+The rings successively thrown off by it are now the planets--some large,
+some small--those last thrown off rotating round him comparatively
+quickly, those outside much more slowly. The rings thrown off by the
+planetary gaseous masses as they contracted have now become satellites;
+except one ring which has remained without breaking up, and is to be
+seen rotating round Saturn still.
+
+One other similar ring, an abortive attempt at a planet, is also left
+round the sun (the zone of asteroids).
+
+Such, crudely and baldly, is the famous nebular hypothesis of Laplace.
+It was first stated, as has been said above, by the philosopher Kant,
+but it was elaborated into much fuller detail by the greatest of French
+mathematicians and astronomers.
+
+The contracting masses will condense and generate great quantities of
+heat by their own shrinkage; they will at a certain stage condense to
+liquid, and after a time will begin to cool and congeal with a
+superficial crust, which will get thicker and thicker; but for ages they
+will remain hot, even after they have become thoroughly solid. The small
+ones will cool fastest; the big ones will retain their heat for an
+immense time. Bullets cool quickly, cannon-balls take hours or days to
+cool, planets take millions of years. Our moon may be nearly cold, but
+the earth is still warm--indeed, very hot inside. Jupiter is believed by
+some observers still to glow with a dull red heat; and the high
+temperature of the much larger and still liquid mass of the sun is
+apparent to everybody. Not till it begins to scum over will it be
+perceptibly cooler.
+
+[Illustration: FIG. 81.--Saturn.]
+
+Many things are now known concerning heat which were not known to
+Laplace (in the above paragraph they are only hinted at), and these
+confirm and strengthen the general features of his hypothesis in a
+striking way; so do the most recent telescopic discoveries. But fresh
+possibilities have now occurred to us, tidal phenomena are seen to have
+an influence then wholly unsuspected, and it will be in a modified and
+amplified form that the philosopher of next century will still hold to
+the main features of this famous old Nebular Hypothesis respecting the
+origin of the sun and planets--the Evolution of the solar system.
+
+
+
+
+NOTES TO LECTURE XII
+
+
+The subject of stellar astronomy was first opened up by Sir William
+Herschel, the greatest observing astronomer.
+
+_Frederick William Herschel_ was born in Hanover in 1738, and brought up
+as a musician. Came to England in 1756. First saw a telescope in 1773.
+Made a great many himself, and began a survey of the heavens. His sister
+Caroline, born in 1750, came to England in 1772, and became his devoted
+assistant to the end of his life. Uranus discovered in 1781. Music
+finally abandoned next year, and the 40-foot telescope begun. Discovered
+two moons of Saturn and two of Uranus. Reviewed, described, and gauged
+all the visible heavens. Discovered and catalogued 2,500 nebul� and 806
+double stars. Speculated concerning the Milky Way, the nebulosity of
+stars, the origin and growth of solar systems. Discovered that the stars
+were in motion, not fixed, and that the sun as one of them was
+journeying towards a point in the constellation Hercules. Died in 1822,
+eighty-four years old. Caroline Herschel discovered eight comets, and
+lived on to the age of ninety-eight.
+
+
+
+
+LECTURE XII
+
+HERSCHEL AND THE MOTION OF THE FIXED STARS
+
+
+We may admit, I think, that, with a few notable exceptions, the work of
+the great men we have been recently considering was rather to complete
+and round off the work of Newton, than to strike out new and original
+lines.
+
+This was the whole tendency of eighteenth century astronomy. It appeared
+to be getting into an adult and uninteresting stage, wherein everything
+could be calculated and predicted. Labour and ingenuity, and a severe
+mathematical training, were necessary to work out the remote
+consequences of known laws, but nothing fresh seemed likely to turn up.
+Consequently men's minds began turning in other directions, and we find
+chemistry and optics largely studied by some of the greatest minds,
+instead of astronomy.
+
+But before the century closed there was destined to arise one remarkable
+exception--a man who was comparatively ignorant of that which had been
+done before--a man unversed in mathematics and the intricacies of
+science, but who possessed such a real and genuine enthusiasm and love
+of Nature that he overcame the force of adverse circumstances, and
+entering the territory of astronomy by a by-path, struck out a new line
+for himself, and infused into the science a healthy spirit of fresh life
+and activity.
+
+This man was William Herschel.
+
+"The rise of Herschel," says Miss Clerke, "is the one conspicuous
+anomaly in the otherwise somewhat quiet and prosy eighteenth century. It
+proved decisive of the course of events in the nineteenth. It was
+unexplained by anything that had gone before, yet all that came after
+hinged upon it. It gave a new direction to effort; it lent a fresh
+impulse to thought. It opened a channel for the widespread public
+interest which was gathering towards astronomical subjects to flow in."
+
+Herschel was born at Hanover in 1738, the son of an oboe player in a
+military regiment. The father was a good musician, and a cultivated man.
+The mother was a German _Frau_ of the period, a strong, active,
+business-like woman, of strong character and profound ignorance. Herself
+unable to write, she set her face against learning and all new-fangled
+notions. The education of the sons she could not altogether control,
+though she lamented over it, but the education of her two daughters she
+strictly limited to cooking, sewing, and household management. These,
+however, she taught them well.
+
+It was a large family, and William was the fourth child. We need only
+remember the names of his younger brother Alexander, and of his much
+younger sister Caroline.
+
+They were all very musical--the youngest boy was once raised upon a
+table to play the violin at a public performance. The girls were
+forbidden to learn music by their mother, but their father sometimes
+taught them a little on the sly. Alexander was besides an ingenious
+mechanician.
+
+At the age of seventeen, William became oboist to the Hanoverian Guards,
+shortly before the regiment was ordered to England. Two years later he
+removed himself from the regiment, with the approval of his parents,
+though probably without the approbation or consent of the commanding
+officer, by whom such removal would be regarded as simple desertion,
+which indeed it was; and George III. long afterwards handed him an
+official pardon for it.
+
+At the age of nineteen, he was thus launched in England with an outfit
+of some French, Latin, and English, picked up by himself; some skill in
+playing the hautboy, the violin, and the organ, as taught by his father;
+and some good linen and clothing, and an immense stock of energy,
+provided by his mother.
+
+He lived as musical instructor to one or two militia bands in Yorkshire,
+and for three years we hear no more than this of him. But, at the end of
+that time, a noted organist, Dr. Miller, of Durham, who had heard his
+playing, proposed that he should come and live with him and play at
+concerts, which he was very glad to do. He next obtained the post of
+organist at Halifax; and some four or five years later he was invited to
+become organist at the Octagon Chapel in Bath, and soon led the musical
+life of that then very fashionable place.
+
+About this time he went on a short visit to his family at Hanover, by
+all of whom he was very much beloved, especially by his young sister
+Caroline, who always regarded him as specially her own brother. It is
+rather pitiful, however, to find that her domestic occupations still
+unfairly repressed and blighted her life. She says:--
+
+     "Of the joys and pleasures which all felt at this long-wished-for
+     meeting with my--let me say my dearest--brother, but a small
+     portion could fall to my share; for with my constant attendance at
+     church and school, besides the time I was employed in doing the
+     drudgery of the scullery, it was but seldom I could make one in the
+     group when the family were assembled together."
+
+While at Bath he wrote many musical pieces--glees, anthems, chants,
+pieces for the harp, and an orchestral symphony. He taught a large
+number of pupils, and lived a hard and successful life. After fourteen
+hours or so spent in teaching and playing, he would retire at night to
+instruct his mind with a study of mathematics, optics, Italian, or
+Greek, in all of which he managed to make some progress. He also about
+this time fell in with some book on astronomy.
+
+In 1763 his father was struck with paralysis, and two years later he
+died.
+
+William then proposed that Alexander should come over from Hanover and
+join him at Bath, which was done. Next they wanted to rescue their
+sister Caroline from her humdrum existence, but this was a more
+difficult matter. Caroline's journal gives an account of her life at
+this time that is instructive. Here are a few extracts from it:--
+
+     "My father wished to give me something like a polished education,
+     but my mother was particularly determined that it should be a
+     rough, but at the same time a useful one; and nothing further she
+     thought was necessary but to send me two or three months to a
+     sempstress to be taught to make household linen....
+
+     "My mother would not consent to my being taught French, ... so all
+     my father could do for me was to indulge me (and please himself)
+     sometimes with a short lesson on the violin, when my mother was
+     either in good humour or out of the way.... She had cause for
+     wishing me not to know more than was necessary for being useful in
+     the family; for it was her certain belief that my brother William
+     would have returned to his country, and my eldest brother not have
+     looked so high, if they had had a little less learning."
+
+However, seven years after the death of their father, William went over
+to Germany and returned to England in triumph, bringing Caroline with
+him: she being then twenty-two.
+
+So now began a busy life in Bath. For Caroline the work must have been
+tremendous. For, besides having to learn singing, she had to learn
+English. She had, moreover, to keep accounts and do the marketing.
+
+When the season at Bath was over, she hoped to get rather more of her
+brother William's society; but he was deep in optics and astronomy, used
+to sleep with the books under his pillow, read them during meals, and
+scarcely ever thought of anything else.
+
+He was determined to see for himself all the astronomical wonders; and
+there being a small Gregorian reflector in one of the shops, he hired
+it. But he was not satisfied with this, and contemplated making a
+telescope 20 feet long. He wrote to opticians inquiring the price of a
+mirror suitable, but found there were none so large, and that even the
+smaller ones were beyond his means. Nothing daunted, he determined to
+make some for himself. Alexander entered into his plans: tools, hones,
+polishers, and all sorts of rubbish were imported into the house, to the
+sister's dismay, who says:--
+
+[Illustration: FIG. 82.--Principle of Newtonian reflector.]
+
+     "And then, to my sorrow, I saw almost every room turned into a
+     workshop. A cabinet-maker making a tube and stands of all
+     descriptions in a handsomely furnished drawing-room; Alex. putting
+     up a huge turning-machine (which he had brought in the autumn from
+     Bristol, where he used to spend the summer) in a bed-room, for
+     turning patterns, grinding glasses, and turning eye-pieces, &c. At
+     the same time music durst not lie entirely dormant during the
+     summer, and my brother had frequent rehearsals at home."
+
+Finally, in 1774, at the age of thirty-six, he had made himself a
+5-1/2-foot telescope, and began to view the heavens. So attached was he
+to the instrument that he would run from the concert-room between the
+parts, and take a look at the stars.
+
+He soon began another telescope, and then another. He must have made
+some dozen different telescopes, always trying to get them bigger and
+bigger; at last he got a 7-foot and then a 10-foot instrument, and began
+a systematic survey of the heavens; he also began to communicate his
+results to the Royal Society.
+
+He now took a larger house, with more room for workshops, and a grass
+plot for a 20-foot telescope, and still he went on grinding
+mirrors--literally hundreds of them.
+
+I read another extract from the diary of his sister, who waited on him
+and obeyed him like a spaniel:--
+
+     "My time was taken up with copying music and practising, besides
+     attendance on my brother when polishing, since by way of keeping
+     him alive I was constantly obliged to feed him by putting the
+     victuals by bits into his mouth. This was once the case when, in
+     order to finish a 7-foot mirror, he had not taken his hands from it
+     for sixteen hours together. In general he was never unemployed at
+     meals, but was always at those times contriving or making drawings
+     of whatever came in his mind. Generally I was obliged to read to
+     him whilst he was at the turning-lathe, or polishing mirrors--_Don
+     Quixote_, _Arabian Nights' Entertainments_, the novels of Sterne,
+     Fielding, &c.; serving tea and supper without interrupting the work
+     with which he was engaged, ... and sometimes lending a hand. I
+     became, in time, as useful a member of the workshop as a boy might
+     be to his master in the first year of his apprenticeship.... But as
+     I was to take a part the next year in the oratorios, I had, for a
+     whole twelvemonth, two lessons per week from Miss Fleming, the
+     celebrated dancing-mistress, to drill me for a gentlewoman (God
+     knows how she succeeded). So we lived on without interruption. My
+     brother Alex. was absent from Bath for some months every summer,
+     but when at home he took much pleasure in executing some turning or
+     clockmaker's work for his brother."
+
+The music, and the astronomy, and the making of telescopes, all went on
+together, each at high pressure, and enough done in each to satisfy any
+ordinary activity. But the Herschels knew no rest. Grinding mirrors by
+day, concerts and oratorios in the evening, star-gazing at night. It is
+strange his health could stand it.
+
+The star-gazing, moreover, was no _dilettante_ work; it was based on a
+serious system--a well thought out plan of observation. It was nothing
+less than this--to pass the whole heavens steadily and in order through
+the telescope, noting and describing and recording every object that
+should be visible, whether previously known or unknown. The operation is
+called sweeping; but it is not a rapid passage from one object to
+another, as the term might suggest; it is a most tedious business, and
+consists in following with the telescope a certain field of view for
+some minutes, so as to be sure that nothing is missed, then shifting it
+to the next overlapping field, and watching again. And whatever object
+appears must be scrutinized anxiously to see what there is peculiar
+about it. If a star, it may be double, or it may be coloured, or it may
+be nebulous; or again it may be variable, and so its brightness must be
+estimated in order to compare with a subsequent observation.
+
+Four distinct times in his life did Herschel thus pass the whole visible
+heavens under review; and each survey occupied him several years. He
+discovered double stars, variable stars, nebul�, and comets; and Mr.
+William Herschel, of Bath, the amateur astronomer, was gradually
+emerging from his obscurity, and becoming a known man.
+
+Tuesday, the 13th of March, 1781, is a date memorable in the annals of
+astronomy. "On this night," he writes to the Royal Society, "in
+examining the small stars near _[eta]_ Geminorum, I perceived one
+visibly larger than the rest. Struck with its uncommon appearance, I
+compared it to _[eta]_ Geminorum and another star, and finding it so
+much larger than either, I suspected it to be a comet."
+
+The "comet" was immediately observed by professional astronomers, and
+its orbit was computed by some of them. It was thus found to move in
+nearly a circle instead of an elongated ellipse, and to be nearly twice
+as far from the sun as Saturn. It was no comet, it was a new planet;
+more than 100 times as big as the earth, and nearly twice as far away as
+Saturn. It was presently christened "Uranus."
+
+This was a most striking discovery, and the news sped over Europe. To
+understand the interest it excited we must remember that such a
+discovery was unique. Since the most ancient times of which men had any
+knowledge, the planets Mercury, Venus, Mars, Jupiter, Saturn, had been
+known, and there had been no addition to their number. Galileo and
+others had discovered satellites indeed, but a new primary planet was an
+entire and utterly unsuspected novelty.
+
+One of the most immediate consequences of the event was the discovery of
+Herschel himself. The Royal Society made him a Fellow the same year. The
+University of Oxford dubbed him a doctor; and the King sent for him to
+bring his telescope and show it at Court. So to London and Windsor he
+went, taking with him his best telescope. Maskelyne, the then
+Astronomer-Royal, compared it with the National one at Greenwich, and
+found Herschel's home-made instrument far the better of the two. He had
+a stand made after Herschel's pattern, but was so disgusted with his own
+instrument now that he scarcely thought it worthy of the stand when it
+was made. At Windsor, George III. was very civil, and Mr. Herschel was
+in great request to show the ladies of the Court Saturn and other
+objects of interest. Mr. Herschel exhibited a piece of worldly wisdom
+under these circumstances, that recalls faintly the behaviour of Tycho
+Brah� under similar circumstances. The evening when the exhibition was
+to take place threatened to become cloudy and wet, so Herschel rigged up
+an artificial Saturn, constructed of card and tissue paper, with a lamp
+behind it, in the distant wall of a garden; and, when the time came, his
+new titled friends were regaled with a view of this imitation Saturn
+through the telescope--the real one not being visible. They went away
+much pleased.
+
+He stayed hovering between Windsor and Greenwich, and uncertain what was
+to be the outcome of all this regal patronizing. He writes to his sister
+that he would much rather be back grinding mirrors at Bath. And she
+writes begging him to come, for his musical pupils were getting
+impatient. They had to get the better of their impatience, however, for
+the King ultimately appointed him astronomer or rather telescope-maker
+to himself, and so Caroline and the whole household were sent for, and
+established in a small house at Datchet.
+
+From being a star-gazing musician, Herschel thus became a practical
+astronomer. Henceforth he lived in his observatory; only on wet and
+moonlight nights could he be torn away from it. The day-time he devoted
+to making his long-contemplated 20-foot telescope.
+
+Not yet, however, were all their difficulties removed. The house at
+Datchet was a tumble-down barn of a place, chosen rather as a workshop
+and observatory than as a dwelling-house. And the salary allowed him by
+George III. was scarcely a princely one. It was, as a matter of fact,
+�200 a year. The idea was that he would earn his living by making
+telescopes, and so indeed he did. He made altogether some hundreds.
+Among others, four for the King. But this eternal making of telescopes
+for other people to use or play with was a weariness to the flesh. What
+he wanted was to observe, observe, observe.
+
+Sir William Watson, an old friend of his, and of some influence at
+Court, expressed his mind pretty plainly concerning Herschel's position;
+and as soon as the King got to understand that there was anything the
+matter, he immediately offered �2,000 for a gigantic telescope to be
+made for Herschel's own use. Nothing better did he want in life. The
+whole army of carpenters and craftsmen resident in Datchet were pressed
+into the service. Furnaces for the speculum metal were built, stands
+erected, and the 40-foot telescope fairly begun. It cost �4,000 before
+it was finished, but the King paid the whole.
+
+[Illustration: FIG. 83.--Herschel's 40-foot telescope.]
+
+With it he discovered two more satellites to Saturn (five hitherto had
+been known), and two moons to his own planet Uranus. These two are now
+known as Oberon and Titania. They were not seen again till some forty
+years after, when his son, Sir John Herschel, reobserved them. And in
+1847, Mr. Lassell, at his house, "Starfield," near Liverpool, discovered
+two more, called Ariel and Umbriel, making the number four, as now
+known. Mr. Lassell also discovered, with a telescope of his own making,
+an eighth satellite of Saturn--Hyperion--and a satellite to Neptune.
+
+A letter from a foreign astronomer about this period describes Herschel
+and his sister's method of work:--
+
+     "I spent the night of the 6th of January at Herschel's, in Datchet,
+     near Windsor, and had the good luck to hit on a fine evening. He
+     has his 20-foot Newtonian telescope in the open air, and mounted in
+     his garden very simply and conveniently. It is moved by an
+     assistant, who stands below it.... Near the instrument is a clock
+     regulated to sidereal time.... In the room near it sits Herschel's
+     sister, and she has Flamsteed's atlas open before her. As he gives
+     her the word, she writes down the declination and right ascension,
+     and the other circumstances of the observation. In this way
+     Herschel examines the whole sky without omitting the least part. He
+     commonly observes with a magnifying power of one hundred and fifty,
+     and is sure that after four or five years he will have passed in
+     review every object above our horizon. He showed me the book in
+     which his observations up to this time are written, and I am
+     astonished at the great number of them. Each sweep covers 2� 15' in
+     declination, and he lets each star pass at least three times
+     through the field of his telescope, so that it is impossible that
+     anything can escape him. He has already found about 900 double
+     stars, and almost as many nebul�. I went to bed about one o'clock,
+     and up to that time he had found that night four or five new
+     nebul�. The thermometer in the garden stood at 13� Fahrenheit; but,
+     in spite of this, Herschel observes the whole night through, except
+     that he stops every three or four hours and goes into the room for
+     a few moments. For some years Herschel has observed the heavens
+     every hour when the weather is clear, and this always in the open
+     air, because he says that the telescope only performs well when it
+     is at the same temperature as the air. He protects himself against
+     the weather by putting on more clothing. He has an excellent
+     constitution, and thinks about nothing else in the world but the
+     celestial bodies. He has promised me in the most cordial way,
+     entirely in the service of astronomy, and without thinking of his
+     own interest, to see to the telescopes I have ordered for European
+     observatories, and he will himself attend to the preparation of the
+     mirrors."
+
+[Illustration: _Painted by Abbott._
+
+_Engraved by Ryder._
+
+FIG. 84.--WILLIAM HERSCHEL.
+
+_From an Original Picture in the Possession of_ WM. WATSON, M.D.,
+F.R.S.]
+
+In 1783, Herschel married an estimable lady who sympathized with his
+pursuits. She was the only daughter of a City magnate, so his pecuniary
+difficulties, such as they were (they were never very troublesome to
+him), came to an end. They moved now into a more commodious house at
+Slough. Their one son, afterwards the famous Sir John Herschel, was
+born some nine years later. But the marriage was rather a blow to his
+devoted sister: henceforth she lived in lodgings, and went over at
+night-time to help him observe. For it must be remarked that this family
+literally turned night into day. Whatever sleep they got was in the
+day-time. Every fine night without exception was spent in observing: and
+the quite incredible fierceness of the pursuit is illustrated, as
+strongly as it can be, by the following sentence out of Caroline's
+diary, at the time of the move from Datchet to Slough: "The last night
+at Datchet was spent in sweeping till daylight, and by the next evening
+the telescope stood ready for observation at Slough."
+
+Caroline was now often allowed to sweep with a small telescope on her
+own account. In this way she picked up a good many nebul� in the course
+of her life, and eight comets, four of which were quite new, and one of
+which, known since as Encke's comet, has become very famous.
+
+The work they got through between them is something astonishing. He made
+with his own hands 430 parabolic mirrors for reflecting telescopes,
+besides a great number of complete instruments. He was forty-two when he
+began contributing to the Royal Society; yet before he died he had sent
+them sixty-nine long and elaborate treatises. One of these memoirs is a
+catalogue of 1000 nebul�. Fifteen years after he sends in another 1000;
+and some years later another 500. He also discovered 806 double stars,
+which he proved were really corrected from the fact that they revolved
+round each other (p. 309). He lived to see some of them perform half a
+revolution. For him the stars were not fixed: they moved slowly among
+themselves. He detected their proper motions. He passed the whole
+northern firmament in review four distinct times; counted the stars in
+3,400 gauge-fields, and estimated the brightness of hundreds of stars.
+He also measured as accurately as he could their proper motions,
+devising for this purpose the method which still to this day remains in
+use.
+
+And what is the outcome of it all? It is not Uranus, nor the satellites,
+nor even the double stars and the nebul� considered as mere objects: it
+is the beginning of a science of the stars.
+
+[Illustration: FIG. 85.--CAROLINE HERSCHEL.
+
+_From a Drawing from Life, by_ GEORGE M�LLER, 1847.]
+
+Hitherto the stars had only been observed for nautical and practical
+purposes. Their times of rising and southing and setting had been noted;
+they had been treated as a clock or piece of dead mechanism, and as
+fixed points of reference. All the energies of astronomers had gone out
+towards the solar system. It was the planets that had been observed.
+Tycho had observed and tabulated their positions. Kepler had found out
+some laws of their motion. Galileo had discovered their peculiarities
+and attendants. Newton and Laplace had perceived every detail of their
+laws.
+
+But for the stars--the old Ptolemaic system might still have been true.
+They might still be mere dots in a vast crystalline sphere, all set at
+about one distance, and subservient to the uses of the earth.
+
+Herschel changed all this. Instead of sameness, he found variety;
+instead of uniformity of distance, limitless and utterly limitless
+fields and boundless distances; instead of rest and quiescence, motion
+and activity; instead of stagnation, life.
+
+[Illustration: FIG. 86.--The double-double star [epsilon] Lyr� as seen
+under three different powers.]
+
+Yes, that is what Herschel discovered--the life and activity of the
+whole visible universe. No longer was our little solar system to be the
+one object of regard, no longer were its phenomena to be alone
+interesting to man. With Herschel every star was a solar system. And
+more than that: he found suns revolving round suns, at distances such as
+the mind reels at, still obeying the same law of gravitation as pulls an
+apple from a tree. He tried hard to estimate the distance of the stars
+from the earth, but there he failed: it was too hopeless a problem. It
+was solved some time after his death by Bessel, and the distances of
+many stars are now known but these distances are awful and unspeakable.
+Our distance from the sun shrinks up into a mere speck--the whole solar
+system into a mere unit of measurement, to be repeated hundreds of
+thousands of times before we reach the stars.
+
+Yet their motion is visible--yes, to very accurate measurement quite
+plain. One star, known as 61 Cygni, was then and is now rushing along at
+the rate of 100 miles every second. Not that you must imagine that this
+makes any obvious and apparent change in its position. No, for all
+ordinary and practical purposes they are still fixed stars; thousands of
+years will show us no obvious change; "Adam" saw precisely the same
+constellations as we do: it is only by refined micrometric measurement
+with high magnifying power that their flight can be detected.
+
+But the sun is one of the stars--not by any means a specially large or
+bright one; Sirius we now know to be twenty times as big as the sun. The
+sun is one of the stars: then is it at rest? Herschel asked this
+question and endeavoured to answer it. He succeeded in the most
+astonishing manner. It is, perhaps, his most remarkable discovery, and
+savours of intuition. This is how it happened. With imperfect optical
+means and his own eyesight to guide him, he considered and pondered over
+the proper motion of the stars as he had observed it, till he discovered
+a kind of uniformity running through it all. Mixed up with
+irregularities and individualities, he found that in a certain part of
+the heavens the stars were on the whole opening out--separating slowly
+from each other; on the opposite side of the heavens they were on the
+average closing up--getting slightly nearer to each other; while in
+directions at right angles to this they were fairly preserving their
+customary distances asunder.
+
+Now, what is the moral to be drawn from such uniformity of behaviour
+among unconnected bodies? Surely that this part of their motion is only
+apparent--that it is we who are moving. Travelling over a prairie
+bounded by a belt of trees, we should see the trees in our line of
+advance opening out, and those behind closing up; we should see in fact
+the same kind of apparent motion as Herschel was able to detect among
+the stars: the opening out being most marked near the constellation
+Hercules. The conclusion is obvious: the sun, with all its planets, must
+be steadily moving towards a point in the constellation Hercules. The
+most accurate modern research has been hardly able to improve upon this
+statement of Herschel's. Possibly the solar system may ultimately be
+found to revolve round some other body, but what that is no one knows.
+All one can tell is the present direction of the majestic motion: since
+it was discovered it has continued unchanged, and will probably so
+continue for thousands of years.
+
+[Illustration: FIG. 87.--Old drawing of the cluster in Hercules.]
+
+And, finally, concerning the nebul�. These mysterious objects exercised
+a strong fascination for Herschel, and many are the speculations he
+indulges in concerning them. At one time he regards them all as clusters
+of stars, and the Milky Way as our cluster; the others he regards as
+other universes almost infinitely distant; and he proceeds to gauge and
+estimate the shape of our own universe or galaxy of suns, the Milky Way.
+
+Later on, however, he pictures to himself the nebul� as nascent suns:
+solar systems before they are formed. Some he thinks have begun to
+aggregate, while some are still glowing gas.
+
+[Illustration: FIG. 88.--Old drawing of the Andromeda nebula.]
+
+He likens the heavens to a garden in which there are plants growing in
+all manner of different stages: some shooting, some in leaf, some in
+flower, some bearing seed, some decaying; and thus at one inspection we
+have before us the whole life-history of the plant.
+
+Just so he thinks the heavens contain worlds, some old, some dead, some
+young and vigorous, and some in the act of being formed. The nebul� are
+these latter, and the nebulous stars are a further stage in the
+condensation towards a sun.
+
+And thus, by simple observation, he is led towards something very like
+the nebular hypothesis of Laplace; and his position, whether it be true
+or false, is substantially the same as is held to-day.
+
+[Illustration: FIG. 89.--The great nebula in Orion.]
+
+We _know_ now that many of the nebul� consist of innumerable isolated
+particles and may be spoken of as gas. We know that some are in a state
+of whirling motion. We know also that such gas left to itself will
+slowly as it cools condense and shrink, so as to form a central solid
+nucleus; and also, if it were in whirling motion, that it would send off
+rings from itself, and that these rings could break up into planets. In
+two familiar cases the ring has not yet thus aggregated into planet or
+satellite--the zone of asteroids, and Saturn's ring.
+
+The whole of this could not have been asserted in Herschel's time: for
+further information the world had to wait.
+
+These are the problems of modern astronomy--these and many others, which
+are the growth of this century, aye, and the growth of the last thirty
+or forty, and indeed of the last ten years. Even as I write, new and
+very confirmatory discoveries are being announced. The Milky Way _does_
+seem to have some affinity with our sun. And the chief stars of the
+constellation of Orion constitute another family, and are enveloped in
+the great nebula, now by photography perceived to be far greater than
+had ever been imagined.
+
+What is to be the outcome of it all I know not; but sure I am of this,
+that the largest views of the universe that we are able to frame, and
+the grandest manner of its construction that we can conceive, are
+certain to pale and shrink and become inadequate when confronted with
+the truth.
+
+
+
+
+NOTES TO LECTURE XIII
+
+
+BODE'S LAW.--Write down the series 0, 3, 6, 12, 24, 48, &c.; add 4 to
+each, and divide by 10; you get the series:
+
+    �4       �7    1�0    1�6    2�8    5�2     10�0    19�6   38�8
+  Mercury  Venus  Earth   Mars  ----  Jupiter  Saturn  Uranus  ----
+
+numbers which very fairly represent the distances of the then known
+planets from the sun in the order specified.
+
+Ceres was discovered on the 1st of January, 1801, by Piazzi; Pallas in
+March, 1802, by Olbers; Juno in 1804, by Harding; and Vesta in 1807, by
+Olbers. No more asteroids were discovered till 1845, but there are now
+several hundreds known. Their diameters range from 500 to 20 miles.
+
+Neptune was discovered from the perturbations of Uranus by sheer
+calculation, carried on simultaneously and independently by Leverrier in
+Paris, and Adams in Cambridge. It was first knowingly seen by Galle, of
+Berlin, on the 23rd of September, 1846.
+
+
+
+
+LECTURE XIII
+
+THE DISCOVERY OF THE ASTEROIDS
+
+
+Up to the time of Herschel, astronomical interest centred on the solar
+system. Since that time it has been divided, and a great part of our
+attention has been given to the more distant celestial bodies. The solar
+system has by no means lost its interest--it has indeed gained in
+interest continually, as we gain in knowledge concerning it; but in
+order to follow the course of science it will be necessary for us to
+oscillate to and fro, sometimes attending to the solar system--the
+planets and their satellites--sometimes extending our vision to the
+enormously more distant stellar spaces.
+
+Those who have read the third lecture in Part I. will remember the
+speculation in which Kepler indulged respecting the arrangements of the
+planets, the order in which they succeeded one another in space, and the
+law of their respective distances from the sun; and his fanciful guess
+about the five regular solids inscribed and circumscribed about their
+orbits.
+
+The rude coincidences were, however, accidental, and he failed to
+discover any true law. No thoroughly satisfactory law is known at the
+present day. And yet, if the nebular hypothesis or anything like it be
+true, there must be some law to be discovered hereafter, though it may
+be a very complicated one.
+
+An empirical relation is, however, known: it was suggested by Tatius,
+and published by Bode, of Berlin, in 1772. It is always known as Bode's
+law.
+
+     Bode's law asserts that the distance of each planet is
+     approximately double the distance of the inner adjacent planet from
+     the sun, but that the rate of increase is distinctly slower than
+     this for the inner ones; consequently a better approximation will
+     be obtained by adding a constant to each term of an appropriate
+     geometrical progression. Thus, form a doubling series like this,
+     1-1/2, 3, 6, 12, 24, &c. doubling each time; then add 4 to each,
+     and you get a series which expresses very fairly the relative
+     distances of the successive planets from the sun, except that the
+     number for Mercury is rather erroneous, and we now know that at the
+     other extreme the number for Neptune is erroneous too.
+
+     I have stated it in the notes above in a form calculated to give
+     the law every chance, and a form that was probably fashionable
+     after the discovery of Uranus; but to call the first term of the
+     doubling series 0 is evidently not quite fair, though it puts
+     Mercury's distance right. Neptune's distance, however, turns out to
+     be more nearly 30 times the earth's distance than 38�8. The others
+     are very nearly right: compare column D of the table preceding
+     Lecture III. on p. 57, with the numbers in the notes on p. 294.
+
+The discovery of Uranus a few years afterwards, in 1781, at 19�2 times
+the earth's distance from the sun, lent great _�cl�t_ to the law, and
+seemed to establish its right to be regarded as at least a close
+approximation to the truth.
+
+The gap between Mars and Jupiter, which had often been noticed, and
+which Kepler filled with a hypothetical planet too small to see, comes
+into great prominence by this law of Bode. So much so, that towards the
+end of last century an enthusiastic German, von Zach, after some search
+himself for the expected planet, arranged a committee of observing
+astronomers, or, as he termed it, a body of astronomical detective
+police, to begin a systematic search for this missing subject of the
+sun.
+
+[Illustration: FIG. 90.--Planetary orbits to scale; showing the
+Asteroidal region between Jupiter and Mars. (The orbits of satellites
+are exaggerated.)]
+
+In 1800 the preliminaries were settled: the heavens near the zodiac
+were divided into twenty-four regions, each of which was intrusted to
+one observer to be swept. Meanwhile, however, quite independently of
+these arrangements in Germany, and entirely unknown to this committee, a
+quiet astronomer in Sicily, Piazzi, was engaged in making a catalogue of
+the stars. His attention was directed to a certain region in Taurus by
+an error in a previous catalogue, which contained a star really
+non-existent.
+
+In the course of his scrutiny, on the 1st of January, 1801, he noticed a
+small star which next evening appeared to have shifted. He watched it
+anxiously for successive evenings, and by the 24th of January he was
+quite sure he had got hold of some moving body, not a star: probably, he
+thought, a comet. It was very small, only of the eighth magnitude; and
+he wrote to two astronomers (one of them Bode himself) saying what he
+had observed. He continued to observe till the 11th of February, when he
+was attacked by illness and compelled to cease.
+
+His letters did not reach their destination till the end of March.
+Directly Bode opened his letter he jumped to the conclusion that this
+must be the missing planet. But unfortunately he was unable to verify
+the guess, for the object, whatever it was, had now got too near the sun
+to be seen. It would not be likely to be out again before September, and
+by that time it would be hopelessly lost again, and have just as much to
+be rediscovered as if it had never been seen.
+
+Mathematical astronomers tried to calculate a possible orbit for the
+body from the observations of Piazzi, but the observed places were so
+desperately few and close together. It was like having to determine a
+curve from three points close together. Three observations ought to
+serve,[27] but if they are taken with insufficient interval between
+them it is extremely difficult to construct the whole circumstances of
+the orbit from them. All the calculations gave different results, and
+none were of the slightest use.
+
+The difficulty as it turned out was most fortunate. It resulted in the
+discovery of one of the greatest mathematicians, perhaps the greatest,
+that Germany has ever produced--Gauss. He was then a young man of
+twenty-five, eking out a living by tuition. He had invented but not
+published several powerful mathematical methods (one of them now known
+as "the method of least squares"), and he applied them to Piazzi's
+observations. He was thus able to calculate an orbit, and to predict a
+place where, by the end of the year, the planet should be visible. On
+the 31st of December of that same year, very near the place predicted by
+Gauss, von Zach rediscovered it, and Olbers discovered it also the next
+evening. Piazzi called it Ceres, after the tutelary goddess of Sicily.
+
+Its distance from the sun as determined by Gauss was 2�767 times the
+earth's distance. Bode's law made it 2�8. It was undoubtedly the missing
+planet. But it was only one hundred and fifty or two hundred miles in
+diameter--the smallest heavenly body known at the time of its discovery.
+It revolves the same way as other planets, but the plane of its orbit is
+tilted 10� to the plane of the ecliptic, which was an exceptionally
+large amount.
+
+Very soon, a more surprising discovery followed. Olbers, while searching
+for Ceres, had carefully mapped the part of the heavens where it was
+expected; and in March, 1802, he saw in this place a star he had not
+previously noticed. In two hours he detected its motion, and in a month
+he sent his observations to Gauss, who returned as answer the calculated
+orbit. It was distant 2�67, like Ceres, and was a little smaller, but it
+had a very excentric orbit: its plane being tilted 34-1/2�, an
+extraordinary inclination. This was called Pallas.
+
+Olbers at once surmised that these two planets were fragments of a
+larger one, and kept an eager look out for other fragments.
+
+In two years another was seen, in the course of charting the region of
+the heavens traversed by Ceres and Pallas. It was smaller than either,
+and was called Juno.
+
+In 1807 the persevering search of Olbers resulted in the discovery of
+another, with a very oblique orbit, which Gauss named Vesta. Vesta is
+bigger than any of the others, being five hundred miles in diameter, and
+shines like a star of the sixth magnitude. Gauss by this time had become
+so practised in the difficult computations that he worked out the
+complete orbit of Vesta within ten hours of receiving the observational
+data from Olbers.
+
+For many weary years Olbers kept up a patient and unremitting search for
+more of these small bodies, or fragments of the large planet as he
+thought them; but his patience went unrewarded, and he died in 1840
+without seeing or knowing of any more. In 1845 another was found,
+however, in Germany, and a few weeks later two others by Mr. Hind in
+England. Since then there seems no end to them; numbers have been
+discovered in America, where Professors Peters and Watson have made a
+specialty of them, and have themselves found something like a hundred.
+
+Vesta is the largest--its area being about the same as that of Central
+Europe, without Russia or Spain--and the smallest known is about twenty
+miles in diameter, or with a surface about the size of Kent. The whole
+of them together do not nearly equal the earth in bulk.
+
+The main interest of these bodies to us lies in the question, What is
+their history? Can they have been once a single planet broken up? or are
+they rather an abortive attempt at a planet never yet formed into one?
+
+The question is not _entirely_ settled, but I can tell you which way
+opinion strongly tends at the present time.
+
+Imagine a shell travelling in an elliptic orbit round the earth to
+suddenly explode: the centre of gravity of all its fragments would
+continue moving along precisely the same path as had been traversed by
+the centre of the shell before explosion, and would complete its orbit
+quite undisturbed. Each fragment would describe an orbit of its own,
+because it would be affected by a different initial velocity; but every
+orbit would be a simple ellipse, and consequently every piece would in
+time return through its starting-point--viz. the place at which the
+explosion occurred. If the zone of asteroids had a common point through
+which they all successively passed, they could be unhesitatingly
+asserted to be the remains of an exploded planet. But they have nothing
+of the kind; their orbits are scattered within a certain broad zone--a
+zone everywhere as broad as the earth's distance from the sun,
+92,000,000 miles--with no sort of law indicating an origin of this kind.
+
+It must be admitted, however, that the fragments of our supposed shell
+might in the course of ages, if left to themselves, mutually perturb
+each other into a different arrangement of orbits from that with which
+they began. But their perturbations would be very minute, and moreover,
+on Laplace's theory, would only result in periodic changes, provided
+each mass were rigid. It is probable that the asteroids were at one time
+not rigid, and hence it is difficult to say what may have happened to
+them; but there is not the least reason to believe that their present
+arrangement is derivable in any way from an explosion, and it is certain
+that an enormous time must have elapsed since such an event if it ever
+occurred.
+
+It is far more probable that they never constituted one body at all, but
+are the remains of a cloudy ring thrown off by the solar system in
+shrinking past that point: a small ring after the immense effort which
+produced Jupiter and his satellites: a ring which has aggregated into a
+multitude of little lumps instead of a few big ones. Such an event is
+not unique in the solar system; there is a similar ring round Saturn.
+At first sight, and to ordinary careful inspection, this differs from
+the zone of asteroids in being a solid lump of matter, like a quoit. But
+it is easy to show from the theory of gravitation, that a solid ring
+could not possibly be stable, but would before long get precipitated
+excentrically upon the body of the planet. Devices have been invented,
+such as artfully distributed irregularities calculated to act as
+satellites and maintain stability; but none of these things really work.
+Nor will it do to imagine the rings fluid; they too would destroy each
+other. The mechanical behaviour of a system of rings, on different
+hypotheses as to their constitution, has been worked out with consummate
+skill by Clerk Maxwell; who finds that the only possible constitution
+for Saturn's assemblage of rings is a multitude of discrete particles
+each pursuing its independent orbit. Saturn's ring is, in fact, a very
+concentrated zone of minor asteroids, and there is every reason to
+conclude that the origin of the solar asteroids cannot be very unlike
+the origin of the Saturnian ones. The nebular hypothesis lends itself
+readily to both.
+
+The interlockings and motions of the particles in Saturn's rings are
+most beautiful, and have been worked out and stated by Maxwell with
+marvellous completeness. His paper constituted what is called "The Adams
+Prize Essay" for 1856. Sir George Airy, one of the adjudicators
+(recently Astronomer-Royal), characterized it as "one of the most
+remarkable applications of mathematics to physics that I have ever
+seen."
+
+There are several distinct constituent rings in the entire Saturnian
+zone, and each perturbs the other, with the result that they ripple and
+pulse in concord. The waves thus formed absorb the effect of the mutual
+perturbations, and prevent an accumulation which would be dangerous to
+the persistence of the whole.
+
+The only effect of gravitational perturbation and of collisions is
+gradually to broaden out the whole ring, enlarging its outer and
+diminishing its inner diameter. But if there were any frictional
+resistance in the medium through which the rings spin, then other
+effects would slowly occur, which ought to be looked for with interest.
+So complete and intimate is the way Maxwell works out and describes the
+whole circumstances of the motion of such an assemblage of particles,
+and so cogent his argument as to the necessity that they must move
+precisely so, and no otherwise, else the rings would not be stable, that
+it was a Cambridge joke concerning him that he paid a visit to Saturn
+one evening, and made his observations on the spot.
+
+
+
+
+NOTES TO LECTURE XIV
+
+
+The total number of stars in the heavens visible to a good eye is about
+5,000. The total number at present seen by telescope is about
+50,000,000. The number able to impress a photographic plate has not yet
+been estimated; but it is enormously greater still. Of those which we
+can see in these latitudes, about 14 are of the first magnitude, 48 of
+the second, 152 of the third, 313 of the fourth, 854 of the fifth, and
+2,010 of the sixth; total, 3,391.
+
+The quickest-moving stars known are a double star of the sixth
+magnitude, called 61 Cygni, and one of the seventh magnitude, called
+Groombridge 1830. The velocity of the latter is 200 miles a second. The
+nearest known stars are 61 Cygni and [alpha] Centauri. The distance
+of these from us is about 400,000 times the distance of the sun. Their
+parallax is accordingly half a second of arc. Sirius is more than a
+million times further from us than our sun is, and twenty times as big;
+many of the brightest stars are at more than double this distance. The
+distance of Arcturus is too great to measure even now. Stellar parallax
+was first securely detected in 1838, by Bessel, for 61 Cygni. Bessel was
+born in 1784, and died in 1846, shortly before the discovery of Neptune.
+
+The stars are suns, and are most likely surrounded by planets. One
+planet belonging to Sirius has been discovered. It was predicted by
+Bessel, its position calculated by Peters, and seen by Alvan Clark in
+1862. Another predicted one, belonging to Procyon, has not yet been
+seen.
+
+A velocity of 5 miles a second could carry a projectile right round the
+earth. A velocity of 7 miles a second would carry it away from the
+earth, and round the sun. A velocity of 27 miles a second would carry a
+projectile right out of the solar system never to return.
+
+
+
+
+LECTURE XIV
+
+BESSEL--THE DISTANCES OF THE STARS, AND THE DISCOVERY OF STELLAR PLANETS
+
+
+We will now leave the solar system for a time, and hastily sketch the
+history of stellar astronomy from the time of Sir William Herschel.
+
+You remember how greatly Herschel had changed the aspect of the heavens
+for man,--how he had found that none of the stars were really fixed, but
+were moving in all manner of ways: some of this motion only apparent,
+much of it real. Nevertheless, so enormously distant are they, that if
+we could be transported back to the days of the old Chald�an
+astronomers, or to the days of Noah, we should still see the heavens
+with precisely the same aspect as they wear now. Only by refined
+apparatus could any change be discoverable in all those centuries. For
+all practical purposes, therefore, the stars may still be well called
+fixed.
+
+Another thing one may notice, as showing their enormous distances, is
+that from every planet of the solar system the aspect of the heavens
+will be precisely the same. Inhabitants of Mars, or Jupiter, or Saturn,
+or Uranus, will see exactly the same constellations as we do. The whole
+dimensions of the solar system shrink up into a speck when so
+contemplated. And from the stars none of the planetary orbs of our
+system are visible at all; nothing but the sun is visible, and that
+merely as a twinkling star, brighter than some, but fainter than many
+others.
+
+The sun and the stars are one. Try to realize this distinctly, and keep
+it in mind. I find it often difficult to drive this idea home. After
+some talk on the subject a friendly auditor will report, "the lecturer
+then described the stars, including that greatest and most magnificent
+of all stars, the sun." It would be difficult more completely to
+misapprehend the entire statement. When I say the sun is one of the
+stars, I mean one among the others; we are a long way from them, they
+are a long way from each other. They need be no more closely packed
+among each other than we are closely packed among them; except that some
+of them are double or multiple, and we are not double.
+
+     It is highly desirable to acquire an intimate knowledge of the
+     constellations and a nodding acquaintance with their principal
+     stars. A description of their peculiarities is dull and
+     uninteresting unless they are at least familiar by name. A little
+     _viv� voce_ help to begin with, supplemented by patient night
+     scrutiny with a celestial globe or star maps under a tent or shed,
+     is perhaps the easiest way: a very convenient instrument for the
+     purpose of learning the constellations is the form of map called a
+     "planisphere," because it can be made to show all the
+     constellations visible at a given time at a given date, and no
+     others. The Greek alphabet also is a thing that should be learnt by
+     everybody. The increased difficulty in teaching science owing to
+     the modern ignorance of even a smattering of Greek is becoming
+     grotesque. The stars are named from their ancient grouping into
+     constellations, and by the prefix of a Greek letter to the larger
+     ones, and of numerals to the smaller ones. The biggest of all have
+     special Arabic names as well. The brightest stars are called of
+     "the first magnitude," the next are of "the second magnitude," and
+     so on. But this arrangement into magnitudes has become technical
+     and precise, and intermediate or fractional magnitudes are
+     inserted. Those brighter than the ordinary first magnitude are
+     therefore now spoken of as of magnitude 1/2, for instance, or �6,
+     which is rather confusing. Small telescopic stars are often only
+     named by their numbers in some specified catalogue--a dull but
+     sufficient method.
+
+     Here is a list of the stars visible from these latitudes, which are
+     popularly considered as of the first magnitude. All of them should
+     be familiarly recognized in the heavens, whenever seen.
+
+  Star.                Constellation.
+
+  Sirius               Canis major
+  Procyon              Canis minor
+  Rigel                Orion
+  Betelgeux            Orion
+  Castor               Gemini
+  Pollux               Gemini
+  Aldebaran            Taurus
+  Arcturus             Bo�tes
+  Vega                 Lyra
+  Capella              Auriga
+  Regulus              Leo
+  Altair               Aquila
+  Fomalhaut            Southern Fish
+  Spica                Virgo
+
+     [alpha] Cygni is a little below the first magnitude. So,
+     perhaps, is Castor. In the southern heavens, Canopus and [alpha]
+     Centauri rank next after Sirius in brightness.
+
+[Illustration: FIG. 91.--Diagram illustrating Parallax.]
+
+The distances of the fixed stars had, we know, been a perennial problem,
+and many had been the attempts to solve it. All the methods of any
+precision have depended on the Copernican fact that the earth in June
+was 184 million miles away from its position in December, and that
+accordingly the grouping and aspect of the heavens should be somewhat
+different when seen from so different a point of view. An apparent
+change of this sort is called generally parallax; _the_ parallax of a
+star being technically defined as the angle subtended at the star by the
+radius of the earth's orbit: that is to say, the angle E[sigma]S;
+where E is the earth, S the sun, and [sigma] a star (Fig. 91).
+
+Plainly, the further off [sigma] is, the more nearly parallel will
+the two lines to it become. And the difficulty of determining the
+parallax was just this, that the more accurately the observations were
+made, the more nearly parallel did those lines become. The angle was, in
+fact, just as likely to turn out negative as positive--an absurd result,
+of course, to be attributed to unavoidable very minute inaccuracies.
+
+For a long time absolute methods of determining parallax were attempted;
+for instance, by observing the position of the star with respect to the
+zenith at different seasons of the year. And many of these
+determinations appeared to result in success. Hooke fancied he had
+measured a parallax for Vega in this way, amounting to 30" of arc.
+Flamsteed obtained 40" for [gamma] Draconis. Roemer made a serious
+attempt by comparing observations of Vega and Sirius, stars almost the
+antipodes of each other in the celestial vault; hoping to detect some
+effect due to the size of the earth's orbit, which should apparently
+displace them with the season of the year. All these fancied results
+however, were shown to be spurious, and their real cause assigned, by
+the great discovery of the aberration of light by Bradley.
+
+After this discovery it was possible to watch for still outstanding very
+minute discrepancies; and so the problem of stellar parallax was
+attacked with fresh vigour by Piazzi, by Brinkley, and by Struve. But
+when results were obtained, they were traced after long discussion to
+age and gradual wear of the instrument, or to some other minute
+inaccuracy. The more carefully the observation was made, the more nearly
+zero became the parallax--the more nearly infinite the distance of the
+stars. The brightest stars were the ones commonly chosen for the
+investigation, and Vega was a favourite, because, going near the zenith,
+it was far removed from the fluctuating and tiresome disturbances of
+atmospheric refraction. The reason bright stars were chosen was because
+they were presumably nearer than the others; and indeed a rough guess at
+their probable distance was made by supposing them to be of the same
+size as the sun, and estimating their light in comparison with sunlight.
+By this confessedly unsatisfactory method it had been estimated that
+Sirius must be 140,000 times further away than the sun is, if he be
+equally big. We now know that Sirius is much further off than this; and
+accordingly that he is much brighter, perhaps sixty times as bright,
+though not necessarily sixty times as big, as our sun. But even
+supposing him of the same light-giving power as the sun, his parallax
+was estimated as 1"�8, a quantity very difficult to be sure of in any
+absolute determination.
+
+Relative methods were, however, also employed, and the advantages of one
+of these (which seems to have been suggested by Galileo) so impressed
+themselves upon William Herschel that he made a serious attempt to
+compass the problem by its means. The method was to take two stars in
+the same telescopic field and carefully to estimate their apparent
+angular distance from each other at different seasons of the year. All
+such disturbances as precession, aberration, nutation, refraction, and
+the like, would affect them both equally, and could thus be eliminated.
+If they were at the same distance from the solar system, relative
+parallax would, indeed, also be eliminated; but if, as was probable,
+they were at different distances, then they would apparently shift
+relatively to one another, and the amount of shift, if it could be
+observed, would measure, not indeed the distance of either from the
+earth, but their distance from each other. And this at any rate would be
+a step. It might be completed by similarly treating other stars in the
+same field, taking them in pairs together. A bright and a faint star
+would naturally be suitable, because their distances were likely to be
+unequal; and so Herschel fixed upon a number of doublets which he knew
+of, containing one bright and one faint component. For up to that time
+it had been supposed that such grouping in occasional pairs or triplets
+was chance coincidence, the two being optically foreshortened together,
+but having no real connection or proximity. Herschel failed in what he
+was looking for, but instead of that he discovered the real connection
+of a number of these doublets, for he found that they were slowly
+revolving round each other. There are a certain number of merely optical
+or accidental doublets, but the majority of them are real pairs of suns
+revolving round each other.
+
+This relative method of mapping micrometrically a field of neighbouring
+stars, and comparing their configuration now and six months hence, was,
+however, the method ultimately destined to succeed; and it is, I
+believe, the only method which has succeeded down to the present day.
+Certainly it is the method regularly employed, at Dunsink, at the Cape
+of Good Hope, and everywhere else where stellar parallax is part of the
+work.
+
+Between 1830 and 1840 the question was ripe for settlement, and, as
+frequently happens with a long-matured difficulty, it gave way in three
+places at once. Bessel, Henderson, and Struve almost simultaneously
+announced a stellar parallax which could reasonably be accepted. Bessel
+was a little the earliest, and by far the most accurate. His, indeed,
+was the result which commanded confidence, and to him the palm must be
+awarded.
+
+He was largely a self-taught student, having begun life in a
+counting-house, and having abandoned business for astronomy. But
+notwithstanding these disadvantages, he became a highly competent
+mathematician as well as a skilful practical astronomer. He was
+appointed to superintend the construction of Germany's first great
+astronomical observatory, that of K�nigsberg, which, by his system,
+zeal, and genius, he rapidly made a place of the first importance.
+
+Struve at Dorpat, Bessel at K�nigsberg, and Henderson at the Cape of
+Good Hope--all of them at newly-equipped observatories--were severally
+engaged at the same problem.
+
+But the Russian and German observers had the advantage of the work of
+one of the most brilliant opticians--I suppose the most brilliant--that
+has yet appeared: Fraunhofer, of Munich. An orphan lad, apprenticed to a
+maker of looking-glasses, and subject to hard struggles and privations
+in early life, he struggled upwards, and ultimately became head of the
+optical department of a Munich firm of telescope-makers. Here he
+constructed the famous "Dorpat refractor" for Struve, which is still at
+work; and designed the "K�nigsberg heliometer" for Bessel. He also made
+a long and most skilful research into the solar spectrum, which has
+immortalized his name. But his health was broken by early trials, and he
+died at the age of thirty-nine, while planning new and still more
+important optical achievements.
+
+A heliometer is the most accurate astronomical instrument for relative
+measurements of position, as a transit circle is the most accurate for
+absolute determinations. It consists of an equatorial telescope with
+object-glass cut right across, and each half movable by a sliding
+movement one past the other, the amount by which the two halves are
+dislocated being read off by a refined method, and the whole instrument
+having a multitude of appendages conducive to convenience and accuracy.
+Its use is to act as a micrometer or measurer of small distances.[28]
+Each half of the object-glass gives a distinct image, which may be
+allowed to coincide or may be separated as occasion requires. If it be
+the components of a double star that are being examined, each component
+will in general be seen double, so that four images will be seen
+altogether; but by careful adjustment it will be possible to arrange
+that one image of each pair shall be superposed on or coincide with each
+other, in which case only three images are visible; the amount of
+dislocation of the halves of the object-glass necessary to accomplish
+this is what is read off. The adjustment is one that can be performed
+with extreme accuracy, and by performing it again and again with all
+possible modifications, an extremely accurate determination of the
+angular distance between the two components is obtained.
+
+[Illustration: FIG. 92.--Heliometer.]
+
+Bessel determined to apply this beautiful instrument to the problem of
+stellar parallax; and he began by considering carefully the kind of star
+for which success was most likely. Hitherto the brightest had been most
+attended to, but Bessel thought that quickness of proper motion would be
+a still better test of nearness. Not that either criterion is conclusive
+as to distance, but there was a presumption in favour of either a very
+bright or an obviously moving star being nearer than a faint or a
+stationary one; and as the "bright" criterion had already been often
+applied without result, he decided to try the other. He had already
+called attention to a record by Piazzi in 1792 of a double star in
+Cygnus whose proper motion was five seconds of arc every year--a motion
+which caused this telescopic object, 61 Cygni, to be known as "the
+flying star." Its motion is not really very perceptible, for it will
+only have traversed one-third of a lunar diameter in the course of a
+century; still it was the quickest moving star then known. The position
+of this interesting double he compared with two other stars which were
+seen simultaneously in the field of the heliometer, by the method I have
+described, throughout the whole year 1838; and in the last month of that
+year he was able to announce with confidence a distinct though very
+small parallax; substantiating it with a mass of detailed evidence which
+commanded the assent of astronomers. The amount of it he gave as
+one-third of a second. We know now that he was very nearly right, though
+modern research makes it more like half a second.[29]
+
+Soon afterwards, Struve announced a quarter of a second as the parallax
+of Vega, but that is distinctly too great; and Henderson announced for
+[alpha] Centauri (then thought to be a double) a parallax of one
+second, which, if correct, would make it quite the nearest of all the
+stars, but the result is now believed to be about twice too big.
+
+Knowing the distance of 61 Cygni, we can at once tell its real rate of
+travel--at least, its rate across our line of sight: it is rather over
+three million miles a day.
+
+Now just consider the smallness of the half second of arc, thus
+triumphantly though only approximately measured. It is the angle
+subtended by twenty-six feet at a distance of 2,000 miles. If a
+telescope planted at New York could be directed to a house in England,
+and be then turned so as to set its cross-wire first on one end of an
+ordinary room and then on the other end of the same room, it would have
+turned through half a second, the angle of greatest stellar parallax.
+Or, putting it another way. If the star were as near us as New York is,
+the sun, on the same scale, would be nine paces off. As twenty-six feet
+is to the distance of New York, so is ninety-two million miles to the
+distance of the nearest fixed star.
+
+Suppose you could arrange some sort of telegraphic vehicle able to carry
+you from here to New York in the tenth part of a second--_i.e._ in the
+time required to drop two inches--such a vehicle would carry you to the
+moon in twelve seconds, to the sun in an hour and a quarter. Travelling
+thus continually, in twenty-four hours you would leave the last member
+of the solar system behind you, and begin your plunge into the depths of
+space. How long would it be before you encountered another object? A
+month, should you guess? Twenty years you must journey with that
+prodigious speed before you reach the nearest star, and then another
+twenty years before you reach another. At these awful distances from one
+another the stars are scattered in space, and were they not brilliantly
+self-luminous and glowing like our sun, they would be hopelessly
+invisible.
+
+I have spoken of 61 Cygni as a flying star, but there is another which
+goes still quicker, a faint star, 1830 in Groombridge's Catalogue. Its
+distance is far greater than that of 61 Cygni, and yet it is seen to
+move almost as quickly. Its actual speed is about 200 miles a
+second--greater than the whole visible firmament of fifty million stars
+can control; and unless the universe is immensely larger than anything
+we can see with the most powerful telescopes, or unless there are crowds
+of invisible non-luminous stars mixed up with the others, it can only be
+a temporary visitor to this frame of things; it is rushing from an
+infinite distance to an infinite distance; it is passing through our
+visible universe for the first and only time--it will never return. But
+so gigantic is the extent of visible space, that even with its amazing
+speed of 200 miles every second, this star will take two or three
+million years to get out of sight of our present telescopes, and several
+thousand years before it gets perceptibly fainter than it is now.
+
+Have we any reason for supposing that the stars we see are all there
+are? In other words, have we any reason for supposing all celestial
+objects to be sufficiently luminous to be visible? We have every ground
+for believing the contrary. Every body in the solar system is dull and
+dark except the sun, though probably Jupiter is still red-hot. Why may
+not some of the stars be dark too? The genius of Bessel surmised this,
+and consistently upheld the doctrine that the astronomy of the future
+would have to concern itself with dark and invisible bodies; he preached
+"an astronomy of the invisible." Moreover he predicted the presence of
+two such dark bodies--one a companion of Sirius, the other of Procyon.
+He noticed certain irregularities in the motions of these stars which he
+asserted must be caused by their revolving round other bodies in a
+period of half a century. He announced in 1844 that both Sirius and
+Procyon were double stars, but that their companions, though large, were
+dark, and therefore invisible.
+
+No one accepted this view, till Peters, in America, found in 1851 that
+the hypothesis accurately explained the anomalous motion of Sirius, and,
+in fact, indicated an exact place where the companion ought to be. The
+obscure companion of Sirius became now a recognized celestial object,
+although it had never been seen, and it was held to revolve round Sirius
+in fifty years, and to be about half as big.
+
+In 1862, the firm of Alvan Clark and Sons, of New York, were completing
+a magnificent 18-inch refractor, and the younger Clark was trying it on
+Sirius, when he said: "Why, father, the star has a companion!" The elder
+Clark also looked, and sure enough there was a faint companion due east
+of the bright star, and in just the position required by theory. Not
+that the Clarks knew anything about the theory. They were keen-sighted
+and most skilful instrument-makers, and they made the discovery by
+accident. After it had once been seen, it was found that several of the
+large telescopes of the world were able to show it. It is half as big,
+but it only gives 1/10000th part of the light that Sirius gives. No
+doubt it shines partly with a borrowed light and partly with a dull heat
+of its own. It is a real planet, but as yet too hot to live on. It will
+cool down in time, as our earth has cooled and as Jupiter is cooling,
+and no doubt become habitable enough. It does revolve round Sirius in a
+period of 49�4 years--almost exactly what Bessel assigned to it.
+
+But Bessel also assigned a dark companion to Procyon. It and its
+luminous neighbour are considered to revolve round each other in a
+period of forty years, and astronomers feel perfectly assured of its
+existence, though at present it has not been seen by man.
+
+
+
+
+LECTURE XV
+
+THE DISCOVERY OF NEPTUNE
+
+
+We approach to-night perhaps the greatest, certainly the most
+conspicuous, triumphs of the theory of gravitation. The explanation by
+Newton of the observed facts of the motion of the moon, the way he
+accounted for precession and nutation and for the tides, the way in
+which Laplace explained every detail of the planetary motions--these
+achievements may seem to the professional astronomer equally, if not
+more, striking and wonderful; but of the facts to be explained in these
+cases the general public are necessarily more or less ignorant, and so
+no beauty or thoroughness of treatment appeals to them, nor can excite
+their imaginations. But to predict in the solitude of the study, with no
+weapons other than pen, ink, and paper, an unknown and enormously
+distant world, to calculate its orbit when as yet it had never been
+seen, and to be able to say to a practical astronomer, "Point your
+telescope in such a direction at such a time, and you will see a new
+planet hitherto unknown to man"--this must always appeal to the
+imagination with dramatic intensity, and must awaken some interest in
+almost the dullest.
+
+Prediction is no novelty in science; and in astronomy least of all is it
+a novelty. Thousands of years ago, Thales, and others whose very names
+we have forgotten, could predict eclipses with some certainty, though
+with only rough accuracy. And many other phenomena were capable of
+prediction by accumulated experience. We have seen, for instance (coming
+to later times), how a gap between Mars and Jupiter caused a missing
+planet to be suspected and looked for, and to be found in a hundred
+pieces. We have seen, also, how the abnormal proper-motion of Sirius
+suggested to Bessel the existence of an unseen companion. And these last
+instances seem to approach very near the same class of prediction as
+that of the discovery of Neptune. Wherein, then, lies the difference?
+How comes it that some classes of prediction--such as that if you put
+your finger in fire it will get burnt--are childishly easy and
+commonplace, while others excite in the keenest intellects the highest
+feelings of admiration? Mainly, the difference lies, first, in the
+grounds on which the prediction is based; second, on the difficulty of
+the investigation whereby it is accomplished; third, in the completeness
+and the accuracy with which it can be verified. In all these points, the
+discovery of Neptune stands out pre-eminently among the verified
+predictions of science, and the circumstances surrounding it are of
+singular interest.
+
+*      *      *      *      *
+
+In 1781, Sir William Herschel discovered the planet Uranus. Now you know
+that three distinct observations suffice to determine the orbit of a
+planet completely, and that it is well to have the three observations as
+far apart as possible so as to minimize the effects of minute but
+necessary errors of observation. (See p. 298.) Directly Uranus was
+found, therefore, old records of stellar observations were ransacked,
+with the object of discovering whether it had ever been unwittingly seen
+before. If seen, it had been thought of course to be a star (for it
+shines like a star of the sixth magnitude, and can therefore be just
+seen without a telescope if one knows precisely where to look for it,
+and if one has good sight), but if it had been seen and catalogued as a
+star it would have moved from its place, and the catalogue would by that
+entry be wrong. The thing to detect, therefore, was errors in the
+catalogues: to examine all entries, and see if the stars entered
+actually existed, or were any of them missing. If a wrong entry were
+discovered, it might of course have been due to some clerical error,
+though that is hardly probable considering the care taken over these
+things, or it might have been some tailless comet or other, or it might
+have been the newly found planet.
+
+So the next thing was to calculate backwards, and see if by any
+possibility the planet could have been in that place at that time.
+Examined in this way the tabulated observations of Flamsteed showed that
+he had unwittingly observed Uranus five distinct times, the first time
+in 1690, nearly a century before Herschel discovered its true nature.
+But more remarkable still, Le Monnier, of Paris, had observed it eight
+times in one month, cataloguing it each time as a different star. If
+only he had reduced and compared his observations, he would have
+anticipated Herschel by twelve years. As it was, he missed it
+altogether. It was seen once by Bradley also. Altogether it had been
+seen twenty times.
+
+These old observations of Flamsteed and those of Le Monnier, combined
+with those made after Herschel's discovery, were very useful in
+determining an exact orbit for the new planet, and its motion was
+considered thoroughly known. It was not an _exact_ ellipse, of course:
+none of the planets describe _exact_ ellipses--each perturbs all the
+rest, and these small perturbations must be taken into account, those of
+Jupiter and Saturn being by far the most important.
+
+For a time Uranus seemed to travel regularly and as expected, in the
+orbit which had been calculated for it; but early in the present century
+it began to be slightly refractory, and by 1820 its actual place showed
+quite a distinct discrepancy from its position as calculated with the
+aid of the old observations. It was at first thought that this
+discrepancy must be due to inaccuracies in the older observations, and
+they were accordingly rejected, and tables prepared for the planet based
+on the newer and more accurate observations only. But by 1830 it became
+apparent that it would not accurately obey even these. The error
+amounted to some 20". By 1840 it was as much as 90', or a minute and a
+half. This discrepancy is quite distinct, but still it is very small,
+and had two objects been in the heavens at once, the actual Uranus and
+the theoretical Uranus, no unaided eye could possibly have distinguished
+them or detected that they were other than a single star.
+
+[Illustration: FIG. 93.--Perturbations of Uranus.
+
+The chance observations by Flamsteed, by Le Monnier, and others, are
+plotted in this diagram, as well as the modern determinations made after
+Herschel had discovered the nature of the planet. The decades are laid
+off horizontally. Vertical distance represents the difference between
+observed and subsequently calculated longitudes--in other words, the
+principal perturbations caused by Neptune. To show the scale, a number
+of standard things are represented too by lengths measured upwards from
+the line of time, viz: the smallest quantity perceptible to the naked
+eye,--the maximum angle of aberration, of nutation, and of stellar
+parallax; though this last is too small to be properly indicated. The
+perturbations are much bigger than these; but compared with what can be
+seen without a telescope they are small--the distance between the
+component pairs of [epsilon] Lyr� (210") (see fig. 86, page 288), which
+a few keen-eyed persons can see as a simple double star, being about
+twice the greatest perturbation.]
+
+The diagram shows all the irregularities plotted in the light of our
+present knowledge; and, to compare with their amounts, a few standard
+things are placed on the same scale, such as the smallest interval
+capable of being detected with the unaided eye, the distance of the
+component stars in [epsilon] Lyr�, the constants of aberration, of
+nutation, and of stellar parallax.
+
+The errors of Uranus therefore, though small, were enormously greater
+than things which had certainly been observed; there was an unmistakable
+discrepancy between theory and observation. Some cause was evidently at
+work on this distant planet, causing it to disagree with its motion as
+calculated according to the law of gravitation. Some thought that the
+exact law of gravitation did not apply to so distant a body. Others
+surmised the presence of some foreign and unknown body, some comet, or
+some still more distant planet perhaps, whose gravitative attraction for
+Uranus was the cause of the whole difficulty--some perturbations, in
+fact, which had not been taken into account because of our ignorance of
+the existence of the body which caused them.
+
+But though such an idea was mentioned among astronomers, it was not
+regarded with any special favour, and was considered merely as one among
+a number of hypotheses which could be suggested as fairly probable.
+
+It is perfectly right not to attach much importance to unelaborated
+guesses. Not until the consequences of an hypothesis have been
+laboriously worked out--not until it can be shown capable of producing
+the effect quantitatively as well as qualitatively--does its statement
+rise above the level of a guess, and attain the dignity of a theory. A
+later stage still occurs when the theory has been actually and
+completely verified by agreement with observation.
+
+     Now the errors in the motion of Uranus, _i.e._ the discrepancy
+     between its observed and calculated longitudes--all known
+     disturbing causes, such as Jupiter and Saturn, being allowed
+     for--are as follows (as quoted by Dr. Haughton) in seconds of
+     arc:--
+
+  ANCIENT OBSERVATIONS (casually made, as of a star).
+
+  Flamsteed         1690    +61�2
+      "             1712    +92�7
+      "             1715    +73�8
+  Le Monnier        1750    -47�6
+  Bradley           1753    -39�5
+  Mayer             1756    -45�7
+  Le Monnier        1764    -34�9
+     "              1769    -19�3
+     "              1771     -2�3
+
+  MODERN OBSERVATIONS.
+
+  1780                 +3�46
+  1783                 +8�45
+  1786                +12�36
+  1789                +19�02
+  1801                +22�21
+  1810                +23�16
+  1822                +20�97
+  1825                +18�16
+  1828                +10�82
+  1831                 -3�98
+  1834                -20�80
+  1837                -42�66
+  1840                -66�64
+
+     These are the numbers plotted in the above diagram (Fig. 92), where
+     H marks the discovery of the planet and the beginning of its
+     regular observation.
+
+Something was evidently the matter with the planet. If the law of
+gravitation held exactly at so great a distance from the sun, there must
+be some perturbing force acting on it besides all those known ones which
+had been fully taken into account. Could it be an outer planet? The
+question occurred to several, and one or two tried if they could solve
+the problem, but were soon stopped by the tremendous difficulties of
+calculation.
+
+The ordinary problem of perturbation is difficult enough: Given a
+disturbing planet in such and such a position, to find the perturbations
+it produces. This problem it was that Laplace worked out in the
+_M�canique C�leste_.
+
+But the inverse problem: Given the perturbations, to find the planet
+which causes them--such a problem had never yet been attacked, and by
+only a few had its possibility been conceived. Bessel made preparations
+for trying what he could do at it in 1840, but he was prevented by fatal
+illness.
+
+In 1841 the difficulties of the problem presented by these residual
+perturbations of Uranus excited the imagination of a young student, an
+undergraduate of St. John's College, Cambridge--John Couch Adams by
+name--and he determined to have a try at it as soon as he was through
+his Tripos. In January, 1843, he graduated as Senior Wrangler, and
+shortly afterwards he set to work. In less than two years he reached a
+definite conclusion; and in October, 1845, he wrote to the
+Astronomer-Royal, at Greenwich, Professor Airy, saying that the
+perturbations of Uranus would be explained by assuming the existence of
+an outer planet, which he reckoned was now situated in a specified
+latitude and longitude.
+
+We know now that had the Astronomer-Royal put sufficient faith in this
+result to point his big telescope to the spot indicated and commence
+sweeping for a planet, he would have detected it within 1-3/4� of the
+place assigned to it by Mr. Adams. But any one in the position of the
+Astronomer-Royal knows that almost every post brings an absurd letter
+from some ambitious correspondent or other, some of them having just
+discovered perpetual motion, or squared the circle, or proved the earth
+flat, or discovered the constitution of the moon, or of ether, or of
+electricity; and out of this mass of rubbish it requires great skill and
+patience to detect such gems of value as there may be.
+
+Now this letter of Mr. Adams's was indeed a jewel of the first water,
+and no doubt bore on its face a very different appearance from the
+chaff of which I have spoken; but still Mr. Adams was an unknown man: he
+had graduated as Senior Wrangler it is true, but somebody must graduate
+as Senior Wrangler every year, and every year by no means produces a
+first-rate mathematician. Those behind the scenes, as Professor Airy of
+course was, having been a Senior Wrangler himself, knew perfectly well
+that the labelling of a young man on taking his degree is much more
+worthless as a testimony to his genius and ability than the general
+public are apt to suppose.
+
+Was it likely that a young and unknown man should have successfully
+solved so extremely difficult a problem? It was altogether unlikely.
+Still, he would test him: he would ask for further explanations
+concerning some of the perturbations which he himself had specially
+noticed, and see if Mr. Adams could explain these also by his
+hypothesis. If he could, there might be something in his theory. If he
+failed--well, there was an end of it. The questions were not difficult.
+They concerned the error of the radius vector. Mr. Adams could have
+answered them with perfect ease; but sad to say, though a brilliant
+mathematician, he was not a man of business. He did not answer Professor
+Airy's letter.
+
+It may to many seem a pity that the Greenwich Equatoreal was not pointed
+to the place, just to see whether any foreign object did happen to be in
+that neighbourhood; but it is no light matter to derange the work of an
+Observatory, and alter the work mapped out for the staff into a sudden
+sweep for a new planet, on the strength of a mathematical investigation
+just received by post. If observatories were conducted on these
+unsystematic and spasmodic principles, they would not be the calm,
+accurate, satisfactory places they are.
+
+Of course, if any one could have known that a new planet was to be had
+for the looking, _any_ course would have been justified; but no one
+could know this. I do not suppose that Mr. Adams himself could feel all
+that confidence in his attempted prediction. So there the matter
+dropped. Mr. Adams's communication was pigeon-holed, and remained in
+seclusion for eight or nine months.
+
+Meanwhile, and quite independently, something of the same sort was going
+on in France. A brilliant young mathematician, born in Normandy in 1811,
+had accepted the post of Astronomical Professor at the �cole
+Polytechnique, then recently founded by Napoleon. His first published
+papers directed attention to his wonderful powers; and the official head
+of astronomy in France, the famous Arago, suggested to him the
+unexplained perturbations of Uranus as a worthy object for his fresh and
+well-armed vigour.
+
+At once he set to work in a thorough and systematic way. He first
+considered whether the discrepancies could be due to errors in the
+tables or errors in the old observations. He discussed them with minute
+care, and came to the conclusion that they were not thus to be explained
+away. This part of the work he published in November, 1845.
+
+He then set to work to consider the perturbations produced by Jupiter
+and Saturn, to see if they had been with perfect accuracy allowed for,
+or whether some minute improvements could be made sufficient to destroy
+the irregularities. He introduced several fresh terms into these
+perturbations, but none of them of sufficient magnitude to do more than
+slightly lessen the unexplained perturbations.
+
+He next examined the various hypotheses that had been suggested to
+account for them:--Was it a failure in the law of gravitation? Was it
+due to the presence of a resisting medium? Was it due to some unseen but
+large satellite? Or was it due to a collision with some comet?
+
+All these he examined and dismissed for various reasons one after the
+other. It was due to some steady continuous cause--for instance, some
+unknown planet. Could this planet be inside the orbit of Uranus? No, for
+then it would perturb Saturn and Jupiter also, and they were not
+perturbed by it. It must, therefore, be some planet outside the orbit of
+Uranus, and in all probability, according to Bode's empirical law, at
+nearly double the distance from the sun that Uranus is. Lastly he
+proceeded to examine where this planet was, and what its orbit must be
+to produce the observed disturbances.
+
+[Illustration: FIG. 94.--Uranus's and Neptune's relative positions.
+
+The above diagram, drawn to scale by Dr. Haughton, shows the paths of
+Uranus and Neptune, and their positions from 1781 to 1840, and
+illustrates the _direction_ of their mutual perturbing force. In 1822
+the planets were in conjunction, and the force would then perturb the
+radius vector (or distance from the sun), but not the longitude (or
+place in orbit). Before that date Uranus had been hurried along, and
+after that date it had been retarded, by the pull of Neptune, and thus
+the observed discrepancies from its computed place were produced. The
+problem was first to disentangle the outstanding perturbations from
+those which would be caused by Jupiter and Saturn and all other known
+causes, and then to assign the place of an outer planet able to produce
+precisely those perturbations in Uranus.]
+
+Not without failures and disheartening complications was this part of
+the process completed. This was, after all, the real tug of war. So many
+unknown quantities: its mass, its distance, its excentricity, the
+obliquity of its orbit, its position at any time--nothing known, in
+fact, about the planet except the microscopic disturbance it caused in
+Uranus, some thousand million miles away from it.
+
+Without going into further detail, suffice it to say that in June, 1846,
+he published his last paper, and in it announced to the world his
+theoretical position for the planet.
+
+Professor Airy received a copy of this paper before the end of the
+month, and was astonished to find that Leverrier's theoretical place for
+the planet was within 1� of the place Mr. Adams had assigned to it eight
+months before. So striking a coincidence seemed sufficient to justify a
+Herschelian "sweep" for a week or two.
+
+But a sweep for so distant a planet would be no easy matter. When seen
+in a large telescope it would still only look like a star, and it would
+require considerable labour and watching to sift it out from the other
+stars surrounding it. We know that Uranus had been seen twenty times,
+and thought to be a star, before its true nature was by Herschel
+discovered; and Uranus is only about half as far away as Neptune is.
+
+Neither in Paris nor yet at Greenwich was any optical search undertaken;
+but Professor Airy wrote to ask M. Leverrier the same old question as he
+had fruitlessly put to Mr. Adams: Did the new theory explain the errors
+of the radius vector or not? The reply of Leverrier was both prompt and
+satisfactory--these errors were explained, as well as all the others.
+The existence of the object was then for the first time officially
+believed in.
+
+The British Association met that year at Southampton, and Sir John
+Herschel was one of its Sectional Presidents. In his inaugural address,
+on September 10th, 1846, he called attention to the researches of
+Leverrier and Adams in these memorable words:--
+
+     "The past year has given to us the new [minor] planet Astr�a; it
+     has done more--it has given us the probable prospect of another.
+     We see it as Columbus saw America from the shores of Spain. Its
+     movements have been felt trembling along the far-reaching line of
+     our analysis with a certainty hardly inferior to ocular
+     demonstration."
+
+It was about time to begin to look for it. So the Astronomer-Royal
+thought on reading Leverrier's paper. But as the national telescope at
+Greenwich was otherwise occupied, he wrote to Professor Challis, at
+Cambridge, to know if he would permit a search to be made for it with
+the Northumberland Equatoreal, the large telescope of Cambridge
+University, presented to it by one of the Dukes of Northumberland.
+
+Professor Challis said he would conduct the search himself; and shortly
+commenced a leisurely and dignified series of sweeps round about the
+place assigned by theory, cataloguing all the stars which he observed,
+intending afterwards to sort out his observations, compare one with
+another, and find out whether any one star had changed its position;
+because if it had it must be the planet. He thus, without giving an
+excessive time to the business, accumulated a host of observations,
+which he intended afterwards to reduce and sift at his leisure.
+
+The wretched man thus actually saw the planet twice--on August 4th and
+August 12th, 1846--without knowing it. If only he had had a map of the
+heavens containing telescopic stars down to the tenth magnitude, and if
+he had compared his observations with this map as they were made, the
+process would have been easy, and the discovery quick. But he had no
+such map. Nevertheless one was in existence: it had just been completed
+in that country of enlightened method and industry--Germany. Dr.
+Bremiker had not, indeed, completed his great work--a chart of the whole
+zodiac down to stars of the tenth magnitude--but portions of it were
+completed, and the special region where the new planet was expected
+happened to be among the portions already just done. But in England
+this was not known.
+
+Meanwhile, Mr. Adams wrote to the Astronomer-Royal several additional
+communications, making improvements in his theory, and giving what he
+considered nearer and nearer approximations for the place of the planet.
+He also now answered quite satisfactorily, but too late, the question
+about the radius vector sent to him months before.
+
+Let us return to Leverrier. This great man was likewise engaged in
+improving his theory and in considering how best the optical search
+could be conducted. Actuated, probably, by the knowledge that in such
+matters as cataloguing and mapping Germany was then, as now, far ahead
+of all the other nations of the world, he wrote in September (the same
+September as Sir John Herschel delivered his eloquent address at
+Southampton) to Berlin. Leverrier wrote, I say, to Dr. Galle, head of
+the Observatory at Berlin, saying to him, clearly and decidedly, that
+the new planet was now in or close to such and such a position, and that
+if he would point his telescope to that part of the heavens he would see
+it; and, moreover, that he would be able to tell it from a star by its
+having a sensible magnitude, or disk, instead of being a mere point.
+
+Galle got the letter on the 23rd of September, 1846. That same evening
+he did point his telescope to the place Leverrier told him, and he saw
+the planet that very night. He recognized it first by its appearance. To
+his practised eye it did seem to have a small disk, and not quite the
+same aspect as an ordinary star. He then consulted Bremiker's great star
+chart, the part just engraved and finished, and sure enough on that
+chart there was no such star there. Undoubtedly it was the planet.
+
+The news flashed over Europe at the maximum speed with which news could
+travel at that date (which was not very fast); and by the 1st of October
+Professor Challis and Mr. Adams heard it at Cambridge, and had the
+pleasure of knowing that they were forestalled, and that England was
+out of the race.
+
+It was an unconscious race to all concerned, however. Those in France
+knew nothing of the search going on in England. Mr. Adams's papers had
+never been published; and very annoyed the French were when a claim was
+set up on his behalf to a share in this magnificent discovery.
+Controversies and recriminations, excuses and justifications, followed;
+but the discussion has now settled down. All the world honours the
+bright genius and mathematical skill of Mr. Adams, and recognizes that
+he first solved the problem by calculation. All the world, too,
+perceives clearly the no less eminent mathematical talents of M.
+Leverrier, but it recognizes in him something more than the mere
+mathematician--the man of energy, decision, and character.
+
+
+
+
+LECTURE XVI
+
+COMETS AND METEORS
+
+
+We have now considered the solar system in several aspects, and we have
+passed in review something of what is known about the stars. We have
+seen how each star is itself, in all probability, the centre of another
+and distinct solar system, the constituents of which are too dark and
+far off to be visible to us; nothing visible here but the central sun
+alone, and that only as a twinkling speck.
+
+But between our solar system and these other suns--between each of these
+suns and all the rest--there exist vast empty spaces, apparently devoid
+of matter.
+
+We have now to ask, Are these spaces really empty? Is there really
+nothing in space but the nebul�, the suns, their planets, and their
+satellites? Are all the bodies in space of this gigantic size? May there
+not be an infinitude of small bodies as well?
+
+The answer to this question is in the affirmative. There appears to be
+no special size suited to the vastness of space; we find, as a matter of
+fact, bodies of all manner of sizes, ranging by gradations from the most
+tremendous suns, like Sirius, down through ordinary suns to smaller
+ones, then to planets of all sizes, satellites still smaller, then the
+asteroids, till we come to the smallest satellite of Mars, only about
+ten miles in diameter, and weighing only some billion tons--the smallest
+of the regular bodies belonging to the solar system known.
+
+But, besides all these, there are found to occur other masses, not much
+bigger and some probably smaller, and these we call comets when we see
+them. Below these, again, we find masses varying from a few tons in
+weight down to only a few pounds or ounces, and these when we see them,
+which is not often, we call meteors or shooting-stars; and to the size
+of these meteorites there would appear to be no limit: some may be
+literal grains of dust. There seems to be a regular gradation of size,
+therefore, ranging from Sirius to dust; and apparently we must regard
+all space as full of these cosmic particles--stray fragments, as it
+were, perhaps of some older world, perhaps going to help to form a new
+one some day. As Kepler said, there are more "comets" in the sky than
+fish in the sea. Not that they are at all crowded together, else they
+would make a cosmic haze. The transparency of space shows that there
+must be an enormous proportion of clear space between each, and they are
+probably much more concentrated near one of the big bodies than they are
+in interstellar space.[30] Even during the furious hail of meteors in
+November 1866 it was estimated that their average distance apart in the
+thickest of the shower was 35 miles.
+
+Consider the nature of a meteor or shooting-star. We ordinarily see them
+as a mere streak of light; sometimes they leave a luminous tail behind
+them; occasionally they appear as an actual fire-ball, accompanied by an
+explosion; sometimes, but very seldom, they are seen to drop, and may
+subsequently be dug up as a lump of iron or rock, showing signs of rough
+treatment by excoriation and heat. These last are the meteorites, or
+siderites, or a�rolites, or bolides, of our museums. They are popularly
+spoken of as thunderbolts, though they have nothing whatever to do with
+atmospheric electricity.
+
+[Illustration: FIG. 95.--Meteorite.]
+
+They appear to be travelling rocky or metallic fragments which in their
+journey through space are caught in the earth's atmosphere and
+instantaneously ignited by the friction. Far away in the depths of space
+one of these bodies felt the attracting power of the sun, and began
+moving towards him. As it approached, its speed grew gradually quicker
+and quicker continually, until by the time it has approached to within
+the distance of the earth, it whizzes past with the velocity of
+twenty-six miles a second. The earth is moving on its own account
+nineteen miles every second. If the two bodies happened to be moving in
+opposite directions, the combined speed would be terrific; and the
+faintest trace of atmosphere, miles above the earth's surface, would
+exert a furious grinding action on the stone. A stream of particles
+would be torn off; if of iron, they would burn like a shower of filings
+from a firework, thus forming a trail; and the mass itself would be
+dissipated, shattered to fragments in an instant.
+
+[Illustration: FIG. 96.--Meteor stream crossing field of telescope.]
+
+[Illustration: FIG. 97.--Diagram of direction of earth's orbital
+motion, showing that after midnight, _i.e._ between midnight and noon,
+more asteroids are likely to be swept up by any locality than between
+noon and midnight. [From Sir R.S. Ball.]]
+
+Even if the earth were moving laterally, the same thing would occur. But
+if earth and stone happened to be moving in the same direction, there
+would be only the differential velocity of seven miles a second; and
+though this is in all conscience great enough, yet there might be a
+chance for a residue of the nucleus to escape entire destruction, though
+it would be scraped, heated, and superficially molten by the friction;
+but so much of its speed would be rubbed out of it, that on striking
+the earth it might bury itself only a few feet or yards in the soil, so
+that it could be dug out. The number of those which thus reach the earth
+is comparatively infinitesimal. Nearly all get ground up and dissipated
+by the atmosphere; and fortunate it is for us that they are so. This
+bombardment of the exposed face of the moon must be something
+terrible.[31]
+
+Thus, then, every shooting-star we see, and all the myriads that we do
+not and cannot see because they occur in the day-time, all these bright
+flashes or streaks, represent the death and burial of one of these
+flying stones. It had been careering on its own account through space
+for untold ages, till it meets a planet. It cannot strike the actual
+body of the planet--the atmosphere is a sufficient screen; the
+tremendous friction reduces it to dust in an instant, and this dust then
+quietly and leisurely settles down on to the surface.
+
+Evidence of the settlement of meteoric dust is not easy to obtain in
+such a place as England, where the dust which accumulates is seldom of a
+celestial character; but on the snow-fields of Greenland or the
+Himalayas dust can be found; and by a Committee of the British
+Association distinct evidence of molten globules of iron and other
+materials appropriate to a�rolites has been obtained, by the simple
+process of collecting, melting, and filtering long exposed snow.
+Volcanic ash may be mingled with it, but under the microscope the
+volcanic and the meteoric constituents have each a distinctive
+character.
+
+The quantity of meteoric material which reaches the earth as dust must
+be immensely in excess of the minute quantity which arrives in the form
+of lumps. Hundreds or thousands of tons per annum must be received; and
+the accretion must, one would think, in the course of ages be able to
+exert some influence on the period of the earth's rotation--the length
+of the day. It is too small, however, to have been yet certainly
+detected. Possibly, it is altogether negligible.
+
+It has been suggested that those stones which actually fall are not the
+true cosmic wanderers, but are merely fragments of our own earth, cast
+up by powerful volcanoes long ago when the igneous power of the earth
+was more vigorous than now--cast up with a speed of close upon seven
+miles a second; and now in these quiet times gradually being swept up by
+the earth, and so returning whence they came.
+
+I confess I am unable to draw a clear distinction between one set and
+the other. Some falling stars may have had an origin of this sort, but
+certainly others have not; and it would seem very unlikely that one set
+only should fall bodily upon the earth, while the others should always
+be rubbed to powder. Still, it is a possibility to be borne in mind.
+
+We have spoken of these cosmic visitors as wandering masses of stone or
+iron; but we should be wrong if we associated with the term "wandering"
+any ideas of lawlessness and irregularity of path. These small lumps of
+matter are as obedient to the law of gravity as any large ones can be.
+They must all, therefore, have definite orbits, and these orbits will
+have reference to the main attracting power of our system--they will, in
+fact, be nearly all careering round the sun.
+
+Each planet may, in truth, have a certain following of its own. Within
+the limited sphere of the earth's predominant attraction, for instance,
+extending some way beyond the moon, we may have a number of satellites
+that we never see, all revolving regularly in elliptic orbits round the
+earth. But, comparatively speaking, these satellite meteorites are few.
+The great bulk of them will be of a planetary character--they will be
+attendant upon the sun.
+
+It may seem strange that such minute bodies should have regular orbits
+and obey Kepler's laws, but they must. All three laws must be as
+rigorously obeyed by them as by the planets themselves. There is nothing
+in the smallness of a particle to excuse it from implicit obedience to
+law. The only consequence of their smallness is their inability to
+perturb others. They cannot appreciably perturb either the planets they
+approach or each other. The attracting power of a lump one million tons
+in weight is very minute. A pound, on the surface of such a body of the
+same density as the earth, would be only pulled to it with a force equal
+to that with which the earth pulls a grain. So the perturbing power of
+such a mass on distant bodies is imperceptible. It is a good thing it is
+so: accurate astronomy would be impossible if we had to take into
+account the perturbations caused by a crowd of invisible bodies.
+Astronomy would then approach in complexity some of the problems of
+physics.
+
+But though we may be convinced from the facts of gravitation that these
+meteoric stones, and all other bodies flying through space near our
+solar system, must be constrained by the sun to obey Kepler's laws, and
+fly round it in some regular elliptic or hyperbolic orbit, what chance
+have we of determining that orbit? At first sight, a very poor chance,
+for we never see them except for the instant when they splash into our
+atmosphere; and for them that instant is instant death. It is unlikely
+that any escape that ordeal, and even if they do, their career and orbit
+are effectually changed. Henceforward they must become attendants on the
+earth. They may drop on to its surface, or they may duck out of our
+atmosphere again, and revolve round us unseen in the clear space between
+earth and moon.
+
+Nevertheless, although the problem of determining the original orbit of
+any given set of shooting-stars before it struck us would seem nearly
+insoluble, it has been solved, and solved with some approach to
+accuracy; being done by the help of observations of certain other
+bodies. The bodies by whose help this difficult problem has been
+attacked and resolved are comets. What are comets?
+
+I must tell you that the scientific world is not entirely and completely
+decided on the structure of comets. There are many floating ideas on the
+subject, and some certain knowledge. But the subject is still, in many
+respects, an open one, and the ideas I propose to advocate you will
+accept for no more than they are worth, viz. as worthy to be compared
+with other and different views.
+
+Up to the time of Newton, the nature of comets was entirely unknown.
+They were regarded with superstitious awe as fiery portents, and were
+supposed to be connected with the death of some king, or with some
+national catastrophe.
+
+Even so late as the first edition of the _Principia_ the problem of
+comets was unsolved, and their theory is not given; but between the
+first and the second editions a large comet appeared, in 1680, and
+Newton speculated on its appearance and behaviour. It rushed down very
+close to the sun, spun half round him very quickly, and then receded
+from him again. If it were a material substance, to which the law of
+gravitation applied, it must be moving in a conic section with the sun
+in one focus, and its radius vector must sweep out equal areas in equal
+times. Examining the record of its positions made at observatories, he
+found its observed path quite accordant with theory; and the motion of
+comets was from that time understood. Up to that time no one had
+attempted to calculate an orbit for a comet. They had been thought
+irregular and lawless bodies. Now they were recognized as perfectly
+obedient to the law of gravitation, and revolving round the sun like
+everything else--as members, in fact, of our solar system, though not
+necessarily permanent members.
+
+But the orbit of a comet is very different from a planetary one. The
+excentricity of its orbit is enormous--in other words, it is either a
+very elongated ellipse or a parabola. The comet of 1680, Newton found
+to move in an orbit so nearly a parabola that the time of describing it
+must be reckoned in hundreds of years at the least. It is now thought
+possible that it may not be quite a parabola, but an ellipse so
+elongated that it will not return till 2255. Until that date arrives,
+however, uncertainty will prevail as to whether it is a periodic comet,
+or one of those that only visit our system once. If it be periodic, as
+suspected, it is the same as appeared when Julius C�sar was killed, and
+which likewise appeared in the years 531 and 1106 A.D. Should it appear
+in 2255, our posterity will probably regard it as a memorial of Newton.
+
+[Illustration: FIG. 98.--Parabolic and elliptic orbits. The _a b_
+(visible) portions are indistinguishable.]
+
+The next comet discussed in the light of the theory of gravitation was
+the famous one of Halley. You know something of the history of this.
+Its period is 75-1/2 years. Halley saw it in 1682, and predicted its
+return in 1758 or 1759--the first cometary prediction. Clairaut
+calculated its return right within a month (p. 219). It has been back
+once more, in 1835; and this time its date was correctly predicted
+within three days, because Uranus was now known. It was away at its
+furthest point in 1873. It will be back again in 1911.
+
+[Illustration: FIG. 99.--Orbit of Halley's comet.]
+
+Coming to recent times, we have the great comets of 1843 and of 1858,
+the history of neither being known. Quite possibly they arrived then for
+the first time. Possibly the second will appear again in 3808. But
+besides these great comets, there are a multitude of telescopic ones,
+which do not show these striking features, and have no gigantic tail.
+Some have no tail at all, others have at best a few insignificant
+streamers, and others show a faint haze looking like a microscopic
+nebula.
+
+All these comets are of considerable extent--some millions of miles
+thick usually, and yet stars are clearly visible through them. Hence
+they must be matter of very small density; their tails can be nothing
+more dense than a filmy mist, but their nucleus must be something more
+solid and substantial.
+
+[Illustration: FIG. 100.--Various appearances of Halley's comet when
+last seen.]
+
+I have said that comets arrive from the depths of space, rush towards
+and round the sun, whizzing past the earth with a speed of twenty-six
+miles a second, on round the sun with a far greater velocity than that,
+and then rush off again. Now, all the time they are away from the sun
+they are invisible. It is only as they get near him that they begin to
+expand and throw off tails and other appendages. The sun's heat is
+evidently evaporating them, and driving away a cloud of mist and
+volatile matter. This is when they can be seen. The comet is most
+gorgeous when it is near the sun, and as soon as it gets a reasonable
+distance away from him it is perfectly invisible.
+
+The matter evaporated from the comet by the sun's heat does not
+return--it is lost to the comet; and hence, after a few such journeys,
+its volatile matter gets appreciably diminished, and so old-established
+periodic comets have no tails to speak of. But the new visitants, coming
+from the depths of space for the first time--these have great supplies
+of volatile matter, and these are they which show the most magnificent
+tails.
+
+[Illustration: FIG. 101.--Head of Donati's comet of 1858.]
+
+The tail of a comet is always directed away from the sun as if it were
+repelled. To this rule there is no exception. It is suggested, and held
+as most probable, that the tail and sun are similarly electrified, and
+that the repulsion of the tail is electrical repulsion. Some great force
+is obviously at work to account for the enormous distance to which the
+tail is shot in a few hours. The pressure of the sun's light can do
+something, and is a force that must not be ignored when small particles
+are being dealt with. (Cf. _Modern Views of Electricity_, 2nd edition,
+p. 363.)
+
+Now just think what analogies there are between comets and meteors. Both
+are bodies travelling in orbits round the sun, and both are mostly
+invisible, but both become visible to us under certain circumstances.
+Meteors become visible when they plunge into the extreme limits of our
+atmosphere. Comets become visible when they approach the sun. Is it
+possible that comets are large meteors which dip into the solar
+atmosphere, and are thus rendered conspicuously luminous? Certainly they
+do not dip into the actual main atmosphere of the sun, else they would
+be utterly destroyed; but it is possible that the sun has a faint trace
+of atmosphere extending far beyond this, and into this perhaps these
+meteors dip, and glow with the friction. The particles thrown off might
+be, also by friction, electrified; and the vaporous tail might be thus
+accounted for.
+
+[Illustration: FIG. 102.--Halley's Comet.]
+
+Let us make this hypothesis provisionally--that comets are large
+meteors, or a compact swarm of meteors, which, coming near the sun, find
+a highly rarefied sort of atmosphere, in which they get heated and
+partly vaporized, just as ordinary meteorites do when they dip into the
+atmosphere of the earth. And let us see whether any facts bear out the
+analogy and justify the hypothesis.
+
+I must tell you now the history of three bodies, and you will see that
+some intimate connection between comets and meteors is proved. The
+three bodies are known as, first, Encke's comet; second, Biela's comet;
+third, the November swarm of meteors.
+
+Encke's comet (one of those discovered by Miss Herschel) is an
+insignificant-looking telescopic comet of small period, the orbit of
+which was well known, and which was carefully observed at each
+reappearance after Encke had calculated its orbit. It was the quickest
+of the comets, returning every 3-1/2 years.
+
+[Illustration: FIG. 103.--Encke's comet.]
+
+It was found, however, that its period was not quite constant; it kept
+on getting slightly shorter. The comet, in fact, returned to the sun
+slightly before its time. Now this effect is exactly what friction
+against a solar atmosphere would bring about. Every time it passed near
+the sun a little velocity would be rubbed out of it. But the velocity is
+that which carries it away, hence it would not go quite so far, and
+therefore would return a little sooner. Any revolving body subject to
+friction must revolve quicker and quicker, and get nearer and nearer
+its central body, until, if the process goes on long enough, it must
+drop upon its surface. This seems the kind of thing happening to Encke's
+comet. The effect is very small, and not thoroughly proved; but, so far
+as it goes, the evidence points to a greatly extended rare solar
+atmosphere, which rubs some energy out of it at every perihelion
+passage.
+
+[Illustration: FIG. 104.--Biela's comet as last seen, in two portions.]
+
+Next, Biela's comet. This also was a well known and carefully observed
+telescopic comet, with a period of six years. In one of its distant
+excursions, it was calculated that it must pass very near Jupiter, and
+much curiosity was excited as to what would happen to it in consequence
+of the perturbation it must experience. As I have said, comets are only
+visible as they approach the sun, and a watch was kept for it about its
+appointed time. It was late, but it did ultimately arrive.
+
+The singular thing about it, however, was that it was now double. It had
+apparently separated into two. This was in 1846. It was looked for again
+in 1852, and this time the components were further separated. Sometimes
+one was brighter, sometimes the other. Next time it ought to have come
+round no one could find either portion. The comet seemed to have wholly
+disappeared. It has never been seen since. It was then recorded and
+advertised as the missing comet.
+
+But now comes the interesting part of the story. The orbit of this Biela
+comet was well known, and it was found that on a certain night in 1872
+the earth would cross the orbit, and had some chance of encountering the
+comet. Not a very likely chance, because it need not be in that part of
+its orbit at the time; but it was suspected not to be far off--if still
+existent. Well, the night arrived, the earth did cross the orbit, and
+there was seen, not the comet, but a number of shooting-stars. Not one
+body, nor yet two, but a multitude of bodies--in fact, a swarm of
+meteors. Not a very great swarm, such as sometimes occurs, but still a
+quite noticeable one; and this shower of meteors is definitely
+recognized as flying along the track of Biela's comet. They are known as
+the Andromedes.
+
+This observation has been generalized. Every cometary orbit is marked by
+a ring of meteoric stones travelling round it, and whenever a number of
+shooting-stars are seen quickly one after the other, it is an evidence
+that we are crossing the track of some comet. But suppose instead of
+only crossing the track of a comet we were to pass close to the comet
+itself, we should then expect to see an extraordinary swarm--a multitude
+of shooting-stars. Such phenomena have occurred. The most famous are
+those known as the November meteors, or Leonids.
+
+This is the third of those bodies whose history I had to tell you.
+Professor H.A. Newton, of America, by examining ancient records arrived
+at the conclusion that the earth passed through a certain definite
+meteor shoal every thirty-three years. He found, in fact, that every
+thirty-three years an unusual flight of shooting-stars was witnessed in
+November, the earliest record being 599 A.D. Their last appearance had
+been in 1833, and he therefore predicted their return in 1866 or 1867.
+Sure enough, in November, 1866, they appeared; and many must remember
+seeing that glorious display. Although their hail was almost continuous,
+it is estimated that their average distance apart was thirty-five miles!
+Their radiant point was and always is in the constellation Leo, and
+hence their name Leonids.
+
+[Illustration: FIG. 105.--Radiant point perspective. The arrows
+represent a number of approximately parallel meteor-streaks
+foreshortened from a common vanishing-point.]
+
+     A parallel stream fixed in space necessarily exhibits a definite
+     aspect with reference to the fixed stars. Its aspect with respect
+     to the earth will be very changeable, because of the rotation and
+     revolution of that body, but its position with respect to
+     constellations will be steady. Hence each meteor swarm, being a
+     steady parallel stream of rushing masses, always strikes us from
+     the same point in stellar space, and by this point (or radiant) it
+     is identified and named.
+
+     The paths do not appear to us to be parallel, because of
+     perspective: they seem to radiate and spread in all directions from
+     a fixed centre like spokes, but all these diverging streaks are
+     really parallel lines optically foreshortened by different amounts
+     so as to produce the radiant impression.
+
+     The annexed diagram (Fig. 105) clearly illustrates the fact that
+     the "radiant" is the vanishing point of a number of parallel lines.
+
+[Illustration: FIG. 106.--Orbit of November meteors.]
+
+This swarm is specially interesting to us from the fact that we cross
+its orbit every year. Its orbit and the earth's intersect. Every
+November we go through it, and hence every November we see a few
+stragglers of this immense swarm. The swarm itself takes thirty-three
+years on its revolution round the sun, and hence we only encounter it
+every thirty-three years.
+
+The swarm is of immense size. In breadth it is such that the earth,
+flying nineteen miles a second, takes four or five hours to cross it,
+and this is therefore the time the display lasts. But in length it is
+far more enormous. The speed with which it travels is twenty-five miles
+a second, (for its orbit extends as far as Uranus, although by no means
+parabolic), and yet it takes more than a year to pass. Imagine a
+procession 200,000 miles broad, every individual rushing along at the
+rate of twenty-five miles every second, and the whole procession so long
+that it takes more than a year to pass. It is like a gigantic shoal of
+herrings swimming round and round the sun every thirty-three years, and
+travelling past the earth with that tremendous velocity of twenty-five
+miles a second. The earth dashes through the swarm and sweeps up
+myriads. Think of the countless numbers swept up by the whole earth in
+crossing such a shoal as that! But heaps more remain, and probably the
+millions which are destroyed every thirty-three years have not yet made
+any very important difference to the numbers still remaining.
+
+The earth never misses this swarm. Every thirty-three years it is bound
+to pass through some part of them, for the shoal is so long that if the
+head is just missed one November the tail will be encountered next
+November. This is a plain and obvious result of its enormous length. It
+may be likened to a two-foot length of sewing silk swimming round and
+round an oval sixty feet in circumference. But, you will say, although
+the numbers are so great that destroying a few millions or so every
+thirty-three years makes but little difference to them, yet, if this
+process has been going on from all eternity, they ought to be all swept
+up. Granted; and no doubt the most ancient swarms have already all or
+nearly all been swept up.
+
+[Illustration: FIG. 107.--Orbit of November meteors; showing their
+probable parabolic orbit previous to 126 A.D., and its sudden conversion
+into an elliptic orbit by the violent perturbation caused by Uranus,
+which at that date occupied the position shown.]
+
+The August meteors, or Perseids, are an example. Every August we cross
+their path, and we have a small meteoric display radiating from the
+sword-hand of Perseus, but never specially more in one August than
+another. It would seem as if the main shoal has disappeared, and nothing
+is now left but the stragglers; or perhaps it is that the shoal has
+gradually become uniformly distributed all along the path. Anyhow, these
+August meteors are reckoned much more ancient members of the solar
+system than are the November meteors. The November meteors are believed
+to have entered the solar system in the year 126 A.D.
+
+This may seem an extraordinary statement. It is not final, but it is
+based on the calculations of Leverrier--confirmed recently by Mr. Adams.
+A few moments will suffice to make the grounds of it clear. Leverrier
+calculated the orbit of the November meteors, and found them to be an
+oval extending beyond Uranus. It was perturbed by the outer planets near
+which it went, so that in past times it must have moved in a slightly
+different orbit. Calculating back to their past positions, it was found
+that in a certain year it must have gone very near to Uranus, and that
+by the perturbation of this planet its path had been completely changed.
+Originally it had in all probability been a comet, flying in a parabolic
+orbit towards the sun like many others. This one, encountering Uranus,
+was pulled to pieces as it were, and its orbit made elliptical as shown
+in Fig. 107. It was no longer free to escape and go away into the depths
+of space: it was enchained and made a member of the solar system. It
+also ceased to be a comet; it was degraded into a shoal of meteors.
+
+This is believed to be the past history of this splendid swarm. Since
+its introduction to the solar system it has made 52 revolutions: its
+next return is due in November, 1899, and I hope that it may occur in
+the English dusk, and (see Fig. 97) in a cloudless after-midnight sky,
+as it did in 1866.
+
+
+
+
+NOTES FOR LECTURE XVII
+
+
+The tide-generating force of one body on another is directly as the mass
+of the one body and inversely as the cube of the distance between them.
+Hence the moon is more effective in producing terrestrial tides than the
+sun.
+
+The tidal wave directly produced by the moon in the open ocean is about
+5 feet high, that produced by the sun is about 2 feet. Hence the average
+spring tide is to the average neap as about 7 to 3. The lunar tide
+varies between apogee and perigee from 4�3 to 5�9.
+
+The solar tide varies between aphelion and perihelion from 1�9 to 2�1.
+Hence the highest spring tide is to the lowest neap as 5�9 + 2�1 is to
+4�3 -2�1, or as 8 to 2�2.
+
+The semi-synchronous oscillation of the Southern Ocean raises the
+magnitude of oceanic tides somewhat above these directly generated
+values.
+
+Oceanic tides are true waves, not currents. Coast tides are currents.
+The momentum of the water, when the tidal wave breaks upon a continent
+and rushes up channels, raises coast tides to a much greater height--in
+some places up to 50 or 60 feet, or even more.
+
+Early observed connections between moon and tides would be these:--
+
+     1st. Spring tides at new and full moon.
+
+     2nd. Average interval between tide and tide is half a lunar, not a
+     solar, day--a lunar day being the interval between two successive
+     returns of the moon to the meridian: 24 hours and 50 minutes.
+
+     3rd. The tides of a given place at new and full moon occur always
+     at the same time of day whatever the season of the year.
+
+
+
+
+LECTURE XVII
+
+THE TIDES
+
+
+Persons accustomed to make use of the Mersey landing-stages can hardly
+fail to have been struck with two obvious phenomena. One is that the
+gangways thereto are sometimes almost level, and at other times very
+steep; another is that the water often rushes past the stage rather
+violently, sometimes south towards Garston, sometimes north towards the
+sea. They observe, in fact, that the water has two periodic motions--one
+up and down, the other to and fro--a vertical and a horizontal motion.
+They may further observe, if they take the trouble, that a complete
+swing of the water, up and down, or to and fro, takes place about every
+twelve and a half hours; moreover, that soon after high and low water
+there is no current--the water is stationary, whereas about half-way
+between high and low it is rushing with maximum speed either up or down
+the river.
+
+To both these motions of the water the name _tide_ is given, and both
+are extremely important. Sailors usually pay most attention to the
+horizontal motion, and on charts you find the tide-races marked; and the
+places where there is but a small horizontal rush of the water are
+labelled "very little tide here." Landsmen, or, at any rate, such of the
+more philosophic sort as pay any attention to the matter at all, think
+most of the vertical motion of the water--its amount of rise and fall.
+
+Dwellers in some low-lying districts in London are compelled to pay
+attention to the extra high tides of the Thames, because it is, or was,
+very liable to overflow its banks and inundate their basements.
+
+Sailors, however, on nearing a port are also greatly affected by the
+time and amount of high water there, especially when they are in a big
+ship; and we know well enough how frequently Atlantic liners, after
+having accomplished their voyage with good speed, have to hang around
+for hours waiting till there is enough water to lift them over the
+Bar--that standing obstruction, one feels inclined to say disgrace, to
+the Liverpool harbour.
+
+[Illustration: FIG. 108.--The Mersey]
+
+To us in Liverpool the tides are of supreme importance--upon them the
+very existence of the city depends--for without them Liverpool would not
+be a port. It may be familiar to many of you how this is, and yet it is
+a matter that cannot be passed over in silence. I will therefore call
+your attention to the Ordnance Survey of the estuaries of the Mersey and
+the Dee. You see first that there is a great tendency for sand-banks to
+accumulate all about this coast, from North Wales right away round to
+Southport. You see next that the port of Chester has been practically
+silted up by the deposits of sand in the wide-mouthed Dee, while the
+port of Liverpool remains open owing to the scouring action of the tide
+in its peculiarly shaped channel. Without the tides the Mersey would be
+a wretched dribble not much bigger than it is at Warrington. With them,
+this splendid basin is kept open, and a channel is cut of such depth
+that the _Great Eastern_ easily rode in it in all states of the water.
+
+The basin is filled with water every twelve hours through its narrow
+neck. The amount of water stored up in this basin at high tide I
+estimate as 600 million tons. All this quantity flows through the neck
+in six hours, and flows out again in the next six, scouring and
+cleansing and carrying mud and sand far out to sea. Just at present the
+currents set strongest on the Birkenhead side of the river, and
+accordingly a "Pluckington bank" unfortunately grows under the Liverpool
+stage. Should this tendency to silt up the gates of our docks increase,
+land can be reclaimed on the other side of the river between Tranmere
+and Rock Ferry, and an embankment made so as to deflect the water over
+Liverpool way, and give us a fairer proportion of the current. After
+passing New Brighton the water spreads out again to the left; its
+velocity forward diminishes; and after a few miles it has no power to
+cut away that sandbank known as the Bar. Should it be thought desirable
+to make it accomplish this, and sweep the Bar further out to sea into
+deeper water, it is probable that a rude training wall (say of old
+hulks, or other removable partial obstruction) on the west of Queen's
+Channel, arranged so as to check the spreading out over all this useless
+area, may be quite sufficient to retain the needed extra impetus in the
+water, perhaps even without choking up the useful old Rock Channel,
+through which smaller ships still find convenient exit.
+
+Now, although the horizontal rush of the tide is necessary to our
+existence as a port, it does not follow that the accompanying rise and
+fall of the water is an unmixed blessing. To it is due the need for all
+the expensive arrangements of docks and gates wherewith to store up the
+high-level water. Quebec and New York are cities on such magnificent
+rivers that the current required to keep open channel is supplied
+without any tidal action, although Quebec is nearly 1,000 miles from the
+open ocean; and accordingly, Atlantic liners do not hover in mid-river
+and discharge passengers by tender, but they proceed straight to the
+side of the quays lining the river, or, as at New York, they dive into
+one of the pockets belonging to the company running the ship, and there
+discharge passengers and cargo without further trouble, and with no need
+for docks or gates. However, rivers like the St. Lawrence and the Hudson
+are the natural property of a gigantic continent; and we in England may
+be well contented with the possession of such tidal estuaries as the
+Mersey, the Thames, and the Humber. That by pertinacious dredging the
+citizens of Glasgow manage to get large ships right up their small
+river, the Clyde, to the quays of the town, is a remarkable fact, and
+redounds very highly to their credit.
+
+We will now proceed to consider the connection existing between the
+horizontal rush of water and its vertical elevation, and ask, Which is
+cause and which is effect? Does the elevation of the ocean cause the
+tidal flow, or does the tidal flow cause the elevation? The answer is
+twofold: both statements are in some sense true. The prime cause of the
+tide is undoubtedly a vertical elevation of the ocean, a tidal wave or
+hump produced by the attraction of the moon. This hump as it passes the
+various channels opening into the ocean raises their level, and causes
+water to flow up them. But this simple oceanic tide, although the cause
+of all tide, is itself but a small affair. It seldom rises above six or
+seven feet, and tides on islands in mid-ocean have about this value or
+less. But the tides on our coasts are far greater than this--they rise
+twenty or thirty feet, or even fifty feet occasionally, at some places,
+as at Bristol. Why is this? The horizontal motion of the water gives it
+such an impetus or momentum that its motion far transcends that of the
+original impulse given to it, just as a push given to a pendulum may
+cause it to swing over a much greater arc than that through which the
+force acts. The inrushing water flowing up the English Channel or the
+Bristol Channel or St. George's Channel has such an impetus that it
+propels itself some twenty or thirty feet high before it has exhausted
+its momentum and begins to descend. In the Bristol Channel the gradual
+narrowing of the opening so much assists this action that the tides
+often rise forty feet, occasionally fifty feet, and rush still further
+up the Severn in a precipitous and extraordinary hill of water called
+"the bore."
+
+Some places are subject to considerable rise and fall of water with very
+little horizontal flow; others possess strong tidal races, but very
+little elevation and depression. The effect observed at any given place
+entirely depends on whether the place has the general character of a
+terminus, or whether it lies _en route_ to some great basin.
+
+You must understand, then, that all tide takes its rise in the free and
+open ocean under the action of the moon. No ordinary-sized sea like the
+North Sea, or even the Mediterranean, is big enough for more than a just
+appreciable tide to be generated in it. The Pacific, the Atlantic, and
+the Southern Oceans are the great tidal reservoirs, and in them the
+tides of the earth are generated as low flat humps of gigantic area,
+though only a few feet high, oscillating up and down in the period of
+approximately twelve hours. The tides we, and other coast-possessing
+nations, experience are the overflow or back-wash of these oceanic
+humps, and I will now show you in what manner the great Atlantic
+tide-wave reaches the British Isles twice a day.
+
+[Illustration: FIG. 109.--Co-tidal lines.]
+
+Fig. 109 shows the contour lines of the great wave as it rolls in east
+from the Atlantic, getting split by the Land's End and by Ireland into
+three portions; one of which rushes up the English Channel and through
+the Straits of Dover. Another rolls up the Irish Sea, with a minor
+offshoot up the Bristol Channel, and, curling round Anglesey, flows
+along the North Wales coast and fills Liverpool Bay and the Mersey. The
+third branch streams round the north coast of Ireland, past the Mull of
+Cantyre and Rathlin Island; part fills up the Firth of Clyde, while the
+rest flows south, and, swirling round the west side of the Isle of Man,
+helps the southern current to fill the Bay of Liverpool. The rest of the
+great wave impinges on the coast of Scotland, and, curling round it,
+fills up the North Sea right away to the Norway coast, and then flows
+down below Denmark, joining the southern and earlier arriving stream.
+The diagram I show you is a rough chart of cotidal lines, which I made
+out of the information contained in _Whitaker's Almanac_.
+
+A place may thus be fed with tide by two distinct channels, and many
+curious phenomena occur in certain places from this cause. Thus it may
+happen that one channel is six hours longer than the other, in which
+case a flow will arrive by one at the same time as an ebb arrives by the
+other; and the result will be that the place will have hardly any tide
+at all, one tide interfering with and neutralizing the other. This is
+more markedly observed at other parts of the world than in the British
+Isles. Whenever a place is reached by two channels of different length,
+its tides are sure to be peculiar, and probably small.
+
+Another cause of small tide is the way the wave surges to and fro in a
+channel. The tidal wave surging up the English Channel, for instance,
+gets largely reflected by the constriction at Dover, and so a crest
+surges back again, as we may see waves reflected in a long trough or
+tilted bath. The result is that Southampton has two high tides rapidly
+succeeding one another, and for three hours the high-water level varies
+but slightly--a fact of evident convenience to the port.
+
+Places on a nodal line, so to speak, about the middle of the length of
+the channel, have a minimum of rise and fall, though the water rushes
+past them first violently up towards Dover, where the rise is
+considerable, and then back again towards the ocean. At Portland, for
+instance, the total rise and fall is very small: it is practically on a
+node. Yarmouth, again, is near a less marked node in the North Sea,
+where stationary waves likewise surge to and fro, and accordingly the
+tidal rise and fall at Yarmouth is only about five feet (varying from
+four and a half to six), whereas at London it is twenty or thirty feet,
+and at Flamborough Head or Leith it is from twelve to sixteen feet.
+
+It is generally supposed that water never flows up-hill, but in these
+cases of oscillation it flows up-hill for three hours together. The
+water is rushing up the English Channel towards Dover long after it is
+highest at the Dover end; it goes on piling itself up, until its
+momentum is checked by the pressure, and then it surges back. It
+behaves, in fact, very like the bob of a pendulum, which rises against
+gravity at every quarter swing.
+
+To get a very large tide, the place ought to be directly accessible by a
+long sweep of a channel to the open ocean, and if it is situate on a
+gradually converging opening, the ebb and flow may be enormous. The
+Severn is the best example of this on the British Isles; but the largest
+tides in the world are found, I believe, in the Bay of Fundy, on the
+coast of North America, where they sometimes rise one hundred and twenty
+feet. Excessive or extra tides may be produced occasionally in any place
+by the propelling force of a high wind driving the water towards the
+shore; also by a low barometer, _i.e._ by a local decrease in the
+pressure of the air.
+
+Well, now, leaving these topographical details concerning tides, which
+we see to be due to great oceanic humps (great in area that is, though
+small in height), let us proceed to ask what causes these humps; and if
+it be the moon that does it, how does it do it?
+
+The statement that the moon causes the tides sounds at first rather an
+absurdity, and a mere popular superstition. Galileo chaffed Kepler for
+believing it. Who it was that discovered the connection between moon and
+tides we know not--probably it is a thing which has been several times
+rediscovered by observant sailors or coast-dwellers--and it is certainly
+a very ancient piece of information.
+
+Probably the first connection observed was that about full moon and
+about new moon the tides are extra high, being called spring tides,
+whereas about half-moon the tides are much less, and are called neap
+tides. The word spring in this connection has no reference to the season
+of the year; except that both words probably represent the same idea of
+energetic uprising or upspringing, while the word neap comes from nip,
+and means pinched, scanty, nipped tide.
+
+The next connection likely to be observed would be that the interval
+between two day tides was not exactly a solar day of twenty-four hours,
+but a lunar day of fifty minutes longer. For by reason of the moon's
+monthly motion it lags behind the sun about fifty minutes a day, and the
+tides do the same, and so perpetually occur later and later, about fifty
+minutes a day later, or 12 hours and 25 minutes on the average between
+tide and tide.
+
+A third and still more striking connection was also discovered by some
+of the ancient great navigators and philosophers--viz. that the time of
+high water at a given place at full moon is always the same, or very
+nearly so. In other words, the highest or spring tides always occur
+nearly at the same time of day at a given place. For instance, at
+Liverpool this time is noon and midnight. London is about two hours and
+a half later. Each port has its own time for receiving a given tide, and
+the time is called the "establishment" of the port. Look out a day when
+the moon is full, and you will find the Liverpool high tide occurs at
+half-past eleven, or close upon it. The same happens when the moon is
+new. A day after full or new moon the spring tides rise to their
+highest, and these extra high tides always occur in Liverpool at noon
+and at midnight, whatever the season of the year. About the equinoxes
+they are liable to be extraordinarily high. The extra low tides here are
+therefore at 6 a.m. and 6 p.m., and the 6 p.m. low tide is a nuisance to
+the river steamers. The spring tides at London are highest about
+half-past two.
+
+*      *      *      *      *
+
+It is, therefore, quite clear that the moon has to do with the tides. It
+and the sun together are, in fact, the whole cause of them; and the mode
+in which these bodies act by gravitative attraction was first made out
+and explained in remarkably full detail by Sir Isaac Newton. You will
+find his account of the tides in the second and third books of the
+_Principia_; and though the theory does not occupy more than a few pages
+of that immortal work, he succeeds not only in explaining the local
+tidal peculiarities, much as I have done to-night, but also in
+calculating the approximate height of mid-ocean solar tide; and from the
+observed lunar tide he shows how to determine the then quite unknown
+mass of the moon. This was a quite extraordinary achievement, the
+difficulty of which it is not easy for a person unused to similar
+discussions fully to appreciate. It is, indeed, but a small part of what
+Newton accomplished, but by itself it is sufficient to confer
+immortality upon any ordinary philosopher, and to place him in a front
+rank.
+
+[Illustration: FIG. 110.--Whirling earth model.]
+
+To make intelligible Newton's theory of the tides, I must not attempt to
+go into too great detail. I will consider only the salient points.
+First, you know that every mass of matter attracts every other piece of
+matter; second, that the moon revolves round the earth, or rather that
+the earth and moon revolve round their common centre of gravity once a
+month; third, that the earth spins on its own axis once a day; fourth,
+that when a thing is whirled round, it tends to fly out from the centre
+and requires a force to hold it in. These are the principles involved.
+You can whirl a bucket full of water vertically round without spilling
+it. Make an elastic globe rotate, and it bulges out into an oblate or
+orange shape; as illustrated by the model shown in Fig. 110. This is
+exactly what the earth does, and Newton calculated the bulging of it as
+fourteen miles all round the equator. Make an elastic globe revolve
+round a fixed centre outside itself, and it gets pulled into a prolate
+or lemon shape; the simplest illustrative experiment is to attach a
+string to an elastic bag or football full of water, and whirl it round
+and round. Its prolateness is readily visible.
+
+Now consider the earth and moon revolving round each other like a man
+whirling a child round. The child travels furthest, but the man cannot
+merely rotate, he leans back and thus also describes a small circle: so
+does the earth; it revolves round the common centre of gravity of earth
+and moon (_cf._ p. 212). This is a vital point in the comprehension of
+the tides: the earth's centre is not at rest, but is being whirled round
+by the moon, in a circle about 1/80 as big as the circle which the moon
+describes, because the earth weighs eighty times as much as the moon.
+The effect of the revolution is to make both bodies slightly protrude in
+the direction of the line joining them; they become slightly "prolate"
+as it is called--that is, lemon-shaped. Illustrating still by the man
+and child, the child's legs fly outwards so that he is elongated in the
+direction of a radius; the man's coat-tails fly out too, so that he too
+is similarly though less elongated. These elongations or protuberances
+constitute the tides.
+
+[Illustration: FIG. 111.--Earth and moon model, illustrating the
+production of statical or "equilibrium" tides when the whole is whirled
+about the point G.]
+
+Fig. 111 shows a model to illustrate the mechanism. A couple of
+cardboard disks (to represent globes of course), one four times the
+diameter of the other, and each loaded so as to have about the correct
+earth-moon ratio of weights, are fixed at either end of a long stick,
+and they balance about a certain point, which is their common centre of
+gravity. For convenience this point is taken a trifle too far out from
+the centre of the earth--that is, just beyond its surface. Through the
+balancing point G a bradawl is stuck, and on that as pivot the whole
+readily revolves. Now, behind the circular disks, you see, are four
+pieces of card of appropriate shape, which are able to slide out under
+proper forces. They are shown dotted in the figure, and are lettered A,
+B, C, D. The inner pair, B and C, are attached to each other by a bit of
+string, which has to typify the attraction of gravitation; the outer
+pair, A and D, are not attached to anything, but have a certain amount
+of play against friction in slots parallel to the length of the stick.
+The moon-disk is also slotted, so a small amount of motion is possible
+to it along the stick or bar. These things being so arranged, and the
+protuberant pieces of card being all pushed home, so that they are
+hidden behind their respective disks, the whole is spun rapidly round
+the centre of gravity, G. The result of a brief spin is to make A and D
+fly out by centrifugal force and show, as in the figure; while the moon,
+flying out too in its slot, tightens up the string, which causes B and C
+to be pulled out too. Thus all four high tides are produced, two on the
+earth and two on the moon, A and D being caused by centrifugal force, B
+and C by the attraction of gravitation. Each disk has become prolate in
+the same sort of fashion as yielding globes do. Of course the fluid
+ocean takes this shape more easily and more completely than the solid
+earth can, and so here are the very oceanic humps we have been talking
+about, and about three feet high (Fig. 112). If there were a sea on the
+_moon_, its humps would be a good deal bigger; but there probably is no
+sea there, and if there were, the earth's tides are more interesting to
+us, at any rate to begin with.
+
+[Illustration: FIG. 112.--Earth and moon (earth's rotation neglected).]
+
+The humps as so far treated are always protruding in the earth-moon
+line, and are stationary. But now we have to remember that the earth is
+spinning inside them. It is not easy to see what precise effect this
+spin will have upon the humps, even if the world were covered with a
+uniform ocean; but we can see at any rate that however much they may get
+displaced, and they do get displaced a good deal, they cannot possibly
+be carried round and round. The whole explanation we have given of their
+causes shows that they must maintain some steady aspect with respect to
+the moon--in other words, they must remain stationary as the earth spins
+round. Not that the same identical water remains stationary, for in that
+case it would have to be dragged over the earth's equator at the rate of
+1,000 miles an hour, but the hump or wave-crest remains stationary. It
+is a true wave, or form only, and consists of continuously changing
+individual particles. The same is true of all waves, except breaking
+ones.
+
+Given, then, these stationary humps and the earth spinning on its axis,
+we see that a given place on the earth will be carried round and round,
+now past a hump, and six hours later past a depression: another six
+hours and it will be at the antipodal hump, and so on. Thus every six
+hours we shall travel from the region in space where the water is high
+to the region where it is low; and ignoring our own motion we shall say
+that the sea first rises and then falls; and so, with respect to the
+place, it does. Thus the succession of high and low water, and the two
+high tides every twenty-four hours, are easily understood in their
+easiest and most elementary aspect. A more complete account of the
+matter it will be wisest not to attempt: suffice it to say that the
+difficulties soon become formidable when the inertia of the water, its
+natural time of oscillation, the varying obliquity of the moon to the
+ecliptic, its varying distance, and the disturbing action of the sun are
+taken into consideration. When all these things are included, the
+problem becomes to ordinary minds overwhelming. A great many of these
+difficulties were successfully attacked by Laplace. Others remained for
+modern philosophers, among whom are Sir George Airy, Sir William
+Thomson, and Professor George Darwin.
+
+     I may just mention that the main and simplest effect of including
+     the inertia or momentum of the water is to dislocate the obvious
+     and simple connexion between high water and high moon; inertia
+     always tends to make an effect differ in phase by a quarter period
+     from the cause producing it, as may be illustrated by a swinging
+     pendulum. Hence high water is not to be expected when the
+     tide-raising force is a maximum, but six hours later; so that,
+     considering inertia and neglecting friction, there would be low
+     water under the moon. Including friction, something nearer the
+     equilibrium state of things occurs. With _sufficient_ friction the
+     motion becomes dead-beat again, _i.e._ follows closely the force
+     that causes it.
+
+Returning to the elementary discussion, we see that the rotation of the
+earth with respect to the humps will not be performed in exactly
+twenty-four hours, because the humps are travelling slowly after the
+moon, and will complete a revolution in a month in the same direction as
+the earth is rotating. Hence a place on the earth has to catch them up,
+and so each high tide arrives later and later each day--roughly
+speaking, an hour later for each day tide; not by any means a constant
+interval, because of superposed disturbances not here mentioned, but on
+the average about fifty minutes.
+
+We see, then, that as a result of all this we get a pair of humps
+travelling all over the surface of the earth, about once a day. If the
+earth were all ocean (and in the southern hemisphere it is nearly all
+ocean), then they would go travelling across the earth, tidal waves
+three feet high, and constituting the mid-ocean tides. But in the
+northern hemisphere they can only thus journey a little way without
+striking land. As the moon rises at a place on the east shores of the
+Atlantic, for instance, the waters begin to flow in towards this place,
+or the tide begins to rise. This goes on till the moon is overhead and
+for some time afterwards, when the tide is at its highest. The hump then
+follows the moon in its apparent journey across to America, and there
+precipitates itself upon the coast, rushing up all the channels, and
+constituting the land tide. At the same time, the water is dragged away
+from the east shores, and so _our_ tide is at its lowest. The same thing
+repeats itself in a little more than twelve hours again, when the other
+hump passes over the Atlantic, as the moon journeys beneath the earth,
+and so on every day.
+
+     In the free Southern Ocean, where land obstruction is comparatively
+     absent, the water gets up a considerable swing by reason of its
+     accumulated momentum, and this modifies and increases the open
+     ocean tides there. Also for some reason, I suppose because of the
+     natural time of swing of the water, one of the humps is there
+     usually much larger than the other; and so places in the Indian and
+     other offshoots of the Southern Ocean get their really high tide
+     only once every twenty-four hours. These southern tides are in fact
+     much more complicated than those the British Isles receive. Ours
+     are singularly simple. No doubt some trace of the influence of the
+     Southern Ocean is felt in the North Atlantic, but any ocean
+     extending over 90� of longitude is big enough to have its own
+     tides generated; and I imagine that the main tides we feel are thus
+     produced on the spot, and that they are simple because the
+     damping-out being vigorous, and accumulated effects small, we feel
+     the tide-producing forces more directly. But for authoritative
+     statements on tides, other books must be read. I have thought, and
+     still think, it best in an elementary exposition to begin by a
+     consideration of the tide-generating forces as if they acted on a
+     non-rotating earth. It is the tide generating forces, and not the
+     tides themselves, that are really represented in Figs. 112 and 114.
+     The rotation of the earth then comes in as a disturbing cause. A
+     more complete exposition would begin with the rotating earth, and
+     would superpose the attraction of the moon as a disturbing cause,
+     treating it as a problem in planetary perturbation, the ocean being
+     a sort of satellite of the earth. This treatment, introducing
+     inertia but ignoring friction and land obstruction, gives low water
+     in the line of pull, and high water at right angles, or where the
+     pull is zero; in the same sort of way as a pendulum bob is highest
+     where most force is pulling it down, and lowest where no force is
+     acting on it. For a clear treatment of the tides as due to the
+     perturbing forces of sun and moon, see a little book by Mr. T.K.
+     Abbott of Trinity College, Dublin. (Longman.)
+
+[Illustration: FIG. 113.--Maps showing how comparatively free from land
+obstruction the ocean in the Southern Hemisphere is.]
+
+If the moon were the only body that swung the earth round, this is all
+that need be said in an elementary treatment; but it is not the only
+one. The moon swings the earth round once a month, the sun swings it
+round once a year. The circle of swing is bigger, but the speed is so
+much slower that the protuberance produced is only one-third of that
+caused by the monthly whirl; _i.e._ the simple solar tide in the open
+sea, without taking momentum into account, is but a little more than a
+foot high, while the simple lunar tide is about three feet. When the two
+agree, we get a spring tide of four feet; when they oppose each other,
+we get a neap tide of only two feet. They assist each other at full moon
+and at new moon. At half-moon they oppose each other. So we have spring
+tides regularly once a fortnight, with neap tides in between.
+
+[Illustration: FIG. 114.--Spring and neap tides.]
+
+Fig. 114 gives the customary diagrams to illustrate these simple things.
+You see that when the moon and sun act at right angles (_i.e._ at every
+half-moon), the high tides of one coincide with the low tides of the
+other; and so, as a place is carried round by the earth's rotation, it
+always finds either solar or else lunar high water, and only experiences
+the difference of their two effects. Whereas, when the sun and moon act
+in the same line (as they do at new and full moon), their high and low
+tides coincide, and a place feels their effects added together. The tide
+then rises extra high and falls extra low.
+
+[Illustration: FIG. 115.--Tidal clock. The position of the disk B shows
+the height of the tide. The tide represented is a nearly high tide eight
+feet above mean level.]
+
+Utilizing these principles, a very elementary form of tidal-clock, or
+tide-predicter, can be made, and for an open coast station it really
+would not give the tides so very badly. It consists of a sort of clock
+face with two hands, one nearly three times as long as the other. The
+short hand, CA, should revolve round C once in twelve hours, and the
+vertical height of its end A represents the height of the solar tide on
+the scale of horizontal lines ruled across the face of the clock. The
+long hand, AB, should revolve round A once in twelve hours and
+twenty-five minutes, and the height of its end B (if A were fixed on the
+zero line) would represent the lunar tide. The two revolutions are made
+to occur together, either by means of a link-work parallelogram, or,
+what is better in practice, by a string and pulleys, as shown; and the
+height of the end point, B, of the third side or resultant, CB, read off
+on a scale of horizontal parallel lines behind, represents the
+combination or actual tide at the place. Every fortnight the two will
+agree, and you will get spring tides of maximum height CA + AB; every
+other fortnight the two will oppose, and you will get neap tides of
+maximum height CA-AB.
+
+Such a clock, if set properly and driven in the ordinary way, would then
+roughly indicate the state of the tide whenever you chose to look at it
+and read the height of its indicating point. It would not indeed be very
+accurate, especially for such an inclosed station as Liverpool is, and
+that is probably why they are not made. A great number of disturbances,
+some astronomical, some terrestrial, have to be taken into account in
+the complete theory. It is not an easy matter to do this, but it can be,
+and has been, done; and a tide-predicter has not only been constructed,
+but two of them are in regular work, predicting the tides for years
+hence--one, the property of the Indian Government, for coast stations of
+India; the other for various British and foreign stations, wherever the
+necessary preliminary observations have been made. These machines are
+the invention of Sir William Thomson. The tide-tables for Indian ports
+are now always made by means of them.
+
+[Illustration: FIG. 116.--Sir William Thomson (Lord Kelvin).]
+
+[Illustration: FIG. 117.--Tide-gauge for recording local tides, a
+pencil moved up and down by a float writes on a drum driven by
+clockwork.]
+
+The first thing to be done by any port which wishes its tides to be
+predicted is to set up a tide-gauge, or automatic recorder, and keep it
+working for a year or two. The tide-gauge is easy enough to understand:
+it marks the height of the tide at every instant by an irregular curved
+line like a barometer chart (Fig. 117). These observational curves so
+obtained have next to be fed into a fearfully complex machine, which it
+would take a whole lecture to make even partially intelligible, but Fig.
+118 shows its aspect. It consists of ten integrating machines in a row,
+coupled up and working together. This is the "harmonic analyzer," and
+the result of passing the curve through this machine is to give you all
+the constituents of which it is built up, viz. the lunar tide, the solar
+tide, and eight of the sub-tides or disturbances. These ten values are
+then set off into a third machine, the tide-predicter proper. The
+general mode of action of this machine is not difficult to understand.
+It consists of a string wound over and under a set of pulleys, which are
+each set on an excentric, so as to have an up-and-down motion. These
+up-and-down motions are all different, and there are ten of these
+movable pulleys, which by their respective excursions represent the
+lunar tide, the solar tide, and the eight disturbances already analyzed
+out of the tide-gauge curve by the harmonic analyzer. One end of the
+string is fixed, the other carries a pencil which writes a trace on a
+revolving drum of paper--a trace which represents the combined motion of
+all the pulleys, and so predicts the exact height of the tide at the
+place, at any future time you like. The machine can be turned quite
+quickly, so that a year's tides can be run off with every detail in
+about half-an-hour. This is the easiest part of the operation. Nothing
+has to be done but to keep it supplied with paper and pencil, and turn a
+handle as if it were a coffee-mill instead of a tide-mill. (Figs. 119
+and 120.)
+
+[Illustration: FIG. 118.--Harmonic analyzer; for analyzing out the
+constituents from a set of observational curves.]
+
+My subject is not half exhausted. I might go on to discuss the question
+of tidal energy--whether it can be ever utilized for industrial
+purposes; and also the very interesting question whence it comes. Tidal
+energy is almost the only terrestrial form of energy that does not
+directly or indirectly come from the sun. The energy of tides is now
+known to be obtained at the expense of the earth's rotation; and
+accordingly our day must be slowly, very slowly, lengthening. The tides
+of past ages have destroyed the moon's rotation, and so it always turns
+the same face to us. There is every reason to believe that in geologic
+ages the moon was nearer to us than it is now, and that accordingly our
+tides were then far more violent, rising some hundreds of feet instead
+of twenty or thirty, and sweeping every six hours right over the face of
+a country, ploughing down hills, denuding rocks, and producing a copious
+sedimentary deposit.
+
+[Illustration: FIG. 119.--Tide-predicter, for combining the ascertained
+constituents into a tidal curve for the future.]
+
+In thus discovering the probable violent tides of past ages, astronomy
+has, within the last few years, presented geology with the most powerful
+denuding agent known; and the study of the earth's past history cannot
+fail to be greatly affected by the modern study of the intricate and
+refined conditions attending prolonged tidal action on incompletely
+rigid bodies. [Read on this point the last chapter of Sir R. Ball's
+_Story of the Heavens_.]
+
+[Illustration: Fig. 120.--Weekly sheet of curves. Tides for successive
+days are predicted on the same sheet of paper, to economise space.]
+
+I might also point out that the magnitude of our terrestrial tides
+enables us to answer the question as to the internal fluidity of the
+earth. It used to be thought that the earth's crust was comparatively
+thin, and that it contained a molten interior. We now know that this is
+not the case. The interior of the earth is hot indeed, but it is not
+fluid. Or at least, if it be fluid, the amount of fluid is but very
+small compared with the thickness of the unyielding crust. All these,
+and a number of other most interesting questions, fringe the subject of
+the tides; the theoretical study of which, started by Newton, has
+developed, and is destined in the future to further develop, into one of
+the most gigantic and absorbing investigations--having to do with the
+stability or instability of solar systems, and with the construction and
+decay of universes.
+
+These theories are the work of pioneers now living, whose biographies it
+is therefore unsuitable for us to discuss, nor shall I constantly
+mention their names. But Helmholtz, and Thomson, are household words,
+and you well know that in them and their disciples the race of Pioneers
+maintains its ancient glory.
+
+
+
+
+NOTES FOR LECTURE XVIII
+
+
+Tides are due to incomplete rigidity of bodies revolving round each
+other under the action of gravitation, and at the same time spinning on
+their axes.
+
+Two spheres revolving round each other can only remain spherical if
+rigid; if at all plastic they become prolate. If either rotate on its
+axis, in the same or nearly the same plane as it revolves, that one is
+necessarily subject to tides.
+
+The axial rotation tends to carry the humps with it, but the pull of the
+other body keeps them from moving much. Hence the rotation takes place
+against a pull, and is therefore more or less checked and retarded. This
+is the theory of Von Helmholtz.
+
+The attracting force between two such bodies is no longer _exactly_
+towards the centre of revolution, and therefore Kepler's second law is
+no longer precisely obeyed: the rate of description of areas is subject
+to slight acceleration. The effect of this tangential force acting on
+the tide-compelling body is gradually to increase its distance from the
+other body.
+
+Applying these statements to the earth and moon, we see that tidal
+energy is produced at the expense of the earth's rotation, and that the
+length of the day is thereby slowly increasing. Also that the moon's
+rotation relative to the earth has been destroyed by past tidal action
+in it (the only residue of ancient lunar rotation now being a scarcely
+perceptible libration), so that it turns always the same face towards
+us. Moreover, that its distance from the earth is steadily increasing.
+This last is the theory of Professor G.H. Darwin.
+
+Long ago the moon must therefore have been much nearer the earth, and
+the day was much shorter. The tides were then far more violent.
+
+Halving the distance would make them eight times as high; quartering it
+would increase them sixty-four-fold. A most powerful geological denuding
+agent. Trade winds and storms were also more violent.
+
+If ever the moon were close to the earth, it would have to revolve round
+it in about three hours. If the earth rotated on its axis in three
+hours, when fluid or pasty, it would be unstable, and begin to separate
+a portion of itself as a kind of bud, which might then get detached and
+gradually pushed away by the violent tidal action. Hence it is possible
+that this is the history of the moon. If so, it is probably an
+exceptional history. The planets were not formed from the sun in this
+way.
+
+Mars' moons revolve round him more quickly than the planet rotates:
+hence with them the process is inverted, and they must be approaching
+him and may some day crash along his surface. The inner moon is now
+about 4,000 miles away, and revolves in 7-1/2 hours. It appears to be
+about 20 miles in diameter, and weighs therefore, if composed of rock,
+40 billion tons. Mars rotates in 24-1/2 hours.
+
+A similar fate may _possibly_ await our moon ages hence--by reason of
+the action of terrestrial tides produced by the sun.
+
+
+
+
+LECTURE XVIII
+
+THE TIDES, AND PLANETARY EVOLUTION
+
+
+In the last lecture we considered the local peculiarities of the tides,
+the way in which they were formed in open ocean under the action of the
+moon and the sun, and also the means by which their heights and times
+could be calculated and predicted years beforehand. Towards the end I
+stated that the subject was very far from being exhausted, and
+enumerated some of the large and interesting questions which had been
+left untouched. It is with some of these questions that I propose now to
+deal.
+
+I must begin by reminding you of certain well-known facts, a knowledge
+of which I may safely assume.
+
+And first we must remind ourselves of the fact that almost all the rocks
+which form the accessible crust of the earth were deposited by the
+agency of water. Nearly all are arranged in regular strata, and are
+composed of pulverized materials--materials ground down from
+pre-existing rocks by some denuding and grinding action. They nearly all
+contain vestiges of ancient life embedded in them, and these vestiges
+are mainly of marine origin. The strata which were once horizontal are
+now so no longer--they have been tilted and upheaved, bent and
+distorted, in many places. Some of them again have been metamorphosed by
+fire, so that their organic remains have been destroyed, and the traces
+of their aqueous origin almost obliterated. But still, to the eye of the
+geologist, all are of aqueous or sedimentary origin: roughly speaking,
+one may say they were all deposited at the bottom of some ancient sea.
+
+The date of their formation no man yet can tell, but that it was vastly
+distant is certain. For the geological era is not over. Aqueous action
+still goes on: still does frost chip the rocks into fragments; still do
+mountain torrents sweep stone and mud and _d�bris_ down the gulleys and
+watercourses; still do rivers erode their channels, and carry mud and
+silt far out to sea. And, more powerful than any of these agents of
+denudation, the waves and the tides are still at work along every
+coast-line, eating away into the cliffs, undermining gradually and
+submerging acre after acre, and making with the refuse a shingly, or a
+sandy, or a muddy beach--the nucleus of a new geological formation.
+
+Of all denuding agents, there can be no doubt that, to the land exposed
+to them, the waves of the sea are by far the most powerful. Think how
+they beat and tear, and drive and drag, until even the hardest rock,
+like basalt, becomes honeycombed into strange galleries and
+passages--Fingal's Cave, for instance--and the softer parts are crumbled
+away. But the area now exposed to the teeth of the waves is not great.
+The fury of a winter storm may dash them a little higher than usual, but
+they cannot reach cliffs 100 feet high. They can undermine such cliffs
+indeed, and then grind the fragments to powder, but their direct action
+is limited. Not so limited, however, as they would be without the tides.
+Consider for a moment the denudation import of the tides: how does the
+existence of tidal rise and fall affect the geological problem?
+
+The scouring action of the tidal currents themselves is not to be
+despised. It is the tidal ebb and flow which keeps open channel in the
+Mersey, for instance. But few places are so favourably situated as
+Liverpool in this respect, and the direct scouring action of the tides
+in general is not very great. Their geological import mainly consists in
+this--that they raise and lower the surface waves at regular intervals,
+so as to apply them to a considerable stretch of coast. The waves are a
+great planing machine attacking the land, and the tides raise and lower
+this planing machine, so that its denuding tooth is applied, now twenty
+feet vertically above mean level, now twenty feet below.
+
+Making all allowance for the power of winds and waves, currents, tides,
+and watercourses, assisted by glacial ice and frost, it must be apparent
+how slowly the work of forming the rocks is being carried on. It goes on
+steadily, but so slowly that it is estimated to take 6000 years to wear
+away one foot of the American continent by all the denuding causes
+combined. To erode a stratum 5000 feet thick will require at this rate
+thirty million years.
+
+The age of the earth is not at all accurately known, but there are many
+grounds for believing it not to be much older than some thirty million
+years. That is to say, not greatly more than this period of time has
+elapsed since it was in a molten condition. It may be as old as a
+hundred million years, but its age is believed by those most competent
+to judge to be more likely within this limit than beyond it. But if we
+ask what is the thickness of the rocks which in past times have been
+formed, and denuded, and re-formed, over and over again, we get an
+answer, not in feet, but in miles. The Laurentian and Huronian rocks of
+Canada constitute a stratum ten miles thick; and everywhere the rocks at
+the base of our stratified system are of the most stupendous volume and
+thickness.
+
+It has always been a puzzle how known agents could have formed these
+mighty masses, and the only solution offered by geologists was,
+unlimited time. Given unlimited time, they could, of course, be formed,
+no matter how slowly the process went on. But inasmuch as the time
+allowable since the earth was cool enough for water to exist on it
+except as steam is not by any means unlimited, it becomes necessary to
+look for a far more powerful engine than any now existing; there must
+have been some denuding agent in those remote ages--ages far more
+distant from us than the Carboniferous period, far older than any forms
+of life, fossil or otherwise, ages among the oldest known to geology--a
+denuding agent must have then existed, far more powerful than any we now
+know.
+
+Such an agent it has been the privilege of astronomy and physics, within
+the last ten years, to discover. To this discovery I now proceed to lead
+up.
+
+Our fundamental standard of time is the period of the earth's
+rotation--the length of the day. The earth is our one standard clock:
+all time is expressed in terms of it, and if it began to go wrong, or if
+it did not go with perfect uniformity, it would seem a most difficult
+thing to discover its error, and a most puzzling piece of knowledge to
+utilize when found.
+
+That it does not go much wrong is proved by the fact that we can
+calculate back to past astronomical events--ancient eclipses and the
+like--and we find that the record of their occurrence, as made by the
+old magi of Chald�a, is in very close accordance with the result of
+calculation. One of these famous old eclipses was observed in Babylon
+about thirty-six centuries ago, and the Chald�an astronomers have put on
+record the time of its occurrence. Modern astronomers have calculated
+back when it should have occurred, and the observed time agrees very
+closely with the actual, but not exactly. Why not exactly?
+
+Partly because of the acceleration of the moon's mean motion, as
+explained in the lecture on Laplace (p. 262). The orbit of the earth was
+at that time getting rounder, and so, as a secondary result, the speed
+of the moon was slightly increasing. It is of the nature of a
+perturbation, and is therefore a periodic not a progressive or
+continuous change, and in a sufficiently long time it will be reversed.
+Still, for the last few thousand years the moon's motion has been, on
+the whole, accelerated (though there seems to be a very slight retarding
+force in action too).
+
+Laplace thought that this fact accounted for the whole of the
+discrepancy; but recently, in 1853, Professor Adams re-examined the
+matter, and made a correction in the details of the theory which
+diminishes its effect by about one-half, leaving the other half to be
+accounted for in some other way. His calculations have been confirmed by
+Professor Cayley. This residual discrepancy, when every known cause has
+been allowed for, amounts to about one hour.
+
+     The eclipse occurred later than calculation warrants. Now this
+     would have happened from either of two causes, either an
+     acceleration of the moon in her orbit, or a retardation of the
+     earth in her diurnal rotation--a shortening of the month or a
+     lengthening of the day, or both. The total discrepancy being, say,
+     two hours, an acceleration of six seconds-per-century per century
+     will in thirty-six centuries amount to one hour; and this,
+     according to the corrected Laplacian theory, is what has occurred.
+     But to account for the other hour some other cause must be sought,
+     and at present it is considered most probably due to a steady
+     retardation of the earth's rotation--a slow, very slow, lengthening
+     of the day.
+
+     The statement that a solar eclipse thirty-six centuries ago was an
+     hour late, means that a place on the earth's surface came into the
+     shadow one hour behind time--that is, had lagged one twenty-fourth
+     part of a revolution. The earth, therefore, had lost this amount in
+     the course of 3600 � 365-1/4 revolutions. The loss per revolution
+     is exceedingly small, but it accumulates, and at any era the total
+     loss is the sum of all the losses preceding it. It may be worth
+     while just to explain this point further.
+
+     Suppose the earth loses a small piece of time, which I will call an
+     instant, per day; a locality on the earth will come up to a given
+     position one instant late on the first day after an event. On the
+     next day it would come up two instants late by reason of the
+     previous loss; but it also loses another instant during the course
+     of the second day, and so the total lateness by the end of that day
+     amounts to three instants. The day after, it will be going slower
+     from the beginning at the rate of two instants a day, it will lose
+     another instant on the fresh day's own account, and it started
+     three instants late; hence the aggregate loss by the end of the
+     third day is 1 + 2 + 3 = 6. By the end of the fourth day the whole
+     loss will be 1 + 2 + 3 + 4, and so on. Wherefore by merely losing
+     one instant every day the total loss in _n_ days is (1 + 2 + 3 +
+     ... + _n_) instants, which amounts to 1/2_n_ (_n_ + 1) instants;
+     or practically, when _n_ is big, to 1/2n^2. Now in thirty-six
+     centuries there have been 3600 � 365-1/4 days, and the total loss
+     has amounted to an hour; hence the length of "an instant," the loss
+     per diem, can be found from the equation 1/2(3600 � 365)^2 instants
+     = 1 hour; whence one "instant" equals the 240 millionth part of a
+     second. This minute quantity represents the retardation of the
+     earth per day. In a year the aggregate loss mounts up to 1/3600th
+     part of a second, in a century to about three seconds, and in
+     thirty-six centuries to an hour. But even at the end of the
+     thirty-six centuries the day is barely any longer; it is only 3600
+     � 365 instants, that is 1/180th of a second, longer than it was at
+     the beginning. And even a million years ago, unless the rate of
+     loss was different (as it probably was), the day would only be
+     thirty-five minutes shorter, though by that time the aggregate
+     loss, as measured by the apparent lateness of any perfectly
+     punctual event reckoned now, would have amounted to nine years.
+     (These numbers are to be taken as illustrative, not as precisely
+     representing terrestrial fact.)
+
+What can have caused the slowing down? Swelling of the earth by reason
+of accumulation of meteoric dust might do something, but probably very
+little. Contraction of the earth as it goes on cooling would act in the
+opposite direction, and probably more than counterbalance the dust
+effect. The problem is thus not a simple one, for there are several
+disturbing causes, and for none of them are the data enough to base a
+quantitative estimate upon; but one certain agent in lengthening the
+day, and almost certainly the main agent, is to be found in the tides.
+
+Remember that the tidal humps were produced as the prolateness of a
+sphere whirled round and round a fixed centre, like a football whirled
+by a string. These humps are pulled at by the moon, and the earth
+rotates on its axis against this pull. Hence it tends to be constantly,
+though very slightly, dragged back.
+
+In so far as the tidal wave is allowed to oscillate freely, it will
+swing with barely any maintaining force, giving back at one
+quarter-swing what it has received at the previous quarter; but in so
+far as it encounters friction, which it does in all channels where
+there is an actual ebb and flow of the water, it has to receive more
+than it gives back, and the balance of energy has to be made up to it,
+or the tides would cease. The energy of the tides is, in fact,
+continually being dissipated by friction, and all the energy so
+dissipated is taken from the rotation of the earth. If tidal energy were
+utilized by engineers, the machines driven would be really driven at the
+expense of the earth's rotation: it would be a mode of harnessing the
+earth and using the moon as fixed point or fulcrum; the moon pulling at
+the tidal protuberance, and holding it still as the earth rotates, is
+the mechanism whereby the energy is extracted, the handle whereby the
+friction brake is applied.
+
+     Winds and ocean currents have no such effect (as Mr. Fronde in
+     _Oceania_ supposes they have), because they are all accompanied by
+     a precisely equal counter-current somewhere else, and no internal
+     rearrangement of fluid can affect the motion of a mass as a whole;
+     but the tides are in different case, being produced, not by
+     internal inequalities of temperature, but by a straightforward pull
+     from an external body.
+
+The ultimate effect of tidal friction and dissipation of energy will,
+therefore, be to gradually retard the earth till it does not rotate with
+reference to the moon, _i.e._ till it rotates once while the moon
+revolves once; in other words, to make the day and the month equal. The
+same cause must have been in operation, but with eighty-fold greater
+intensity, on the moon. It has ceased now, because the rotation has
+stopped, but if ever the moon rotated on its axis with respect to the
+earth, and if it were either fluid itself or possessed any liquid ocean,
+then the tides caused by the pull of the earth must have been
+prodigious, and would tend to stop its rotation. Have they not
+succeeded? Is it not probable that this is _why_ the moon always now
+turns the same face towards us? It is believed to be almost certainly
+the cause. If so, there was a time when the moon behaved
+differently--when it rotated more quickly than it revolved, and
+exhibited to us its whole surface. And at this era, too, the earth
+itself must have rotated a little faster, for it has been losing speed
+ever since.
+
+We have thus arrived at this fact, that a thousand years ago the day was
+a trifle shorter than it is now. A million years ago it was, perhaps, an
+hour shorter. Twenty million years ago it must have been much shorter.
+Fifty million years ago it may have been only a few hours long. The
+earth may have spun round then quite quickly. But there is a limit. If
+it spun too fast it would fly to pieces. Attach shot by means of wax to
+the whirling earth model, Fig. 110, and at a certain speed the cohesion
+of the wax cannot hold them, so they fly off. The earth is held together
+not by cohesion but by gravitation; it is not difficult to reckon how
+fast the earth must spin for gravity at its surface to be annulled, and
+for portions to fly off. We find it about one revolution in three hours.
+This is a critical speed. If ever the day was three hours long,
+something must have happened. The day can never have been shorter than
+that; for if it were, the earth would have a tendency to fly in pieces,
+or, at least, to separate into two pieces. Remember this, as a natural
+result of a three-hour day, which corresponds to an unstable state of
+things; remember also that in some past epoch a three-hour day is a
+probability.
+
+     If we think of the state of things going on in the earth's
+     atmosphere, if it had an atmosphere at that remote date, we shall
+     recognize the existence of the most fearful tornadoes. The trade
+     winds, which are now peaceful agents of commerce, would then be
+     perpetual hurricanes, and all the denudation agents of the
+     geologist would be in a state of feverish activity. So, too, would
+     the tides: instead of waiting six hours between low and high tide,
+     we should have to wait only three-quarters of an hour. Every
+     hour-and-a-half the water would execute a complete swing from high
+     tide to high again.
+
+Very well, now leave the earth, and think what has been happening to the
+moon all this while.
+
+We have seen that the moon pulls the tidal hump nearest to it back; but
+action and reaction are always equal and opposite--it cannot do that
+without itself getting pulled forward. The pull of the earth on the moon
+will therefore not be quite central, but will be a little in advance of
+its centre; hence, by Kepler's second law, the rate of description of
+areas by its radius vector cannot be constant, but must increase (p.
+208). And the way it increases will be for the radius vector to
+lengthen, so as to sweep out a bigger area. Or, to put it another way,
+the extra speed tending to be gained by the moon will fling it further
+away by extra centrifugal force. This last is not so good a way of
+regarding the matter; though it serves well enough for the case of a
+ball whirled at the end of an elastic string. After having got up the
+whirl, the hand holding the string may remain almost fixed at the centre
+of the circle, and the motion will continue steadily; but if the hand be
+moved so as always to pull the string a little in advance of the centre,
+the speed of whirl will increase, the elastic will be more and more
+stretched, until the whirling ball is describing a much larger circle.
+But in this case it will likewise be going faster--distance and speed
+increase together. This is because it obeys a different law from
+gravitation--the force is not inversely as the square, or any other
+single power, of the distance. It does not obey any of Kepler's laws,
+and so it does not obey the one which now concerns us, viz. the third;
+which practically states that the further a planet is from the centre
+the slower it goes; its velocity varies inversely with the square root
+of its distance (p. 74).
+
+If, instead of a ball held by elastic, it were a satellite held by
+gravity, an increase in distance must be accompanied by a diminution in
+speed. The time of revolution varies as the square of the cube root of
+the distance (Kepler's third law). Hence, the tidal reaction on the
+moon, having as its primary effect, as we have seen, the pulling the
+moon a little forward, has also the secondary or indirect effect of
+making it move slower and go further off. It may seem strange that an
+accelerating pull, directed in front of the centre, and therefore always
+pulling the moon the way it is going, should retard it; and that a
+retarding force like friction, if such a force acted, should hasten it,
+and make it complete its orbit sooner; but so it precisely is.
+
+Gradually, but very slowly, the moon is receding from us, and the month
+is becoming longer. The tides of the earth are pushing it away. This is
+not a periodic disturbance, like the temporary acceleration of its
+motion discovered by Laplace, which in a few centuries, more or less,
+will be reversed; it is a disturbance which always acts one way, and
+which is therefore cumulative. It is superposed upon all periodic
+changes, and, though it seems smaller than they, it is more inexorable.
+In a thousand years it makes scarcely an appreciable change, but in a
+million years its persistence tells very distinctly; and so, in the long
+run, the month is getting longer and the moon further off. Working
+backwards also, we see that in past ages the moon must have been nearer
+to us than it is now, and the month shorter.
+
+Now just note what the effect of the increased nearness of the moon was
+upon our tides. Remember that the tide-generating force varies inversely
+as the cube of distance, wherefore a small change of distance will
+produce a great difference in the tide-force.
+
+The moon's present distance is 240 thousand miles. At a time when it was
+only 190 thousand miles, the earth's tides would have been twice as high
+as they are now. The pushing away action was then a good deal more
+violent, and so the process went on quicker. The moon must at some time
+have been just half its present distance, and the tides would then have
+risen, not 20 or 30 feet, but 160 or 200 feet. A little further back
+still, we have the moon at one-third of its present distance from the
+earth, and the tides 600 feet high. Now just contemplate the effect of a
+600-foot tide. We are here only about 150 feet above the level of the
+sea; hence, the tide would sweep right over us and rush far away inland.
+At high tide we should have some 200 feet of blue water over our heads.
+There would be nothing to stop such a tide as that in this neighbourhood
+till it reached the high lands of Derbyshire. Manchester would be a
+seaport then with a vengeance!
+
+The day was shorter then, and so the interval between tide and tide was
+more like ten than twelve hours. Accordingly, in about five hours, all
+that mass of water would have swept back again, and great tracts of sand
+between here and Ireland would be left dry. Another five hours, and the
+water would come tearing and driving over the country, applying its
+furious waves and currents to the work of denudation, which would
+proceed apace. These high tides of enormously distant past ages
+constitute the denuding agent which the geologist required. They are
+very ancient--more ancient than the Carboniferous period, for instance,
+for no trees could stand the furious storms that must have been
+prevalent at this time. It is doubtful whether any but the very lowest
+forms of life then existed. It is the strata at the bottom of the
+geological scale that are of the most portentous thickness, and the only
+organism suspected in them is the doubtful _Eozoon Canadense_. Sir
+Robert Ball believes, and several geologists agree with him, that the
+mighty tides we are contemplating may have been co�val with this ancient
+Laurentian formation, and others of like nature with it.
+
+But let us leave geology now, and trace the inverted progress of events
+as we recede in imagination back through the geological era, beyond,
+into the dim vista of the past, when the moon was still closer and
+closer to the earth, and was revolving round it quicker and quicker,
+before life or water existed on it, and when the rocks were still
+molten.
+
+Suppose the moon once touched the earth's surface, it is easy to
+calculate, according to the principles of gravitation, and with a
+reasonable estimate of its size as then expanded by heat, how fast it
+must then have revolved round the earth, so as just to save itself from
+falling in. It must have gone round once every three hours. The month
+was only three hours long at this initial epoch.
+
+Remember, however, the initial length of the day. We found that it was
+just possible for the earth to rotate on its axis in three hours, and
+that when it did so, something was liable to separate from it. Here we
+find the moon in contact with it, and going round it in this same
+three-hour period. Surely the two are connected. Surely the moon was a
+part of the earth, and was separating from it.
+
+That is the great discovery--the origin of the moon.
+
+Once, long ages back, at date unknown, but believed to be certainly as
+much as fifty million years ago, and quite possibly one hundred million,
+there was no moon, only the earth as a molten globe, rapidly spinning on
+its axis--spinning in about three hours. Gradually, by reason of some
+disturbing causes, a protuberance, a sort of bud, forms at one side, and
+the great inchoate mass separates into two--one about eighty times as
+big as the other. The bigger one we now call earth, the smaller we now
+call moon. Round and round the two bodies went, pulling each other into
+tremendously elongated or prolate shapes, and so they might have gone on
+for a long time. But they are unstable, and cannot go on thus: they must
+either separate or collapse. Some disturbing cause acts again, and the
+smaller mass begins to revolve less rapidly. Tides at once
+begin--gigantic tides of molten lava hundreds of miles high; tides not
+in free ocean, for there was none then, but in the pasty mass of the
+entire earth. Immediately the series of changes I have described begins,
+the speed of rotation gets slackened, the moon's mass gets pushed
+further and further away, and its time of revolution grows rapidly
+longer. The changes went on rapidly at first, because the tides were so
+gigantic; but gradually, and by slow degrees, the bodies get more
+distant, and the rate of change more moderate. Until, after the lapse of
+ages, we find the day twenty-four hours long, the moon 240,000 miles
+distant, revolving in 27-1/3 days, and the tides only existing in the
+water of the ocean, and only a few feet high. This is the era we call
+"to-day."
+
+The process does not stop here: still the stately march of events goes
+on; and the eye of Science strives to penetrate into the events of the
+future with the same clearness as it has been able to descry the events
+of the past. And what does it see? It will take too long to go into full
+detail: but I will shortly summarize the results. It sees this
+first--the day and the month both again equal, but both now about 1,400
+hours long. Neither of these bodies rotating with respect to each
+other--the two as if joined by a bar--and total cessation of
+tide-generating action between them.
+
+The date of this period is one hundred and fifty millions of years
+hence, but unless some unforeseen catastrophe intervenes, it must
+assuredly come. Yet neither will even this be the final stage; for the
+system is disturbed by the tide-generating force of the sun. It is a
+small effect, but it is cumulative; and gradually, by much slower
+degrees than anything we have yet contemplated, we are presented with a
+picture of the month getting gradually shorter than the day, the moon
+gradually approaching instead of receding, and so, incalculable myriads
+of ages hence, precipitating itself upon the surface of the earth whence
+it arose.
+
+Such a catastrophe is already imminent in a neighbouring planet--Mars.
+Mars' principal moon circulates round him at an absurd pace, completing
+a revolution in 7-1/2 hours, and it is now only 4,000 miles from his
+surface. The planet rotates in twenty-four hours as we do; but its tides
+are following its moon more quickly than it rotates after them; they are
+therefore tending to increase its rate of spin, and to retard the
+revolution of the moon. Mars is therefore slowly but surely pulling its
+moon down on to itself, by a reverse action to that which separated our
+moon. The day shorter than the month forces a moon further away; the
+month shorter than the day tends to draw a satellite nearer.
+
+This moon of Mars is not a large body: it is only twenty or thirty miles
+in diameter, but it weighs some forty billion tons, and will ultimately
+crash along the surface with a velocity of 8,000 miles an hour. Such a
+blow must produce the most astounding effects when it occurs, but I am
+unable to tell you its probable date.
+
+So far we have dealt mainly with the earth and its moon; but is the
+existence of tides limited to these bodies? By no means. No body in the
+solar system is rigid, no body in the stellar universe is rigid. All
+must be susceptible of some tidal deformation, and hence, in all of
+them, agents like those we have traced in the history of the earth and
+moon must be at work: the motion of all must be complicated by the
+phenomena of tides. It is Prof. George Darwin who has worked out the
+astronomical influence of the tides, on the principles of Sir William
+Thomson: it is Sir Robert Ball who has extended Mr. Darwin's results to
+the past history of our own and other worlds.[32]
+
+     Tides are of course produced in the sun by the action of the
+     planets, for the sun rotates in twenty-five days or thereabouts,
+     while the planets revolve in much longer periods than that. The
+     principal tide-generating bodies will be Venus and Jupiter; the
+     greater nearness of one rather more than compensating for the
+     greater mass of the other.
+
+     It may be interesting to tabulate the relative tide-producing
+     powers of the planets on the sun. They are as follows, calling that
+     of the earth 1,000:--
+
+  RELATIVE TIDE-PRODUCING POWERS OF THE PLANETS
+  ON THE SUN.
+
+  Mercury         1,121
+  Venus           2,339
+  Earth           1,000
+  Mars              304
+  Jupiter         2,136
+  Saturn          1,033
+  Uranus             21
+  Neptune             9
+
+     The power of all of them is very feeble, and by acting on different
+     sides they usually partly neutralize each other's action; but
+     occasionally they get all on one side, and in that case some
+     perceptible effect may be produced; the probable effect seems
+     likely to be a gentle heaving tide in the solar surface, with
+     breaking up of any incipient crust; and such an effect may be
+     considered as evidenced periodically by the great increase in the
+     number of solar spots which then break out.
+
+     The solar tides are, however, much too small to appreciably push
+     any planet away, hence we are not to suppose that the planets
+     originated by budding from the sun, in contradiction of the nebular
+     hypothesis. Nor is it necessary to assume that the satellites, as a
+     class, originated in the way ours did; though they may have done
+     so. They were more probably secondary rings. Our moon differs from
+     other satellites in being exceptionally large compared with the
+     size of its primary; it is as big as some of the moons of Jupiter
+     and Saturn. The earth is the only one of the small planets that has
+     an appreciable moon, and hence there is nothing forced or unnatural
+     in supposing that it may have had an exceptional history.
+
+     Evidently, however, tidal phenomena must be taken into
+     consideration in any treatment of the solar system through enormous
+     length of time, and it will probably play a large part in
+     determining its future.
+
+When Laplace and Lagrange investigated the question of the stability or
+instability of the solar system, they did so on the hypothesis that the
+bodies composing it were rigid. They reached a grand conclusion--that
+all the mutual perturbations of the solar system were periodic--that
+whatever changes were going on would reach a maximum and then begin to
+diminish; then increase again, then diminish, and so on. The system was
+stable, and its changes were merely like those of a swinging pendulum.
+
+But this conclusion is not final. The hypothesis that the bodies are
+rigid is not strictly true: and directly tidal deformation is taken into
+consideration it is perceived to be a potent factor, able in the long
+run to upset all their calculations. But it is so utterly and
+inconceivably minute--it only produces an appreciable effect after
+millions of years--whereas the ordinary perturbations go through their
+swings in some hundred thousand years or so at the most. Granted it is
+small, but it is terribly persistent; and it always acts in one
+direction. Never does it cease: never does it begin to act oppositely
+and undo what it has done. It is like the perpetual dropping of water.
+There may be only one drop in a twelvemonth, but leave it long enough,
+and the hardest stone must be worn away at last.
+
+*      *      *      *      *
+
+We have been speaking of millions of years somewhat familiarly; but
+what, after all, is a million years that we should not speak familiarly
+of it? It is longer than our lifetime, it is true. To the ephemeral
+insects whose lifetime is an hour, a year might seem an awful period,
+the mid-day sun might seem an almost stationary body, the changes of the
+seasons would be unknown, everything but the most fleeting and rapid
+changes would appear permanent and at rest. Conversely, if our
+life-period embraced myriads of �ons, things which now seem permanent
+would then appear as in a perpetual state of flux. A continent would be
+sometimes dry, sometimes covered with ocean; the stars we now call fixed
+would be moving visibly before our eyes; the earth would be humming on
+its axis like a top, and the whole of human history might seem as
+fleeting as a cloud of breath on a mirror.
+
+Evolution is always a slow process. To evolve such an animal as a
+greyhound from its remote ancestors, according to Mr. Darwin, needs
+immense tracts of time; and if the evolution of some feeble animal
+crawling on the surface of this planet is slow, shall the stately
+evolution of the planetary orbs themselves be hurried? It may be that we
+are able to trace the history of the solar system for some thousand
+million years or so; but for how much longer time must it not have a
+history--a history, and also a future--entirely beyond our ken?
+
+Those who study the stars have impressed upon them the existence of the
+most immeasurable distances, which yet are swallowed up as nothing in
+the infinitude of space. No less are we compelled to recognize the
+existence of incalculable �ons of time, and yet to perceive that these
+are but as drops in the ocean of eternity.
+
+
+FOOTNOTES:
+
+[1] The following account of Mars's motion is from the excellent small
+manual of astronomy by Dr. Haughton of Trinity College, Dublin:--(P.
+151) "Mars's motion is very unequal; when he first appears in the
+morning emerging from the rays of the sun, his motion is direct and
+rapid; it afterwards becomes slower, and he becomes stationary when at
+an elongation of 137� from the sun; then his motion becomes retrograde,
+and its velocity increases until he is in opposition to the sun at 180�;
+at this time the retrograde motion is most rapid, and afterwards
+diminishes until he is 137� distant from the sun on the other side, when
+Mars again becomes stationary; his motion then becomes direct, and
+increases in velocity until it reaches a maximum, when the planet is
+again in conjunction with the sun. The retrograde motion of this planet
+lasts for 73 days: and its arc of retrogradation is 16�."
+
+[2] It is not so easy to plot the path of the sun among the stars by
+direct observation, as it is to plot the path of a planet; because sun
+and stars are not visible together. Hipparchus used the moon as an
+intermediary; since sun and moon are visible together, and also moon and
+stars.
+
+[3] This is, however, by no means the whole of the matter. The motion is
+not a simple circle nor has it a readily specifiable period. There are
+several disturbing causes. All that is given here is a first rough
+approximation.
+
+[4] The proof is easy, and ought to occur in books on solid geometry. By
+a "regular" solid is meant one with all its faces, edges, angles, &c.,
+absolutely alike: it is of these perfectly symmetrical bodies that there
+are only five. Crystalline forms are practically infinite in number.
+
+[5] Best known to us by his Christian name, as so many others of that
+time are known, _e.g._ Raphael Sanzio, Dante Alighieri, Michael Angelo
+Buonarotti. The rule is not universal. Tasso and Ariosto are surnames.
+
+[6] It would seem that the fact that all bodies of every material tend
+to fall at the same rate is still not clearly known. Confusion is
+introduced by the resistance of the air. But a little thought should
+make it clear that the effect of the air is a mere disturbance, to be
+eliminated as far as possible, since the atmosphere has nothing to do
+with gravitation. The old fashioned "guinea and feather experiment"
+illustrates that in a vacuum things entirely different in specific
+gravity or surface drop at the same pace.
+
+[7] Karl von Gebler (Galileo), p. 13.
+
+[8] It is of course the "silver lining" of clouds that outside observers
+see.
+
+[9] L.U.K., _Life of Galileo_, p. 26.
+
+[10] _Note added September, 1892._ News from the Lick Observatory makes
+a very small fifth satellite not improbable.
+
+[11] They remained there till this century. In 1835 they were quietly
+dropped.
+
+[12] It was invented by van Helmont, a Belgian chemist, who died in
+1644. He suggested two names _gas_ and _blas_, and the first has
+survived. Blas was, I suppose, from _blasen_, to blow, and gas seems to
+be an attempt to get at the Sanskrit root underlying all such words as
+_geist_.
+
+[13] Such as this, among many others:--The duration of a flame under
+different conditions is well worth determining. A spoonful of warm
+spirits of wine burnt 116 pulsations. The same spoonful of spirits of
+wine with addition of one-sixth saltpetre burnt 94 pulsations. With
+one-sixth common salt, 83; with one-sixth gunpowder, 110; a piece of wax
+in the middle of the spirit, 87; a piece of _Kieselstein_, 94; one-sixth
+water, 86; and with equal parts water, only 4 pulse-beats. This, says
+Liebig, is given as an example of a "_licht-bringende Versuch_."
+
+[14] Draper, _History of Civilization in Europe_, vol. ii. p. 259.
+
+[15] Professor Knight's series of Philosophical Classics.
+
+[16] To explain why the entire system, horse and cart together, move
+forward, the forces acting on the ground must be attended to.
+
+[17] The distance being proportional to the _square_ of the time, see p.
+82.
+
+[18] The following letter, recently unearthed and published in _Nature_,
+May 12, 1881, seems to me well worth preserving. The feeling of a
+respiratory interval which it describes is familiar to students during
+the too few periods of really satisfactory occupation. The early guess
+concerning atmospheric electricity is typical of his extraordinary
+instinct for guessing right.
+
+  "LONDON, _Dec. 15, 1716_.
+
+"DEAR DOCTOR,--He that in ye mine of knowledge deepest diggeth, hath,
+like every other miner, ye least breathing time, and must sometimes at
+least come to terr. alt. for air.
+
+"In one of these respiratory intervals I now sit down to write to you,
+my friend.
+
+"You ask me how, with so much study, I manage to retene my health. Ah,
+my dear doctor, you have a better opinion of your lazy friend than he
+hath of himself. Morpheous is my last companion; without 8 or 9 hours of
+him yr correspondent is not worth one scavenger's peruke. My practices
+did at ye first hurt my stomach, but now I eat heartily enou' as y' will
+see when I come down beside you.
+
+"I have been much amused at ye singular [Greek: _phenomena_] resulting
+from bringing of a needle into contact with a piece of amber or resin
+fricated on silke clothe. Ye flame putteth me in mind of sheet lightning
+on a small--how very small--scale. But I shall in my epistles abjure
+Philosophy whereof when I come down to Sakly I'll give you enou'. I
+began to scrawl at 5 mins. from 9 of ye clk. and have in writing consmd.
+10 mins. My Ld. Somerset is announced.
+
+"Farewell, Gd. bless you and help yr sincere friend.
+
+  "ISAAC NEWTON.
+
+  "_To_ DR. LAW, Suffolk."
+
+
+
+[19] Kepler's laws may be called respectively, the law of path, the law
+of speed, and the relationship law. By the "mass" of a body is meant the
+number of pounds or tons in it: the amount of matter it contains. The
+idea is involved in the popular word "massive."
+
+[20] The equation we have to verify is
+
+         4[pi]^2r^3
+  gR^2 = -----------,
+            T^2
+
+with the data that _r_, the moon's distance, is 60 times R, the earth's
+radius, which is 3,963 miles; while T, the time taken to complete the
+moon's orbit, is 27 days, 13 hours, 18 minutes, 37 seconds. Hence,
+suppose we calculate out _g_, the intensity of terrestrial gravity, from
+the above equation, we get
+
+          4[pi]^2               39�92 � 216000 � 3963 miles
+  _g_ = ---------- � (60)^3R = -----------------------------
+             T                  (27 days, 13 hours, &c.)^2
+
+           = 32�57 feet-per-second per second,
+
+which is not far wrong.
+
+[21] The two motions may be roughly compounded into a single motion,
+which for a few centuries may without much error be regarded as a
+conical revolution about a different axis with a different period; and
+Lieutenant-Colonel Drayson writes books emphasizing this simple fact,
+under the impression that it is a discovery.
+
+[22] Members of the Accademia dei Lyncei, the famous old scientific
+Society established in the time of Cosmo de Medici--older than our own
+Royal Society.
+
+[23] Newton suspected that the moon really did so oscillate, and so it
+may have done once; but any real or physical libration, if existing at
+all, is now extremely minute.
+
+[24] An interesting picture in the New Gallery this year (1891),
+attempting to depict "Earth-rise in Moon-land," unfortunately errs in
+several particulars. First of all, the earth does not "rise," but is
+fixed relatively to each place on the moon; and two-fifths of the moon
+never sees it. Next, the earth would not look like a map of the world
+with a haze on its edge. Lastly, whatever animal remains the moon may
+contain would probably be rather in the form of fossils than of
+skeletons. The skeleton is of course intended as an image of death and
+desolation. It is a matter of taste: but a skeleton, it seems to me,
+speaks too recently of life to be as appallingly weird and desolate as a
+blank stone or ice landscape, unshaded by atmosphere or by any trace of
+animal or plant life, could be made.
+
+[25] Five of Jupiter's revolutions occupy 21,663 days; two of Saturn's
+revolutions occupy 21,526 days.
+
+[26] _Excircularity_ is what is meant by this term. It is called
+"excentricity" because the foci (not the centre) of an ellipse are
+regarded as the representatives of the centre of a circle. Their
+distance from the centre, compared with the radius of the unflattened
+circle, is called the excentricity.
+
+[27] A curve of the _n_th degree has 1/2_n_(_n_+3) arbitrary constants
+in its equation, hence this number of points specifically determine it.
+But special points, like focus or vertex, count as two ordinary ones.
+Hence three points plus the focus act as five points, and determine a
+conic or curve of the second degree. Three observations therefore fix an
+orbit round the sun.
+
+[28] Its name suggests a measure of the diameter of the sun's disk, and
+this is one of its functions; but it can likewise measure planetary and
+other disks; and in general behaves as the most elaborate and expensive
+form of micrometer. The K�nigsberg instrument is shewn in fig. 92.
+
+[29] It may be supposed that the terms "minute" and "second" have some
+necessary connection with time, but they are mere abbreviations for
+_partes minut�_ and _partes minut� secund�_, and consequently may be
+applied to the subdivision of degrees just as properly as to the
+subdivision of hours. A "second" of arc means the 3600th part of a
+degree, just as a second of time means the 3600th part of an hour.
+
+[30] A group of flying particles, each one invisible, obstructs light
+singularly little, even when they are close together, as one can tell by
+the transparency of showers and snowstorms. The opacity of haze may be
+due not merely to dust particles, but to little eddies set up by
+radiation above each particle, so that the air becomes turbulent and of
+varying density. (See a similar suggestion by Mr. Poynting in _Nature_,
+vol. 39, p. 323.)
+
+[31] The moon ought to be watched during the next great shower, if the
+line of fire happens to take effect on a visible part of the dark
+portion.
+
+[32] Address to Birmingham Midland Institute, "A Glimpse through the
+Corridors of Time."
+
+
+
+
+INDEX
+
+INDEX
+
+
+A
+
+Abbott, T.K., on tides, 369
+
+Adams, John Couch, 193, 217, 302, 323, 324, 325, 327, 329, 330, 352, 385
+
+Airy, Sir George, 193, 244, 302, 323, 324, 327, 367
+
+Anaxagoras, 15
+
+Appian, 218
+
+Arabs, the, form a link between the old and new science, 9
+
+Archimedes, 7, 8, 84, 87, 144, 177
+
+Aristarchus, 34
+
+Aristotle, 66, 69, 88, 94, 99, 167.
+  He taught that the earth was a sphere, 16;
+  his theories did not allow of the earth's motion, 34;
+  he was regarded as inspired, 89
+
+
+B
+
+Bacon, Francis, 142, 143, 144, 145.
+  His _Novum Organum_, 141
+
+Bacon, Roger, 96, 139, 140.
+  The herald of the dawn of science, 9
+
+Brah�, George, uncle of Tycho Brah�, 39
+
+Brah�, Steno, brother of Tycho Brah�, 39
+
+Brah�, Tycho, 37, 39, 40, 44, 45, 49, 51, 53, 54, 55, 58, 63, 64, 65, 66,
+ 68, 71, 72, 74, 75, 77, 78, 86, 94, 117, 137, 155, 165, 166, 200, 244,
+ 281, 288.
+  He tried to adopt the main features of the Copernican theory without
+   admitting the motion of the earth, 37;
+  he was a poor theorist but a great observer, 38;
+  his medicine, 44;
+  his personal history, 39, _seq._;
+  his observatory, Uraniburg, 47;
+  his greatest invention, 50, note;
+  his maniac Lep, 52;
+  his kindness to Kepler, 63
+
+Ball, Sir R., 391, 394;
+  his _Story of the Heavens_, 377
+
+Barrow, Dr., 165, 187
+
+Bessel, 288, 310, 311, 313, 315, 316, 318, 323
+
+Biela, 345, 346, 347
+
+Bode's Law, 60, 296, 298, 299, 326
+
+Boyle, 139, 188
+
+Bradley, Prof. James, 233, 246, 247, 249, 252, 253, 308, 319
+
+Bremiker, 328, 329
+
+Brewster, on Kepler, 78
+
+Brinkley, 308
+
+Bruno, Giordano, 108, 127
+
+
+C
+
+Castelli, 112, 133
+
+Cayley, Prof., 385
+
+Challis, Prof., 328, 329
+
+Clairut, 193, 216, 217, 219, 234, 341
+
+Clark, Alvan and Sons, 316
+
+Columbus, 9, 144
+
+Copernicus, 7, 10, _seq._, 14, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38,
+ 62, 66, 68, 70, 78, 93, 95, 100, 108, 111, 121, 122, 137, 155, 166, 223,
+ 234, 247, 307;
+  his _De Revolutionibus Orbium Coelestium_, 11, 75, 138;
+  he _proved_ that the earth went round the sun, 13;
+  the influence of his theory on the Church, 13, _seq._;
+  his life-work summarised, 30;
+  his Life by Mr. E.J.C. Morton, 31
+
+Copernican tables, 40;
+  Copernican theory, 59, 60, 125, 144, 167
+
+Copernik, Nicolas; see Copernicus
+
+Cornu, 238
+
+Croll, Dr., his _Climate and Time_, 264
+
+
+D
+
+D'Alembert, 193, 234
+
+Darwin, Charles, 134, 138, 397
+
+Darwin, Prof. George, 367, 394
+
+Delambre, 253
+
+Descartes, 145, 146, 148, 151, 153, 156, 158, 164, 165, 167, 178, 181,
+ 224, 227;
+  his _Discourse on Method_, 142;
+  his dream, 147;
+  his system of algebraic geometry, 149, _seq._;
+  his doctrine of vortices, 151, _seq._;
+  his _Principia Mathematica_, 154;
+  his Life by Mr. Mahaffy, 154
+
+
+E
+
+Earth, the difficulties in the way of believing that it moved, 34, _seq._
+
+"Earth-rise in Moon-land," 258, note
+
+Encke, 345, 346
+
+Epicyclic orbits explained, 23, _seq._
+
+Equinoxes, their precession discovered by Hipparchus, 27
+
+Eudoxus, 19
+
+Euler, 193, 234
+
+
+F
+
+Faraday, 84
+
+Fizeau, 238, 239
+
+Flamsteed, 215, 246, 284, 308, 319
+
+Fraunhofer, 311
+
+Froude, Prof.; his _Oceania_, 387
+
+
+G
+
+Galen, 87
+
+Galileo, Galilei, 63, 75, 84, 88, 90, 92, 93, 97, 98, 101, 104, 106, 107,
+ 108, 109, 110, 112, 114, 116, 117, 118, 120, 121, 122, 123, 125, 127,
+ 133, 134, 137, 144, 145, 153, 154, 157, 165, 166, 167, 168, 177, 188,
+ 200, 224, 227, 256, 281, 288, 309, 361;
+  his youth, 85;
+  his discovery of the pendulum, 86;
+  his first observations about falling bodies, 88, _seq._;
+  he invents a telescope, 95;
+  he adopts the Copernican theory, 94;
+  he conceives "earth-shine," 100;
+  he discovers Jupiter's moons, 103;
+  he studies Saturn, 114, _seq._;
+  his _Dialogues on the Ptolemaic and Copernican Systems_, 124;
+  his abjuration, 130;
+  he becomes blind, 132;
+  he discovered the Laws of Motion, 167, _seq._;
+  he guessed that sight was not instantaneous, 236, 237
+
+Galle, Dr., 245, 329
+
+Gauss, 299, 300
+
+Gilbert, Dr., 139, 140, 157, 188;
+  his _De Magnete_, 140, 144
+
+Greeks, their scientific methods, 7
+
+Groombridge's Catalogue, 315
+
+
+H
+
+Hadley, 185
+
+Halley, 192, 193, 194, 195, 197, 215, 218, 219, 246, 258, 260, 261, 340,
+ 341;
+  he discovered the _Principia_, 194
+
+Harvey, 144, 149
+
+Haughton, Dr., 321;
+  his manual on Astronomy, 21, note
+
+Heliometer, described, 311
+
+Helmholtz, 378
+
+Helmont, Van, invented the word "gas," 141
+
+Henderson, 310, 314
+
+Herschel, Alexander, 275, 277, 278, 279
+
+Herschel, Caroline, 275, 276, 279, 286, 345;
+  her journal quoted, 277, _seq._;
+  her work with William H. described, 284
+
+Herschel, Sir John, 283, 285, 327, 329
+
+Herschel, William, 185, 234, 235, 244, 249, 274, 275, 280, 281, 282, 284,
+ 288, 289, 290, 293, 295, 305, 309, 310, 318, 319, 327;
+  he "sweeps" the heavens, 280;
+  his discovery of Uranus, 281, 287;
+  his artificial Saturn, 281, 282;
+  his methods of work with his sister, described, 284;
+  he founded the science of Astronomy, 287
+
+Hind, 300
+
+Hipparchus, 7, 18, 20, 27, 28, and note, 30, 40, 66, 223, 253;
+  an explanation of his discovery of the precession of the equinoxes,
+  27, seq.
+
+Hippocrates, 87
+
+Homeric Cosmogony, 15, _seq._
+
+Hooke, 139, 188, 192, 193, 196, 197, 308
+
+H�pital, Marquis de l', 228
+
+Horkey, Martin, 106
+
+Horrebow, 244
+
+Huxley, Prof., 149
+
+Huyghens, 86, 166, 185
+
+
+K
+
+Kant, 267, 270
+
+Kelvin, Lord, see Thomson, Sir W.
+
+Kepler, John, 59, 60, 63, 64, 65, 66, 70, 72, 73, 75, 77, 79, 84, 93,
+ 94, 95, 104, 106, 107, 110, 122, 137, 145, 153, 158, 164, 165, 166,
+ 167, 192, 200, 208, 209, 210, 211, 212, 214, 218, 224, 227, 253, 256,
+ 259, 260, 262, 288, 295, 296, 332, 338, 361, 389;
+  he replaced epicycles by an ellipse, 27;
+  he was a pupil of Tycho Brah�, 54;
+  he was a speculator more than an observer, 58;
+  his personal life, 58, _seq._;
+  his theories about the numbers and distances of the planets, 60, 62;
+  he was helped by Tycho, 63;
+  his main work, 65, _seq._;
+  he gave up circular motion, 69;
+  his _Mysterium Cosmographicon_, 105;
+  his Laws, 71, 74, 173, 174, 176, 179, 180, 206, _seq._
+
+
+L
+
+Lagrange, 193, 234, 255, 256, 257, 258, 263
+
+Lagrange and Laplace, 258, 266, 395;
+  they laid the foundations of the planetary theory, 259
+
+Laplace, 68, 193, 218, 234, 255, 261, 262, 267, 268, 269, 270, 272,
+ 288, 301, 317, 384, 385, 390;
+  his nebular hypothesis, 267, 292;
+  his _M�canique C�leste_, 323
+
+Lassell, Mr., 283, 284
+
+Leibnitz, 192, 197, 233
+
+Le Monnier, 319
+
+Leonardo, see Vinci, Leonardo da
+
+Leverrier, 193, 327, 328, 329, 330, 352
+
+Lippershey, Hans, 95
+
+
+M
+
+Maskelyne, 281
+
+Maxwell, Clerk, 302, 303
+
+Molyneux, 248, 249
+
+Morton, Mr. E.J. C, his Life of Copernicus, 31
+
+
+N
+
+Newton, Prof. H.A., 347
+
+Newton, Sir Isaac, 7, 30, 79, 138, 139, 144, 145, 149, 153, 157, 158,
+ 165, 166, 167, 174, 176, 184, 187, 188, 189, 191, 192, 194, 196, 198,
+ 199, 201, 213, 216, 219, 220, 221, 224, 226, 227, 228, 233, 242, 253,
+ 255, 256, 274, 288, 317, 340, 378;
+  his _Principia_, 191, 192, 193, 194, 195, 196, 197, 207, 214, 216, 218,
+   228, 233, 242, 253;
+  his early life, 161, _seq._;
+  his first experiments, 163;
+  his work at Cambridge, 164;
+  his Laws, 168;
+  his application of the Laws of Gravity to Astronomy, 177, 178, 179, 185,
+   190;
+  his reticence, 178;
+  his discoveries in Optics, 181, _seq._;
+  his work summarised, 186;
+  his _Optics_, 189;
+  anecdotes of him, 191;
+  his appearance in a Court of Justice, 195;
+  some of his manuscripts very recently discovered, 217;
+  his theories of the Equinoxes and tides, 223, _seq._, 225, 363, _seq._
+
+
+O
+
+Olbers, 299, 300
+
+
+P
+
+Peters, Prof., 300, 316
+
+Piazzi, 298, 299, 308, 313
+
+Picard, 190, 242, 244, 247
+
+Pioneers, genuine, 7
+
+Planets and days of the week, 18
+
+Poynting, 332
+
+Printing, 9
+
+Ptolemy, 18, 20, 27, 38, 153, 155, 166, 214;
+  his system of the Heavens simplified by Copernicus, 11, 30;
+  his system described, 19, _seq._;
+  his system taught, 34;
+  his harmonies, 74
+
+Pythagoras, 19, 20, 34
+
+
+Q
+
+Quadrant, an early, 42, 43
+
+
+R
+
+Rheiter, 107
+
+Ricci, Ostillio, 86, 87
+
+Roberts, Isaac, 268
+
+Roemer, 239, 240, 242, 244, 249, 251, 308
+
+Rosse, Lord, his telescope, 186, 268
+
+Rudolphine tables, 65
+
+
+S
+
+Scheiner, 107
+
+Sizzi, Francesca, an orthodox astronomer, 106
+
+Snell, Willebrod, and the law of refraction, 65
+
+Solar system, its fate, 265
+
+Stars, a list of, 307
+
+Struve, 308, 310, 311, 313
+
+Stuart, Prof., quoted, 52
+
+
+T
+
+Tatius, 296
+
+Telescopes, early, 96
+
+Thales, 7, 140, 317
+
+Thomson, Sir William, 367, 372, 373, 378, 394
+
+Tide-gauge, described, 373, _seq._
+
+Tides, 354, _seq._
+
+Time, is not exactly uniform, 384
+
+Torricelli, 133, 168
+
+Tycho, see Brah�, Tycho
+
+
+V
+
+Vinci, Leonardo da, 9, 100, 144, 184
+
+Viviani, 133, 168
+
+Voltaire, 181
+
+
+W
+
+Watson, Prof., 300
+
+Whewell, 227
+
+Wren, Sir Christopher, 188, 192, 193, 197
+
+
+Z
+
+Zach, Von, 296, 299
+
+Zone of Asteroids, 300, _seq._
+
+
+  THE END.
+
+  RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.
+
+
+
+
+
+End of the Project Gutenberg EBook of Pioneers of Science, by Oliver Lodge
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+
+The Project Gutenberg EBook of Pioneers of Science, by Oliver Lodge
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Pioneers of Science
+
+Author: Oliver Lodge
+
+Release Date: April 26, 2009 [EBook #28613]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK PIONEERS OF SCIENCE ***
+
+
+
+
+Produced by Audrey Longhurst, Greg Bergquist and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by The Internet Archive/American Libraries.)
+
+
+
+
+
+
+</pre>
+
+
+<div class="tn">
+
+<p class="center"><big><b>Transcriber&#8217;s Note</b></big></p>
+
+<p class="noin">The punctuation and spelling from the original text have been faithfully preserved. Only obvious
+typographical errors have been corrected.</p>
+
+<p class="noin">This text contains a few phrases in Greek, with English transliterations given as mouse hover pop-ups:
+<span class="greek" title="Greek: phenomena">&#966;&#949;&#957;&#8057;&#956;&#949;&#957;&#945;</span><br />
+
+Your browser should be set to read the UTF-8 character set.</p>
+
+
+</div>
+<hr />
+<div class="figcenter" style="width: 400px;">
+<img src="images/cover.jpg" width="400" height="641" alt="" title="Book Cover" />
+</div>
+<hr />
+
+<h3><br /><br />PIONEERS OF SCIENCE<br /><br /></h3>
+<hr />
+<div class="figcenter" style="width: 150px;">
+<img src="images/deco.jpg" width="150" height="108" alt="" title="Decoration" />
+</div>
+<hr  />
+<div class="figcenter" style="width: 450px;">
+<img src="images/afrontis.jpg" width="350" height="476" alt="NEWTON" title="" /><br />
+<span class="caption">NEWTON<br /><i>From the picture by Kneller, 1689, now at Cambridge</i></span>
+</div>
+
+<hr  />
+
+
+
+<h1>PIONEERS OF SCIENCE<br /><br /><br /></h1>
+
+<p class="center"><small>BY</small><br />
+<big>OLIVER LODGE, F.R.S.</big><br />
+<small>PROFESSOR OF PHYSICS IN VICTORIA UNIVERSITY COLLEGE, LIVERPOOL</small><br />
+<br /><br /><br /><br />
+<i>WITH PORTRAITS AND OTHER ILLUSTRATIONS</i><br />
+<br />
+<br /><br /></p>
+<p class="t1">London</p>
+<p class="center">MACMILLAN AND CO.<br />
+<small>AND NEW YORK</small><br />
+1893</p>
+<hr  />
+<p class="center"><br /><br /><br /><br />
+<span class="smcap">Richard Clay and Sons, Limited,</span><br />
+<span class="smcap">LONDON AND BUNGAY.</span><br /><br /><br /><br />
+</p>
+
+
+
+<hr  />
+<h2>PREFACE</h2>
+
+
+<p><span class="smcap">This </span>book takes its origin in a course of lectures on the history and
+progress of Astronomy arranged for me in the year 1887 by three of my
+colleagues (A.C.B., J.M., G.H.R.), one of whom gave the course its name.</p>
+
+<p>The lectures having been found interesting, it was natural to write them
+out in full and publish.</p>
+
+<p>If I may claim for them any merit, I should say it consists in their
+simple statement and explanation of scientific facts and laws. The
+biographical details are compiled from all readily available sources,
+there is no novelty or originality about them; though it is hoped that
+there may be some vividness. I have simply tried to present a living
+figure of each Pioneer in turn, and to trace his influence on the
+progress of thought.</p>
+
+<p>I am indebted to many biographers and writers, among others to Mr.
+E.J.C. Morton, whose excellent set of lives published by the S.P.C.K.
+saved me much trouble in the early part of the course.</p>
+
+<p>As we approach recent times the subject grows more complex, and the men
+more nearly contemporaries; hence the biographical aspect diminishes and
+the scientific treatment becomes fuller, but in no case has it been
+allowed to become technical and generally unreadable.</p>
+
+<p>To the friends (C.C.C., F.W.H.M., E.F.R.) who with great kindness have
+revised the proofs, and have indicated places where the facts could be
+made more readily intelligible by a clearer statement, I express my
+genuine gratitude.</p>
+
+<p>
+<span class="smcap">University College, Liverpool,</span><br />
+<span style="margin-left: 4em;"><i>November, 1892</i>.<br /></span>
+</p>
+
+
+
+<hr  />
+<h2>CONTENTS</h2>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Contents">
+<tr>
+ <td align='center' colspan='2'><big><i><a href="#PART_I">PART I</a></i></big></td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_I">LECTURE I</a></td>
+</tr>
+<tr>
+ <td align='right' colspan='2'><small>PAGE</small></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>COPERNICUS AND THE MOTION OF THE EARTH</small></td>
+ <td align='right'>2</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_II">LECTURE II</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>TYCHO BRAH&Eacute; AND THE EARLIEST OBSERVATORY </small></td>
+ <td align='right'>32</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#SUMMARY_OF_FACTS_FOR_LECTURE_III">LECTURE III</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>KEPLER AND THE LAWS OF PLANETARY MOTION </small></td>
+ <td align='right'>56</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#SUMMARY_OF_FACTS_FOR_LECTURES_IV_AND_V">LECTURE IV</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>GALILEO AND THE INVENTION OF THE TELESCOPE </small></td>
+ <td align='right'>80</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#LECTURE_V">LECTURE V</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>GALILEO AND THE INQUISITION </small></td>
+ <td align='right'>108</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#SUMMARY_OF_FACTS_FOR_LECTURE_VI">LECTURE VI</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>DESCARTES AND HIS THEORY OF VORTICES</small></td>
+ <td align='right'>136</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#SUMMARY_OF_FACTS_FOR_LECTURES_VII_AND_VIII">LECTURE VII</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>SIR ISAAC NEWTON</small></td>
+ <td align='right'>159</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#LECTURE_VIII">LECTURE VIII</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>NEWTON AND THE LAW OF GRAVITATION</small></td>
+ <td align='right'>180</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_FOR_LECTURE_IX">LECTURE IX</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>NEWTON'S "PRINCIPIA"</small></td>
+ <td align='right'>203</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><br /><big><i><a href="#PART_II">PART II</a></i></big></td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_TO_LECTURE_X">LECTURE X</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT</small></td>
+ <td align='right'>232</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_TO_LECTURE_XI">LECTURE XI</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>LAGRANGE AND LAPLACE&mdash;THE STABILITY OF THE SOLAR SYSTEM,
+AND THE NEBULAR HYPOTHESIS</small></td>
+ <td align='right'>254</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_TO_LECTURE_XII">LECTURE XII</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>HERSCHEL AND THE MOTION OF THE FIXED STARS</small></td>
+ <td align='right'>273</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_TO_LECTURE_XIII">LECTURE XIII</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>THE DISCOVERY OF THE ASTEROIDS</small></td>
+ <td align='right'>294</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_TO_LECTURE_XIV">LECTURE XIV</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>BESSEL&mdash;THE DISTANCES OF THE STARS, AND THE DISCOVERY OF
+STELLAR PLANETS</small></td>
+ <td align='right'>304</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#LECTURE_XV">LECTURE XV</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>THE DISCOVERY OF NEPTUNE</small></td>
+ <td align='right'>317</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#LECTURE_XVI">LECTURE XVI</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>COMETS AND METEORS</small></td>
+ <td align='right'>331</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_FOR_LECTURE_XVII">LECTURE XVII</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>THE TIDES</small></td>
+ <td align='right'>353</td>
+</tr>
+<tr class='tr1'>
+ <td align='center'><a href="#NOTES_FOR_LECTURE_XVIII">LECTURE XVIII</a></td>
+</tr>
+<tr class='tr3'>
+ <td align='left'><small>THE TIDES, AND PLANETARY EVOLUTION</small></td>
+ <td align='right'>379</td>
+</tr>
+</table></div>
+
+
+
+<hr  />
+<h2>ILLUSTRATIONS</h2>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Illustrations">
+<tr>
+ <td align='right'><small>FIG.</small></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'>&nbsp;</td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'><small>PAGE</small>
+</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_1">1.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Archimedes</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>8</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_2">2.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Leonardo da Vinci</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>10</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_3">3.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Copernicus</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>12</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_4">4.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Homeric Cosmogony</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>15</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_5">5.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Egyptian Symbol of the Universe</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>16</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_6">6.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Hindoo Earth</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>17</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_7">7.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Order of ancient Planets corresponding to the Days of the Week</span></td>
+ <td align='left'>&nbsp;</td><td align='right'>19</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_8">8.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Ptolemaic System</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>20</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_9">9.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Specimens of Apparent Paths of Venus and of Mars among the stars</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>21</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_10">10.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Apparent Epicyclic Orbits of Jupiter and Saturn</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>22</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_11">11.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Egyptian System</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>24</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_12">12.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">True Orbits of Earth and Jupiter</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>25</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_13">13.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Orbits of Mercury and Earth</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>25</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_14">14.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Copernican System as frequently represented</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>26</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_15">15.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Slow Movement of the North Pole in a Circle among the Stars</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>29</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_16">16.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tychonic system, showing the Sun with all the Planets revolving round the Earth</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>38</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_17">17.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Portrait of Tycho</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>41</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_18">18.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Early out-door Quadrant of Tycho</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>43</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_19">19.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Map of Denmark, showing the Island of Huen</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>45</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_20">20.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Uraniburg</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>46</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_21">21.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Astrolabe</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>47</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_22">22.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tycho's large Sextant</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>48</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_23">23.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">The Quadrant in Uraniburg</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>49</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_24">24.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tycho's Form of Transit Circle</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>50</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_25">25.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">A Modern Transit Circle</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>51</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_26">26.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Orbits of some of the Planets drawn to scale</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>60</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_27">27.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Many-sided Polygon or Approximate Circle enveloped by Straight Lines</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>61</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_28">28.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Kepler's Idea of the Regular Solids</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>62</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_29">29.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Diagram of Equant</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>67</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_30">30.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Excentric Circle supposed to be divided into equal Areas</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>68</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_31">31.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Mode of drawing an Ellipse</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>70</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_32">32.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Kepler's Diagram proving Equable Description of Areas for an Ellipse</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>71</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_33">33.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Diagram of a Planet's Velocity in Different Parts of its Orbit</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>72</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_34">34.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Portrait of Kepler</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>76</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_35">35.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Curve described by a Projectile</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>82</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_36">36.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Two Forms of Pulsilogy</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>87</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_37">37.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tower of Pisa</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>91</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_38">38.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">View of the Half-Moon in small Telescope</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>97</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_39">39.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Portion of the Lunar Surface more highly magnified</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>98</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_40">40.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Another Portion of the Lunar Surface</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>99</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_41">41.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Lunar Landscape showing Earth</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>100</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_42">42.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Galileo's Method of estimating the Height of Lunar Mountain</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>101</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_43">43.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Some Clusters and Nebul&aelig;</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>102</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_44">44.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Stages of the Discovery of Jupiter's Satellites</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>103</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_45">45.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Eclipses of Jupiter's Satellites</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>105</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_46">46.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Old Drawings of Saturn by Different Observers, with the imperfect Instruments of that day</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>111</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_47">47.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Phases of Venus</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>112</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_48">48.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Sunspots as seen with Low Power</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>113</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_49">49.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">A Portion of the Sun's Disk as seen in a powerful modern Telescope</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>114</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_50">50.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Saturn and his Rings</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>115</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_51">51.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Map of Italy</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>118</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_52">52.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Portrait of Galileo</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>126</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_53">53.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Portrait of Descartes</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>148</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_54">54.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Descartes's Eye Diagram</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>151</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_55">55.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Descartes's Diagram of Vortices from his "Principia"</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>152</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_56">56.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Manor-house of Woolsthorpe</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>162</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_57">57.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Projectile Diagram</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>170</td>
+</tr>
+<tr class='tr2'>
+ <td align='right'><a href="#Fig_58">58.</a></td>
+ <td align='right' rowspan='4'><span class="stache">}</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right' rowspan='4'><span class="stache">{</span></td>
+ <td align='right'>171</td>
+</tr>
+<tr class='tr2'>
+ <td align='right'><a href="#Fig_59">59.</a></td>
+ <td align='left'><span class="smcap">Diagrams illustrative of those near the Beginning of Newton's "Principia"</span></td>
+ <td align='right'>174</td>
+</tr>
+<tr class='tr2'>
+ <td align='right'><a href="#Fig_60">60.</a></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>175</td>
+</tr>
+<tr class='tr2'>
+ <td align='right'><a href="#Fig_61">61-2.</a></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>175</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_63">63.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Prismatic Dispersion</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>182</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_64">64.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">A single Constituent of White Light is capable of no more Dispersion</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>183</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_65">65.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Parallel Beam passing through a Lens</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>184</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_66">66.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Newton's Telescope</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>186</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_67">67.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">The Sextant, as now made</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>187</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_68">68.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Newton when young</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>196</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_69">69.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Sir Isaac Newton</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>200</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_70">70.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Another "Principia" Diagram</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>207</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_71">71.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Well-known Model exhibiting the Oblate Spheroidal Form as a Consequence of spinning about a Central Axis</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>219</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_72">72.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Jupiter</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>221</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_73">73.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Diagram of Eye looking at a Light reflected in a Distant Mirror through the Teeth of a revolving Wheel</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>238</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_74">74.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Fizeau's Wheel, showing the appearance of distant Image seen through its Teeth</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>239</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_75">75.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Eclipses of one of Jupiter's Satellites</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>241</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_76">76.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">A Transit instrument for the British Astronomical Expedition, 1874</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>243</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_77">77.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Diagram of equatorially mounted Telescope</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>245</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_78">78.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Aberration Diagram</span></td>
+ <td align='left'>&nbsp;</td><td align='right'>250</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_79">79.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Showing the three Conjunction Places in the Orbits of Jupiter and Saturn</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>259</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_80">80.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Lord Rosse's Drawing of the Spiral Nebula in Canes Venatici</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>269</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_81">81.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Saturn</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>271</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_82">82.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Principle of Newtonian Reflector</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>278</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_83">83.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Herschel's 40-foot telescope</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>283</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_84">84.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">William Herschel</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>285</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_85">85.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Caroline Herschel</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>287</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_86">86.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Double Stars</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>288</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_87">87.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Old Drawing of the Cluster in Hercules</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>290</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_88">88.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Old Drawing of the Andromeda Nebula</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>291</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_89">89.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">The Great Nebula in Orion</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>292</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_90">90.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Planetary Orbits to scale</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>297</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_91">91.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Diagram illustrating Parallax</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>307</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_92">92.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">The K&ouml;nigsberg Heliometer</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>312</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_93">93.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Perturbations of Uranus</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>320</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_94">94.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Uranus' and Neptune's Relative Positions</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>325</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_95">95.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Meteorite</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>333</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_96">96.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Meteor Stream crossing Field of Telescope</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>334</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_97">97.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Diagram of Direction of Earth's Orbital Motion</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>335</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_98">98.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Parabolic and Elliptic Orbits</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>340</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_99">99.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Orbit of Halley's Comet</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>341</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_100">100.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Various Appearances of Halley's Comet when last seen</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>342</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_101">101.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Head of Donati's Comet of 1858</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>343</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_102">102.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Comet</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>344</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_103">103.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Encke's Comet</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>345</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_104">104.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Biela's Comet as last seen in two Portions</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>346</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_105">105.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Radiant Point Perspective</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>348</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_106">106.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Present Orbit of November Meteors</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>349</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_107">107.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Orbit of November Meteors before and after Encounter with Uranus</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>351</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_108">108.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">The Mersey</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>355</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_109">109.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Co-tidal Lines, showing the way the Tidal Wave reaches the British Isles from the Atlantic</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>359</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_110">110.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Whirling Earth Model</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>364</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_111">111.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Earth and Moon Model</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>365</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_112">112.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Earth and Moon (Earth's Rotation Neglected)</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>366</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_113">113.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Maps showing how comparatively Free from Land Obstruction the Ocean in the Southern Hemisphere Is</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>369</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_114">114.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Spring and Neap Tides</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>370</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_115">115.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tidal Clock</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>371</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_116">116.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Sir William Thomson (Lord Kelvin)</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>373</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_117">117.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tide-gauge for recording Local Tides</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>375</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_118">118.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Harmonic Analyzer</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>375</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_119">119.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Tide-predicter</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>376</td>
+</tr>
+<tr class='tr3'>
+ <td align='right'><a href="#Fig_120">120.</a></td>
+ <td align='right'>&nbsp;</td>
+ <td align='left'><span class="smcap">Weekly Sheet of Curves</span></td>
+ <td align='left'>&nbsp;</td>
+ <td align='right'>377</td>
+</tr>
+</table></div>
+
+
+
+<hr />
+<h3>PIONEERS OF SCIENCE</h3>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_3" id="Page_3">[Pg 3]</a></span></p>
+<h3><a name="PART_I" id="PART_I"></a>PART I<br />
+<br />
+<i>FROM DUSK TO DAYLIGHT</i></h3>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_4" id="Page_4">[Pg 4]</a></span></p>
+<h4><a name="DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_I" id="DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_I"></a>DATES AND SUMMARY OF FACTS FOR LECTURE I</h4>
+
+
+<p><i>Physical Science of the Ancients.</i> Thales 640 <span class="ampm">B.C.</span>, Anaximander 610
+<span class="ampm">B.C.</span>, <span class="smcap">Pythagoras</span> 600 <span class="ampm">B.C.</span>, Anaxagoras 500 <span class="ampm">B.C.</span>, Eudoxus 400 <span class="ampm">B.C.</span>,
+<span class="smcap">Aristotle</span> 384 <span class="ampm">B.C.</span>, Aristarchus 300 <span class="ampm">B.C.</span>, <span class="smcap">Archimedes</span> 287 <span class="ampm">B.C.</span>,
+Eratosthenes 276 <span class="ampm">B.C.</span>, <span class="smcap">Hipparchus</span> 160 <span class="ampm">B.C.</span>, Ptolemy 100 <span class="ampm">A.D.</span></p>
+
+<p><i>Science of the Middle Ages.</i> Cultivated only among the Arabs; largely
+in the forms of astrology, alchemy, and algebra.</p>
+
+<p><i>Return of Science to Europe.</i> Roger Bacon 1240, Leonardo da Vinci 1480,
+(Printing 1455), Columbus 1492, Copernicus 1543.</p>
+
+<p><i>A sketch of Copernik's life and work.</i> Born 1473 at Thorn in Poland.
+Studied mathematics at Bologna. Became an ecclesiastic. Lived at
+Frauenburg near mouth of Vistula. Substituted for the apparent motion of
+the heavens the real motion of the earth. Published tables of planetary
+motions. Motion still supposed to be in epicycles. Worked out his ideas
+for 36 years, and finally dedicated his work to the Pope. Died just as
+his book was printed, aged 72, a century before the birth of Newton. A
+colossal statue by Thorwaldsen erected at Warsaw in 1830.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_5" id="Page_5">[Pg 5]</a></span></p>
+<h2>PIONEERS OF SCIENCE</h2>
+
+
+
+<hr />
+<h3><a name="LECTURE_I" id="LECTURE_I"></a>LECTURE I</h3>
+
+<h5>COPERNICUS AND THE MOTION OF THE EARTH</h5>
+
+
+<p><span class="smcap">The</span> ordinary run of men live among phenomena of which they know nothing
+and care less. They see bodies fall to the earth, they hear sounds, they
+kindle fires, they see the heavens roll above them, but of the causes
+and inner working of the whole they are ignorant, and with their
+ignorance they are content.</p>
+
+<p>"Understand the structure of a soap-bubble?" said a cultivated literary
+man whom I know; "I wouldn't cross the street to know it!"</p>
+
+<p>And if this is a prevalent attitude now, what must have been the
+attitude in ancient times, when mankind was emerging from savagery, and
+when history seems composed of harassments by wars abroad and
+revolutions at home? In the most violently disturbed times indeed, those
+with which ordinary history is mainly occupied, science is quite
+impossible. It needs as its condition, in order to flourish, a fairly
+quiet, untroubled state, or else a cloister or university removed from
+the din and bustle of the political and commercial world. In such places
+it has taken its rise, and in such peaceful places and quiet times true
+science will continue to be cultivated.</p>
+
+<p><span class='pagenum'><a name="Page_6" id="Page_6">[Pg 6]</a></span></p><p>The great bulk of mankind must always remain, I suppose, more or less
+careless of scientific research and scientific result, except in so far
+as it affects their modes of locomotion, their health and pleasure, or
+their purse.</p>
+
+<p>But among a people hurried and busy and preoccupied, some in the pursuit
+of riches, some in the pursuit of pleasure, and some, the majority, in
+the struggle for existence, there arise in every generation, here and
+there, one or two great souls&mdash;men who seem of another age and country,
+who look upon the bustle and feverish activity and are not infected by
+it, who watch others achieving prizes of riches and pleasure and are not
+disturbed, who look on the world and the universe they are born in with
+quite other eyes. To them it appears not as a bazaar to buy and to sell
+in; not as a ladder to scramble up (or down) helter-skelter without
+knowing whither or why; but as a fact&mdash;a great and mysterious fact&mdash;to
+be pondered over, studied, and perchance in some small measure
+understood. By the multitude these men were sneered at as eccentric or
+feared as supernatural. Their calm, clear, contemplative attitude seemed
+either insane or diabolic; and accordingly they have been pitied as
+enthusiasts or killed as blasphemers. One of these great souls may have
+been a prophet or preacher, and have called to his generation to bethink
+them of why and what they were, to struggle less and meditate more, to
+search for things of true value and not for dross. Another has been a
+poet or musician, and has uttered in words or in song thoughts dimly
+possible to many men, but by them unutterable and left inarticulate.
+Another has been influenced still more <i>directly</i> by the universe around
+him, has felt at times overpowered by the mystery and solemnity of it
+all, and has been impelled by a force stronger than himself to study it,
+patiently, slowly, diligently; content if he could gather a few crumbs
+of the great harvest of knowledge, happy if he could grasp some great
+generalization or wide-embracing law, and so in some small measure enter
+into<span class='pagenum'><a name="Page_7" id="Page_7">[Pg 7]</a></span> the mind and thought of the Designer of all this wondrous frame of
+things.</p>
+
+<p>These last have been the men of science, the great and heaven-born men
+of science; and they are few. In our own day, amid the throng of
+inventions, there are a multitude of small men using the name of science
+but working for their own ends, jostling and scrambling just as they
+would jostle and scramble in any other trade or profession. These may be
+workers, they may and do advance knowledge, but they are never pioneers.
+Not to them is it given to open out great tracts of unexplored
+territory, or to view the promised land as from a mountain-top. Of them
+we shall not speak; we will concern ourselves only with the greatest,
+the epoch-making men, to whose life and work we and all who come after
+them owe so much. Such a man was Thales. Such was Archimedes,
+Hipparchus, Copernicus. Such pre-eminently was Newton.</p>
+
+<p>Now I am not going to attempt a history of science. Such a work in ten
+lectures would be absurd. I intend to pick out a few salient names here
+and there, and to study these in some detail, rather than by attempting
+to deal with too many to lose individuality and distinctness.</p>
+
+<p>We know so little of the great names of antiquity, that they are for
+this purpose scarcely suitable. In some departments the science of the
+Greeks was remarkable, though it is completely overshadowed by their
+philosophy; yet it was largely based on what has proved to be a wrong
+method of procedure, viz the introspective and conjectural, rather than
+the inductive and experimental methods. They investigated Nature by
+studying their own minds, by considering the meanings of words, rather
+than by studying things and recording phenomena. This wrong (though by
+no means, on the face of it, absurd) method was not pursued exclusively,
+else would their science have been valueless, but the influence it had
+was such as materially to detract from the value of their speculations
+and discoveries. For<span class='pagenum'><a name="Page_8" id="Page_8">[Pg 8]</a></span> when truth and falsehood are inextricably woven
+into a statement, the truth is as hopelessly hidden as if it had never
+been stated, for we have no criterion to distinguish the false from the
+true.</p>
+
+<div class="figcenter" style="width: 300px;"><a name="Fig_1" id="Fig_1"></a>
+<img src="images/fig1.jpg" width="300" height="403" alt="Fig. 1." title="" />
+<span class="caption"><span class="smcap">Fig. 1.</span>&mdash;Archimedes.</span>
+</div>
+
+<p>Besides this, however, many of their discoveries were ultimately lost to
+the world, some, as at Alexandria, by fire&mdash;the bigoted work of a
+Mohammedan conqueror&mdash;some by irruption of barbarians; and all were
+buried so long and so completely by the night of the dark ages, that
+they had to be rediscovered almost as absolutely and completely as
+though they had never been. Some of the names of antiquity we shall have
+occasion to refer to; so I have arranged some of them in chronological
+order on <a href="#Page_4">page 4</a>, and as a representative one I may specially emphasize
+Archimedes, one of the greatest men of science there has ever been, and
+the father of physics.</p>
+
+<p><span class='pagenum'><a name="Page_9" id="Page_9">[Pg 9]</a></span></p><p>The only effective link between the old and the new science is afforded
+by the Arabs. The dark ages come as an utter gap in the scientific
+history of Europe, and for more than a thousand years there was not a
+scientific man of note except in Arabia; and with the Arabs knowledge
+was so mixed up with magic and enchantment that one cannot contemplate
+it with any degree of satisfaction, and little real progress was made.
+In some of the <i>Waverley Novels</i> you can realize the state of matters in
+these times; and you know how the only approach to science is through
+some Arab sorcerer or astrologer, maintained usually by a monarch, and
+consulted upon all great occasions, as the oracles were of old.</p>
+
+<p>In the thirteenth century, however, a really great scientific man
+appeared, who may be said to herald the dawn of modern science in
+Europe. This man was Roger Bacon. He cannot be said to do more than
+herald it, however, for we must wait two hundred years for the next name
+of great magnitude; moreover he was isolated, and so far in advance of
+his time that he left no followers. His own work suffered from the
+prevailing ignorance, for he was persecuted and imprisoned, not for the
+commonplace and natural reason that he frightened the Church, but merely
+because he was eccentric in his habits and knew too much.</p>
+
+<p>The man I spoke of as coming two hundred years later is Leonardo da
+Vinci. True he is best known as an artist, but if you read his works you
+will come to the conclusion that he was the most scientific artist who
+ever lived. He teaches the laws of perspective (then new), of light and
+shade, of colour, of the equilibrium of bodies, and of a multitude of
+other matters where science touches on art&mdash;not always quite correctly
+according to modern ideas, but in beautiful and precise language. For
+clear and conscious power, for wide-embracing knowledge and skill,
+Leonardo is one of the most remarkable men that ever lived.</p>
+
+<p>About this time the tremendous invention of printing was achieved, and
+Columbus unwittingly discovered the New<span class='pagenum'><a name="Page_10" id="Page_10">[Pg 10]</a></span> World. The middle of the next
+century must be taken as the real dawn of modern science; for the year
+1543 marks the publication of the life-work of Copernicus.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_2" id="Fig_2"></a>
+<img src="images/fig2.jpg" width="350" height="418" alt="Fig. 2." title="" />
+<span class="caption"><span class="smcap">Fig. 2.</span>&mdash;Leonardo da Vinci.</span>
+</div>
+
+<p>Nicolas Copernik was his proper name. Copernicus is merely the Latinized
+form of it, according to the then prevailing<span class='pagenum'><a name="Page_11" id="Page_11">[Pg 11]</a></span> fashion. He was born at
+Thorn, in Polish Prussia, in 1473. His father is believed to have been a
+German. He graduated at Cracow as doctor in arts and medicine, and was
+destined for the ecclesiastical profession. The details of his life are
+few; it seems to have been quiet and uneventful, and we know very little
+about it. He was instructed in astronomy at Cracow, and learnt
+mathematics at Bologna. Thence he went to Rome, where he was made
+Professor of Mathematics; and soon afterwards he went into orders. On
+his return home, he took charge of the principal church in his native
+place, and became a canon. At Frauenburg, near the mouth of the Vistula,
+he lived the remainder of his life. We find him reporting on coinage for
+the Government, but otherwise he does not appear as having entered into
+the life of the times.</p>
+
+<p>He was a quiet, scholarly monk of studious habits, and with a reputation
+which drew to him several earnest students, who received <i>viv&acirc; voce</i>
+instruction from him; so, in study and meditation, his life passed.</p>
+
+<p>He compiled tables of the planetary motions which were far more correct
+than any which had hitherto appeared, and which remained serviceable for
+long afterwards. The Ptolemaic system of the heavens, which had been the
+orthodox system all through the Christian era, he endeavoured to improve
+and simplify by the hypothesis that the sun was the centre of the system
+instead of the earth; and the first consequences of this change he
+worked out for many years, producing in the end a great book: his one
+life-work. This famous work, "De Revolutionibus Orbium C&#339;lestium,"
+embodied all his painstaking calculations, applied his new system to
+each of the bodies in the solar system in succession, and treated
+besides of much other recondite matter. Towards the close of his life it
+was put into type. He can scarcely be said to have lived to see it
+appear, for he was stricken with paralysis before its completion;<span class='pagenum'><a name="Page_12" id="Page_12">[Pg 12]</a></span> but a
+printed copy was brought to his bedside and put into his hands, so that
+he might just feel it before he died.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_3" id="Fig_3"></a>
+<img src="images/fig3.jpg" width="350" height="472" alt="Fig. 3." title="" />
+<span class="caption"><span class="smcap">Fig. 3.</span>&mdash;Copernicus.</span>
+</div>
+
+<p>That Copernicus was a giant in intellect or power&mdash;such as had lived in
+the past, and were destined to live in the near future&mdash;I see no reason
+whatever to believe. He was just a quiet, earnest, patient, and
+God-fearing man, a deep<span class='pagenum'><a name="Page_13" id="Page_13">[Pg 13]</a></span> student, an unbiassed thinker, although with no
+specially brilliant or striking gifts; yet to him it was given to effect
+such a revolution in the whole course of man's thoughts as is difficult
+to parallel.</p>
+
+<p>You know what the outcome of his work was. It proved&mdash;he did not merely
+speculate, he proved&mdash;that the earth is a planet like the others, and
+that it revolves round the sun.</p>
+
+<p>Yes, it can be summed up in a sentence, but what a revelation it
+contains. If you have never made an effort to grasp the full
+significance of this discovery you will not appreciate it. The doctrine
+is very familiar to us now, we have heard it, I suppose, since we were
+four years old, but can you realize it? I know it was a long time before
+I could. Think of the solid earth, with trees and houses, cities and
+countries, mountains and seas&mdash;think of the vast tracts of land in Asia,
+Africa, and America&mdash;and then picture the whole mass spinning like a
+top, and rushing along its annual course round the sun at the rate of
+nineteen miles every second.</p>
+
+<p>Were we not accustomed to it, the idea would be staggering. No wonder it
+was received with incredulity. But the difficulties of the conception
+are not only physical, they are still more felt from the speculative and
+theological points of view. With this last, indeed, the reconcilement
+cannot be considered complete even yet. Theologians do not, indeed, now
+<i>deny</i> the fact of the earth's subordination in the scheme of the
+universe, but many of them ignore it and pass it by. So soon as the
+Church awoke to a perception of the tremendous and revolutionary import
+of the new doctrines, it was bound to resist them or be false to its
+traditions. For the whole tenor of men's thought must have been changed
+had they accepted it. If the earth were not the central and
+all-important body in the universe, if the sun and planets and stars
+were not attendant and subsidiary lights, but were other worlds larger
+and perhaps superior to ours, where was man's place in the universe?<span class='pagenum'><a name="Page_14" id="Page_14">[Pg 14]</a></span>
+and where were the doctrines they had maintained as irrefragable? I by
+no means assert that the new doctrines were really utterly
+irreconcilable with the more essential parts of the old dogmas, if only
+theologians had had patience and genius enough to consider the matter
+calmly. I suppose that in that case they might have reached the amount
+of reconciliation at present attained, and not only have left scientific
+truth in peace to spread as it could, but might perhaps themselves have
+joined the band of earnest students and workers, as so many of the
+higher Catholic clergy do at the present day.</p>
+
+<p>But this was too much to expect. Such a revelation was not to be
+accepted in a day or in a century&mdash;the easiest plan was to treat it as a
+heresy, and try to crush it out.</p>
+
+<p>Not in Copernik's life, however, did they perceive the dangerous
+tendency of the doctrine&mdash;partly because it was buried in a ponderous
+and learned treatise not likely to be easily understood; partly,
+perhaps, because its propounder was himself an ecclesiastic; mainly
+because he was a patient and judicious man, not given to loud or
+intolerant assertion, but content to state his views in quiet
+conversation, and to let them gently spread for thirty years before he
+published them. And, when he did publish them, he used the happy device
+of dedicating his great book to the Pope, and a cardinal bore the
+expense of printing it. Thus did the Roman Church stand sponsor to a
+system of truth against which it was destined in the next century to
+hurl its anathemas, and to inflict on its conspicuous adherents torture,
+imprisonment, and death.</p>
+
+<p>To realize the change of thought, the utterly new view of the universe,
+which the Copernican theory introduced, we must go back to preceding
+ages, and try to recall the views which had been held as probable
+concerning the form of the earth and the motion of the heavenly bodies.</p>
+
+<p><span class='pagenum'><a name="Page_15" id="Page_15">[Pg 15]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_4" id="Fig_4"></a>
+<img src="images/fig4.jpg" width="400" height="385" alt="Fig. 4." title="" />
+<span class="caption"><span class="smcap">Fig. 4.</span>&mdash;Homeric Cosmogony.</span>
+</div>
+
+<p>The earliest recorded notion of the earth is the very natural one that
+it is a flat area floating in an illimitable ocean. The sun was a god
+who drove his chariot across the heavens once a day; and Anaxagoras was
+threatened with death and punished with banishment for teaching that the
+sun was only a ball of fire, and that it might perhaps be as big as the
+country of Greece. The obvious difficulty as to how the sun got back to
+the east again every morning was got over&mdash;not by the conjecture that he
+went back in the dark, nor by the idea that there was a fresh sun every
+day; though, indeed, it was once believed that the moon was created once
+a month, and periodically cut up into stars&mdash;but by the doctrine that in
+the northern part of the earth was a high range of mountains, and that
+the sun travelled round on the surface of the sea behind these.<span class='pagenum'><a name="Page_16" id="Page_16">[Pg 16]</a></span>
+Sometimes, indeed, you find a representation of the sun being rowed
+round in a boat. Later on it was perceived to be necessary that the sun
+should be able to travel beneath the earth, and so the earth was
+supposed to be supported on pillars or on roots, or to be a dome-shaped
+body floating in air&mdash;much like Dean Swift's island of Laputa. The
+elephant and tortoise of the Hindu earth are, no doubt, emblematic or
+typical, not literal.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="Fig_5" id="Fig_5"></a>
+<img src="images/fig5.jpg" width="400" height="276" alt="Fig. 5." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 5.</span>&mdash;Egyptian Symbol of the Universe.<br />
+<small>The earth a figure with leaves, the heaven a figure with stars, the
+principle of equilibrium and support, the boats of the rising and
+setting sun.</small></span>
+</div>
+
+<p>Aristotle, however, taught that the earth must be a sphere, and used all
+the orthodox arguments of the present children's geography-books about
+the way you see ships at sea, and about lunar eclipses.</p>
+
+<p>To imagine a possible antipodes must, however, have been a tremendous
+difficulty in the way of this conception<span class='pagenum'><a name="Page_17" id="Page_17">[Pg 17]</a></span> of a sphere, and I scarcely
+suppose that any one can at that time have contemplated the possibility
+of such upside-down regions being inhabited. I find that intelligent
+children invariably feel the greatest difficulty in realizing the
+existence of inhabitants on the opposite side of the earth. Stupid
+children, like stupid persons in general, will of course believe
+anything they are told, and much good may the belief do them; but the
+kind of difficulties felt by intelligent and thoughtful children are
+most instructive, since it is quite certain that the early philosophers
+must have encountered and overcome those very same difficulties by their
+own genius.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_6" id="Fig_6"></a>
+<img src="images/fig6.jpg" width="350" height="245" alt="Fig. 6." title="" />
+<span class="caption"><span class="smcap">Fig. 6.</span>&mdash;Hindoo Earth.</span>
+</div>
+
+<p>However, somehow or other the conception of a spherical earth was
+gradually grasped, and the heavenly bodies were perceived all to revolve
+round it: some moving regularly, as the stars, all fixed together into
+one spherical shell or firmament; some moving irregularly and apparently
+anomalously&mdash;these irregular bodies were therefore called planets [or
+wanderers]. Seven of them were known, viz.<span class='pagenum'><a name="Page_18" id="Page_18">[Pg 18]</a></span> Moon, Mercury, Venus, Sun,
+Mars, Jupiter, Saturn, and there is little doubt that this number seven,
+so suggested, is the origin of the seven days of the week.</p>
+
+<div class="blockquot"><p>The above order of the ancient planets is that of their supposed
+distance from the earth. Not always, however, are they thus quoted
+by the ancients: sometimes the sun is supposed nearer than Mercury
+or Venus. It has always been known that the moon was the nearest of
+the heavenly bodies; and some rough notion of its distance was
+current. Mars, Jupiter, and Saturn were placed in that order
+because that is the order of their apparent motions, and it was
+natural to suppose that the slowest moving bodies were the furthest
+off.</p>
+
+<p>The order of the days of the week shows what astrologers considered
+to be the order of the planets; on their system of each successive
+hour of the day being ruled over by the successive planets taken in
+order. The diagram (fig. 7) shows that if the Sun rule the first
+hour of a certain day (thereby giving its name to the day) Venus
+will rule the second hour, Mercury the third, and so on; the Sun
+will thus be found to rule the eighth, fifteenth, and twenty-second
+hour of that day, Venus the twenty-third, and Mercury the
+twenty-fourth hour; so the Moon will rule the first hour of the
+next day, which will therefore be Monday. On the same principle
+(numbering round the hours successively, with the arrows) the first
+hour of the next day will be found to be ruled by Mars, or by the
+Saxon deity corresponding thereto; the first hour of the day after,
+by Mercury (<i>Mercredi</i>), and so on (following the straight lines of
+the pattern).</p>
+
+<p>The order of the planets round the circle counter-clockwise, <i>i.e.</i>
+the direction of their proper motions, is that quoted above in the
+text. </p></div>
+
+<p>To explain the motion of the planets and reduce them to any sort of law
+was a work of tremendous difficulty. The greatest astronomer of ancient
+times was Hipparchus, and to him the system known as the Ptolemaic
+system is no doubt largely due. But it was delivered to the world mainly
+by Ptolemy, and goes by his name. This was a fine piece of work, and a
+great advance on anything that had gone before; for although it is of
+course saturated with error, still it is based on a large substratum of
+truth. Its superiority to all the previously mentioned systems is
+obvious. And it really did in its more developed form describe the
+observed motions of the planets.</p>
+
+<p><span class='pagenum'><a name="Page_19" id="Page_19">[Pg 19]</a></span></p><p>Each planet was, in the early stages of this system, as taught, say, by
+Eudoxus, supposed to be set in a crystal sphere, which revolved so as to
+carry the planet with it. The sphere had to be of crystal to account for
+the visibility of other planets and the stars through it. Outside the
+seven planetary spheres, arranged one inside the other, was a still
+larger one in which were set the stars. This was believed to turn all
+the others, and was called the <i>primum mobile</i>. The whole system was
+supposed to produce, in its revolution, for the few privileged to hear
+the music of the spheres, a sound as of some magnificent harmony.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_7" id="Fig_7"></a>
+<img src="images/fig7.jpg" width="400" height="385" alt="Fig. 7." title="" />
+<span class="caption"><span class="smcap">Fig. 7.</span>&mdash;Order of ancient planets corresponding to the
+days of the week.</span>
+</div>
+
+<p>The enthusiastic disciples of Pythagoras believed that their master was
+privileged to hear this noble chant; and<span class='pagenum'><a name="Page_20" id="Page_20">[Pg 20]</a></span> far be it from us to doubt
+that the rapt and absorbing pleasure of contemplating the harmony of
+nature, to a man so eminently great as Pythagoras, must be truly and
+adequately represented by some such poetic conception.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_8" id="Fig_8"></a>
+<img src="images/fig8.jpg" width="400" height="415" alt="Fig. 8." title="" />
+<span class="caption"><span class="smcap">Fig. 8.</span>&mdash;Ptolemaic system.</span>
+</div>
+
+<p>The precise kind of motion supposed to be communicated from the <i>primum
+mobile</i> to the other spheres so as to produce the observed motions of
+the planets was modified and improved by various philosophers until it
+developed into the epicyclic train of Hipparchus and of Ptolemy.</p>
+
+<p>It is very instructive to observe a planet (say Mars or Jupiter) night
+after night and plot down its place with<span class='pagenum'><a name="Page_21" id="Page_21">[Pg 21]</a></span> reference to the fixed stars
+on a celestial globe or star-map. Or, instead of direct observation by
+alignment with known stars, it is easier to look out its right ascension
+and declination in <i>Whitaker's Almanac</i>, and plot those down. If this be
+done for a year or two, it will be found that the motion of the planet
+is by no means regular, but that though on the whole it advances it
+sometimes is stationary and sometimes goes back.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a></p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_9" id="Fig_9"></a>
+<img src="images/fig9.jpg" width="400" height="376" alt="Fig. 9." title="" />
+<span class="caption"><span class="smcap">Fig. 9.</span>&mdash;Specimens of Apparent paths of Venus and of Mars
+among the stars.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_22" id="Page_22">[Pg 22]</a></span></p>
+<div class="figcenter" style="width: 400px;"><br /><a name="Fig_10" id="Fig_10"></a>
+<img src="images/fig10.jpg" width="400" height="404" alt="Fig. 10." title="" />
+<div class="caption1"><span class="smcap">Fig. 10.</span>&mdash;Apparent epicyclic orbits of Jupiter and
+Saturn; the Earth being supposed fixed at the centre, with the Sun
+revolving in a small circle. A loop is made by each planet every year.</div>
+</div>
+
+<p>These "stations" and "retrogressions" of the planets were well known to
+the ancients. It was not to be supposed for a moment that the crystal
+spheres were subject to any irregularity, neither was uniform circular
+motion to be readily abandoned; so it was surmised that the main sphere
+carried, not the planet itself, but the centre or axis<span class='pagenum'><a name="Page_23" id="Page_23">[Pg 23]</a></span> of a subordinate
+sphere, and that the planet was carried by this. The minor sphere could
+be allowed to revolve at a different uniform pace from the main sphere,
+and so a curve of some complexity could be obtained.</p>
+
+<p>A curve described in space by a point of a circle or sphere, which
+itself is carried along at the same time, is some kind of cycloid; if
+the centre of the tracing circle travels along a straight line, we get
+the ordinary cycloid, the curve traced in air by a nail on a
+coach-wheel; but if the centre of the tracing circle be carried round
+another circle the curve described is called an epicycloid. By such
+curves the planetary stations and retrogressions could be explained. A
+large sphere would have to revolve once for a "year" of the particular
+planet, carrying with it a subsidiary sphere in which the planet was
+fixed; this latter sphere revolving once for a "year" of the earth. The
+actual looped curve thus described is depicted for Jupiter and Saturn in
+the annexed diagram (fig. 10.)</p>
+
+<div class="blockquot"><p>It was long ago perceived that real material spheres were
+unnecessary; such spheres indeed, though possibly transparent to
+light, would be impermeable to comets: any other epicyclic gearing
+would serve, and as a mere description of the motion it is simpler
+to think of a system of jointed bars, one long arm carrying a
+shorter arm, the two revolving at different rates, and the end of
+the short one carrying the planet. This does all that is needful
+for the first approximation to a planet's motion. In so far as the
+motion cannot be thus truly stated, the short arm may be supposed
+to carry another, and that another, and so on, so that the
+resultant motion of the planet is compounded of a large number of
+circular motions of different periods; by this device any required
+amount of complexity could be attained. We shall return to this at
+greater length in <a href="#SUMMARY_OF_FACTS_FOR_LECTURE_III">Lecture III</a>.</p>
+
+<p>The main features of the motion, as shown in the diagram, required
+only two arms for their expression; one arm revolving with the
+average motion of the planet, and the other revolving with the
+apparent motion of the sun, and always pointing in the same
+direction as the single arm supposed to carry the sun. This last
+fact is of course because the motion to be represented does not
+really belong to the planet at all, but to the earth, and so all
+the main epicyclic motions for the superior planets were the same.
+As for the<span class='pagenum'><a name="Page_24" id="Page_24">[Pg 24]</a></span> inferior planets (Mercury and Venus) they only appear
+to oscillate like the bob of a pendulum about the sun, and so it is
+very obvious that they must be really revolving round it. An
+ancient Egyptian system perceived this truth; but the Ptolemaic
+system imagined them to revolve round the earth like the rest, with
+an artificial system of epicycles to prevent their ever getting far
+away from the neighbourhood of the sun.</p>
+
+<p>It is easy now to see how the Copernican system explains the main
+features of planetary motion, the stations and retrogressions,
+quite naturally and without any complexity.</p></div>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_11" id="Fig_11"></a>
+<img src="images/fig11.jpg" width="400" height="408" alt="Fig. 11." title="" />
+<span class="caption"><span class="smcap">Fig. 11.</span>&mdash;Egyptian system.</span>
+</div>
+
+<div class="blockquot"><p>Let the outer circle represent the orbit of Jupiter, and the inner
+circle the orbit of the earth, which is moving faster than Jupiter
+(since Jupiter takes 4332 days to make one revolution); then
+remember that the apparent position of Jupiter is referred to the
+infinitely distant fixed stars and refer to fig. 12.</p>
+
+<p>Let E<sub>1</sub>, E<sub>2</sub>, &amp;c., be successive positions of the earth; J<sub>1</sub>,
+J<sub>2</sub>, &amp;c., corresponding positions of Jupiter. Produce the lines
+E<sub>1</sub> J<sub>1</sub>, E<sub>2</sub> J<sub>2</sub>, &amp;c., to an enormously greater circle
+outside, and it will be seen that the termination of these lines,
+representing apparent positions of Jupiter among the stars,
+advances while the earth goes from E<sub>1</sub> to E<sub>3</sub>; is almost
+stationary from somewhere about E<sub>3</sub> to E<sub>4</sub>; and recedes from
+E<sub>4</sub> to E<sub>5</sub>; so that evidently the recessions of Jupiter are
+only apparent, and are due to the orbital motion of the earth. The
+apparent complications in the path of Jupiter, shown in <a href="#Fig_10">Fig. 10</a>,
+are seen to be caused simply by the motion of the earth, and to be
+thus completely and easily explained.</p></div>
+
+<p><span class='pagenum'><a name="Page_25" id="Page_25">[Pg 25]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_12" id="Fig_12"></a>
+<img src="images/fig12.jpg" width="400" height="395" alt="Fig. 12." title="" />
+<span class="caption"><span class="smcap">Fig. 12.</span>&mdash;True orbits of Earth and Jupiter.</span>
+</div>
+
+<div class="blockquot"><p>The same thing for an inferior planet, say Mercury, is even still
+more easily seen (<i>vide</i> <a href="#Fig_13">figure 13</a>).</p>
+
+<p>The motion of Mercury is direct from M'' to M''', retrograde from
+M''' to M'', and stationary at M'' and M'''. It appears to
+oscillate, taking 72&middot;5 days for its direct swing, and 43&middot;5 for its
+return swing.</p></div>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_13" id="Fig_13"></a>
+<img src="images/fig13.jpg" width="400" height="363" alt="Fig. 13." title="" />
+<span class="caption"><span class="smcap">Fig. 13.</span>&mdash;Orbit of Mercury and Earth.</span>
+</div>
+
+<div class="blockquot"><p>On this system no artificiality is required to prevent Mercury's
+ever getting far from the sun: the radius of its orbit limits its
+real and apparent excursions. Even if the earth were stationary,
+the motions<span class='pagenum'><a name="Page_26" id="Page_26">[Pg 26]</a></span> of Mercury and Venus would not be <i>essentially</i>
+modified, but the stations and retrogressions of the superior
+planets, Mars, Jupiter, &amp;c., would wholly cease.</p>
+
+<p>The complexity of the old mode of regarding apparent motion may be
+illustrated by the case of a traveller in a railway train unaware
+of his own motion. It is as though trees, hedges, distant objects,
+were all flying past him and contorting themselves as you may see
+the furrows of a ploughed field do when travelling, while you
+yourself seem stationary amidst it all. How great a simplicity
+would be introduced by the hypothesis that, after all, these things
+might be stationary and one's self moving.</p></div>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_14" id="Fig_14"></a>
+<img src="images/fig14.jpg" width="400" height="405" alt="Fig. 14." title="" />
+<div class="caption1"><span class="smcap">Fig. 14.</span>&mdash;Copernican system as frequently represented.
+But the cometary orbit is a much later addition, and no attempt is made
+to show the relative distances of the planets.</div>
+</div>
+
+<p>Now you are not to suppose that the system of Copernicus swept away the
+entire doctrine of epicycles; that doctrine can hardly be said to be
+swept away even now. As a description of a planet's motion it is not
+incorrect, though it is geometrically cumbrous. If you describe the
+motion of a railway train by stating that every point on the rim of each
+wheel describes a cycloid with reference to the earth, and a circle with
+reference to the train, and that the motion of the train is compounded
+of these cycloidal and circular motions, you will not be saying what is
+false, only what is cumbrous.</p>
+
+<p>The Ptolemaic system demanded large epicycles, depending<span class='pagenum'><a name="Page_27" id="Page_27">[Pg 27]</a></span> on the motion
+of the earth, these are what Copernicus overthrew; but to express the
+minuter details of the motion smaller epicycles remained, and grew more
+and more complex as observations increased in accuracy, until a greater
+man than either Copernicus or Ptolemy, viz. Kepler, replaced them all by
+a simple ellipse.</p>
+
+<p>One point I must not omit from this brief notice of the work of
+Copernicus. Hipparchus had, by most sagacious interpretation of certain
+observations of his, discovered a remarkable phenomenon called the
+precession of the equinoxes. It was a discovery of the first magnitude,
+and such as would raise to great fame the man who should have made it in
+any period of the world's history, even the present. It is scarcely
+expressible in popular language, and without some technical terms; but I
+can try.</p>
+
+<p>The plane of the earth's orbit produced into the sky gives the apparent
+path of the sun throughout a year. This path is known as the ecliptic,
+because eclipses only happen when the moon is in it. The sun keeps to it
+accurately, but the planets wander somewhat above and below it (fig. 9),
+and the moon wanders a good deal. It is manifest, however, in order that
+there may be an eclipse of any kind, that a straight line must be able
+to be drawn through earth and moon and sun (not necessarily through
+their centres of course), and this is impossible unless some parts of
+the three bodies are in one plane, viz. the ecliptic, or something very
+near it. The ecliptic is a great circle of the sphere, and is usually
+drawn on both celestial and terrestrial globes.</p>
+
+<p>The earth's equator also produced into the sky, where it may still be
+called the equator (sometimes it is awkwardly called "the equinoctial"),
+gives another great circle inclined to the ecliptic and cutting it at
+two opposite points, labelled respectively &#9800; and &#9806;,
+and together called "the equinoxes." The reason for the
+name is that when the sun is in that part of the ecliptic it is
+temporarily also on the equator, and hence is symmetrically situated
+with respect to the<span class='pagenum'><a name="Page_28" id="Page_28">[Pg 28]</a></span> earth's axis of rotation, and consequently day and
+night are equal all over the earth.</p>
+
+<p>Well, Hipparchus found, by plotting the position of the sun for a long
+time,<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> that these points of intersection, or equinoxes, were not
+stationary from century to century, but slowly moved among the stars,
+moving as it were to meet the sun, so that he gets back to one of these
+points again 20 minutes 23&frac14; seconds before it has really completed a
+revolution, <i>i.e.</i> before the true year is fairly over. This slow
+movement forward of the goal-post is called precession&mdash;the precession
+of the equinoxes. (One result of it is to shorten our years by about 20
+minutes each; for the shortened period has to be called a year, because
+it is on the position of the sun with respect to the earth's axis that
+our seasons depend.) Copernicus perceived that, assuming the motion of
+the earth, a clearer account of this motion could be given. The ordinary
+approximate statement concerning the earth's axis is that it remains
+parallel to itself, <i>i.e.</i> has a fixed direction as the earth moves
+round the sun. But if, instead of being thus fixed, it be supposed to
+have a slow movement of revolution, so that it traces out a cone in the
+course of about 26,000 years, then, since the equator of course goes
+with it, the motion of its intersection with the fixed ecliptic is so
+far accounted for. That is to say, the precession of the equinoxes is
+seen to be dependent on, and caused by, a slow conical movement of the
+earth's axis.</p>
+
+<p>The prolongation of each end of the earth's axis into the sky, or the
+celestial north and south poles, will thus slowly trace out an
+approximate circle among the stars; and the course of the north pole
+during historic time is exhibited in the annexed diagram.</p>
+
+<p>It is now situated near one of the stars of the Lesser Bear,<span class='pagenum'><a name="Page_29" id="Page_29">[Pg 29]</a></span> which we
+therefore call the Pole star; but not always was it so, nor will it be
+so in the future. The position of the north pole 4000 years ago is shown
+in the figure; and a revolution will be completed in something like
+26,000 years.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></p>
+
+<div class="figcenter" style="width: 600px;"><a name="Fig_15" id="Fig_15"></a>
+<img src="images/fig15.jpg" width="400" height="340" alt="Fig. 15." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 15.</span>&mdash;Slow movement of the north pole in a circle
+among the stars.<br />(Copied from Sir R. Ball.)</span>
+</div>
+
+<p>This perception of the conical motion of the earth's axis was a
+beautiful generalization of Copernik's, whereby a multitude of facts
+were grouped into a single phenomenon. Of course he did not explain the
+motion of the axis itself. He stated the fact that it so moved, and I do
+not suppose it ever struck him to seek for an explanation.</p>
+
+<p><span class='pagenum'><a name="Page_30" id="Page_30">[Pg 30]</a></span></p><p>An explanation was given later, and that a most complete one; but the
+idea even of seeking for it is a brilliant and striking one: the
+achievement of the explanation by a single individual in the way it
+actually was accomplished is one of the most astounding things in the
+history of science; and were it not that the same individual
+accomplished a dozen other things, equally and some still more
+extraordinary, we should rank that man as one of the greatest
+astronomers that ever lived.</p>
+
+<p>As it is, he is Sir Isaac Newton.</p>
+
+<p>We are to remember, then, as the life-work of Copernicus, that he placed
+the sun in its true place as the centre of the solar system, instead of
+the earth; that he greatly simplified the theory of planetary motion by
+this step, and also by the simpler epicyclic chain which now sufficed,
+and which he worked out mathematically; that he exhibited the precession
+of the equinoxes (discovered by Hipparchus) as due to a conical motion
+of the earth's axis; and that, by means of his simpler theory and more
+exact planetary tables, he reduced to some sort of order the confused
+chaos of the Ptolemaic system, whose accumulation of complexity and of
+outstanding errors threatened to render astronomy impossible by the mere
+burden of its detail.</p>
+
+<p>There are many imperfections in his system, it is true; but his great
+merit is that he dared to look at the facts of Nature with his own eyes,
+unhampered by the prejudice of centuries. A system venerable with age,
+and supported by great names, was universally believed, and had been
+believed for centuries. To doubt this system, and to seek after another
+and better one, at a time when all men's minds were governed by
+tradition and authority, and when to doubt was sin&mdash;this required a
+great mind and a high character. Such a mind and such a character had
+this monk of Frauenburg. And it is interesting to notice that the
+so-called religious scruples of smaller and less truly religious men did
+not affect Copernicus; it was no dread of<span class='pagenum'><a name="Page_31" id="Page_31">[Pg 31]</a></span> consequences to one form of
+truth that led him to delay the publication of the other form of truth
+specially revealed to him. In his dedication he says:&mdash;</p>
+
+<p>"If there be some babblers who, though ignorant of all mathematics, take
+upon them to judge of these things, and dare to blame and cavil at my
+work, because of some passage of Scripture which they have wrested to
+their own purpose, I regard them not, and will not scruple to hold their
+judgment in contempt."</p>
+
+<p>I will conclude with the words of one of his biographers (Mr. E.J.C.
+Morton):&mdash;</p>
+
+<p>"Copernicus cannot be said to have flooded with light the dark places of
+nature&mdash;in the way that one stupendous mind subsequently did&mdash;but still,
+as we look back through the long vista of the history of science, the
+dim Titanic figure of the old monk seems to rear itself out of the dull
+flats around it, pierces with its head the mists that overshadow them,
+and catches the first gleam of the rising sun,</p>
+
+<p class="poem">
+"'... like some iron peak, by the Creator<br />
+<span style="margin-left: 1em;">Fired with the red glow of the rushing morn.'"</span><br />
+</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_32" id="Page_32">[Pg 32]</a></span></p>
+<h4><a name="DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_II" id="DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_II"></a>DATES AND SUMMARY OF FACTS FOR LECTURE II</h4>
+
+
+<p>Copernicus lived from 1473 to 1543, and was contemporary with Paracelsus
+and Raphael.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Tycho Brahe Contemporaries">
+<tr><td align='left'>Tycho Brah&eacute;</td><td align='left'>from 1546 to 1601.</td></tr>
+<tr><td align='left'>Kepler</td><td align='left'>from 1571 to 1630.</td></tr>
+<tr><td align='left'>Galileo</td><td align='left'>from 1564 to 1642.</td></tr>
+<tr><td align='left'>Gilbert</td><td align='left'>from 1540 to 1603.</td></tr>
+<tr><td align='left'>Francis Bacon</td><td align='left'>from 1561 to 1626.</td></tr>
+<tr><td align='left'>Descartes</td><td align='left'>from 1596 to 1650.</td></tr>
+</table></div>
+
+<p><i>A sketch of Tycho Brah&eacute;'s life and work.</i> Tycho was a Danish noble,
+born on his ancestral estate at Knudstorp, near Helsinborg, in 1546.
+Adopted by his uncle, and sent to the University of Copenhagen to study
+law. Attracted to astronomy by the occurrence of an eclipse on its
+predicted day, August 21st, 1560. Began to construct astronomical
+instruments, especially a quadrant and a sextant. Observed at Augsburg
+and Wittenberg. Studied alchemy, but was recalled to astronomy by the
+appearance of a new star. Overcame his aristocratic prejudices, and
+delivered a course of lectures at Copenhagen, at the request of the
+king. After this he married a peasant girl. Again travelled and observed
+in Germany. In 1576 was sent for to Denmark by Frederick II., and
+established in the island of Huen, with an endowment enabling him to
+devote his life to astronomy. Built Uraniburg, furnished it with
+splendid instruments, and became the founder of accurate instrumental
+astronomy. His theories were poor, but his observations were admirable.
+In 1592 Frederick died, and five years later, Tycho was impoverished and
+practically banished. After wandering till 1599, he was invited to
+Prague by the Emperor Rudolf, and there received John Kepler among other
+pupils. But the sentence of exile was too severe, and he died in 1601,
+aged 54 years.</p>
+
+<p>A man of strong character, untiring energy, and devotion to accuracy,
+his influence on astronomy has been immense.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_33" id="Page_33">[Pg 33]</a></span></p>
+<h3><a name="LECTURE_II" id="LECTURE_II"></a>LECTURE II</h3>
+
+<h5>TYCHO BRAH&Eacute; AND THE EARLIEST OBSERVATORY</h5>
+
+
+<p><span class="smcap">We</span> have seen how Copernicus placed the earth in its true position in the
+solar system, making it merely one of a number of other worlds revolving
+about a central luminary. And observe that there are two phenomena to be
+thus accounted for and explained: first, the diurnal revolution of the
+heavens; second, the annual motion of the sun among the stars.</p>
+
+<p>The effect of the diurnal motion is conspicuous to every one, and
+explains the rising, southing, and setting of the whole visible
+firmament. The effect of the annual motion, <i>i.e.</i> of the apparent
+annual motion, of the sun among the stars, is less obvious, but it may
+be followed easily enough by observing the stars visible at any given
+time of evening at different seasons of the year. At midnight, for
+instance, the position of the sun is definite, viz. due north always,
+but the constellation which at that time is due south or is rising or
+setting varies with the time of year; an interval of one month producing
+just the same effect on the appearance of the constellations as an
+interval of two hours does (because the day contains twice as many hours
+as the year contains months), <i>e.g.</i> the sky looks the same at midnight
+on the 1st of October as it does at 10 p.m. on the 1st of November.</p>
+
+<p>All these simple consequences of the geocentric as opposed to the
+heliocentric point of view were pointed<span class='pagenum'><a name="Page_34" id="Page_34">[Pg 34]</a></span> out by Copernicus, in addition
+to his greater work of constructing improved planetary tables on the
+basis of his theory. But it must be admitted that he himself felt the
+hypothesis of the motion of the earth to be a difficulty. Its acceptance
+is by no means such an easy and childish matter as we are apt now to
+regard it, and the hostility to it is not at all surprising. The human
+race, after having ridiculed and resisted the truth for a long time, is
+apt to end in accepting it so blindly and unimaginatively as to fail to
+recognize the real achievement of its first propounders, or the
+difficulties which they had to overcome. The majority of men at the
+present day have grown accustomed to hear the motion of the earth spoken
+of: their acceptance of it means nothing: the attitude of the paradoxer
+who denies it is more intelligent.</p>
+
+<p>It is not to be supposed that the idea of thus explaining some of the
+phenomena of the heavens, especially the daily motion of the entire
+firmament, by a diurnal rotation of the earth had not struck any one. It
+was often at this time referred to as the Pythagorean theory, and it had
+been taught, I believe, by Aristarchus. But it was new to the modern
+world, and it had the great weight of Aristotle against it.
+Consequently, for long after Copernicus, only a few leading spirits
+could be found to support it, and the long-established venerable
+Ptolemaic system continued to be taught in all Universities.</p>
+
+<p>The main objections to the motion of the earth were such as the
+following:&mdash;</p>
+
+<p>1. The motion is unfelt and difficult to imagine.</p>
+
+<div class="blockquot"><p>That it is unfelt is due to its uniformity, and can be explained
+mechanically. That it is difficult to imagine is and remains true,
+but a most important lesson we have to learn is that difficulty of
+conception is no valid argument against reality. </p></div>
+
+<p>2. That the stars do not alter their relative positions<span class='pagenum'><a name="Page_35" id="Page_35">[Pg 35]</a></span> according to
+the season of the year, but the constellations preserve always the same
+aspect precisely, even to careful measurement.</p>
+
+<div class="blockquot"><p>This is indeed a difficulty, and a great one. In June the earth is
+184 million miles away from where it was in December: how can we
+see precisely the same fixed stars? It is not possible, unless they
+are at a practically infinite distance. That is the only answer
+that can be given. It was the tentative answer given by Copernicus.
+It is the correct answer. Not only from every position of the
+earth, but from every planet of the solar system, the same
+constellations are visible, and the stars have the same aspect. The
+whole immensity of the solar system shrinks to practically a point
+when confronted with the distance of the stars.</p>
+
+<p>Not, however, so entirely a speck as to resist the terrific
+accuracy of the present century, and their microscopic relative
+displacement with the season of the year has now at length been
+detected, and the distance of many thereby measured. </p></div>
+
+<p>3. That, if the earth revolved round the sun, Mercury and Venus ought to
+show phases like the moon.</p>
+
+<div class="blockquot"><p>So they ought. Any globe must show phases if it live nearer the sun
+than we do and if we go round it, for we shall see varying amounts
+of its illuminated half. The only answer that Copernicus could give
+to this was that they might be difficult to see without extra
+powers of sight, but he ventured to predict that the phases would
+be seen if ever our powers of vision should be enhanced. </p></div>
+
+<p>4. That if the earth moved, or even revolved on its own axis, a stone or
+other dropped body ought to be left far behind.</p>
+
+<div class="blockquot"><p>This difficulty is not a real one, like the two last, and it is
+based on an ignorance of the laws of mechanics, which had not at
+that time been formulated. We know now that a ball dropped from a
+high tower, so far from lagging, drops a minute trifle <i>in front</i>
+of the foot of a perpendicular, because the top of the tower is
+moving a trace faster than the<span class='pagenum'><a name="Page_36" id="Page_36">[Pg 36]</a></span> bottom, by reason of the diurnal
+rotation. But, ignoring this, a stone dropped from the lamp of a
+railway carriage drops in the centre of the floor, whether the
+carriage be moving steadily or standing still; a slant direction of
+fall could only be detected if the carriage were being accelerated
+or if the brake were applied. A body dropped from a moving carriage
+shares the motion of the carriage, and starts with that as its
+initial velocity. A ball dropped from a moving balloon does not
+simply drop, but starts off in whatever direction the car was
+moving, its motion being immediately modified by gravity, precisely
+in the same way as that of a thrown ball is modified. This is,
+indeed, the whole philosophy of throwing&mdash;to drop a ball from a
+moving carriage. The carriage is the hand, and, to throw far, a run
+is taken and the body is jerked forward; the arm is also moved as
+rapidly as possible on the shoulder as pivot. The fore-arm can be
+moved still faster, and the wrist-joint gives yet another motion:
+the art of throwing is to bring all these to bear at the same
+instant, and then just as they have all attained their maximum
+velocity to let the ball go. It starts off with the initial
+velocity thus imparted, and is abandoned to gravity. If the vehicle
+were able to continue its motion steadily, as a balloon does, the
+ball when let go from it would appear to the occupant simply to
+drop; and it would strike the ground at a spot vertically under the
+moving vehicle, though by no means vertically below the place where
+it started. The resistance of the air makes observations of this
+kind inaccurate, except when performed inside a carriage so that
+the air shares in the motion. Otherwise a person could toss and
+catch a ball out of a train window just as well as if he were
+stationary; though to a spectator outside he would seem to be using
+great skill to throw the ball in the parabola adapted to bring it
+back to his hand.</p>
+
+<p>The same circumstance enhances the apparent difficulty of the
+circus rider's jumping feats. All he has to do is to jump up and
+down on the horse; the forward motion which carries him through
+hoops belongs to him by virtue of the motion of the horse, without
+effort on his part.</p>
+
+<p>Thus, then, it happens that a stone dropped sixteen feet on the
+earth appears to fall straight down, although its real path in
+space is a very flat trajectory of nineteen miles base and sixteen
+feet height; nineteen miles being the distance<span class='pagenum'><a name="Page_37" id="Page_37">[Pg 37]</a></span> traversed by the
+earth every second in the course of its annual journey round the
+sun.</p>
+
+<p>No wonder that it was thought that bodies must be left behind if
+the earth was subject to such terrific speed as this. All that
+Copernicus could suggest on this head was that perhaps the
+atmosphere might help to carry things forward, and enable them to
+keep pace with the earth. </p></div>
+
+<p>There were thus several outstanding physical difficulties in the way of
+the acceptance of the Copernican theory, besides the Biblical
+difficulty.</p>
+
+<p>It was quite natural that the idea of the earth's motion should be
+repugnant, and take a long time to sink into the minds of men; and as
+scientific progress was vastly slower then than it is now, we find not
+only all priests but even some astronomers one hundred years afterwards
+still imagining the earth to be at rest. And among them was a very
+eminent one, Tycho Brah&eacute;.</p>
+
+<p>It is interesting to note, moreover, that the argument about the motion
+of the earth being contrary to Scripture appealed not only to
+ecclesiastics in those days, but to scientific men also; and Tycho
+Brah&eacute;, being a man of great piety, and highly superstitious also, was so
+much influenced by it, that he endeavoured to devise some scheme by
+which the chief practical advantages of the Copernican system could be
+retained, and yet the earth be kept still at the centre of the whole.
+This was done by making all the celestial sphere, with stars and
+everything, rotate round the earth once a day, as in the Ptolemaic
+scheme; and then besides this making all the planets revolve round the
+sun, and this to revolve round the earth. Such is the Tychonic system.</p>
+
+<p>So far as <i>relative</i> motion is concerned it comes to the same thing;
+just as when you drop a book you may say either that the earth rises to
+meet the book, or that the book falls to meet the earth. Or when a fly
+buzzes round your head, you may say that you are revolving round the<span class='pagenum'><a name="Page_38" id="Page_38">[Pg 38]</a></span>
+fly. But the absurdity of making the whole gigantic system of sun and
+planets and stars revolve round our insignificant earth was too great to
+be swallowed by other astronomers after they had once had a taste of the
+Copernican theory; and accordingly the Tychonic system died a speedy and
+an easy death at the same time as its inventor.</p>
+
+<p>Wherein then lay the magnitude of the man?&mdash;not in his theories, which
+were puerile, but in his observations, which were magnificent. He was
+the first observational astronomer, the founder of the splendid system
+of practical astronomy which has culminated in the present Greenwich
+Observatory.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_16" id="Fig_16"></a>
+<img src="images/fig16.jpg" width="400" height="397" alt="Fig. 16." title="" />
+<span class="caption"><span class="smcap">Fig. 16.</span>&mdash;Tychonic system showing the sun with all the
+planets revolving round the earth.</span>
+</div>
+
+<p>Up to Tycho the only astronomical measurements had been of the rudest
+kind. Copernicus even improved upon what had gone before, with measuring
+rules made with his own hands. Ptolemy's observations could never be
+trusted to half a degree. Tycho introduced accuracy before undreamed of,
+and though his measurements, reckoned by modern ideas, are of course
+almost ludicrously rough (remember no such thing as a telescope or
+microscope was then dreamed of), yet, estimated by the era in which they
+were made, they are marvels of accuracy, and not a single<span class='pagenum'><a name="Page_39" id="Page_39">[Pg 39]</a></span> mistake due
+to carelessness has ever been detected in them. In fact they may be
+depended on almost to minutes of arc, <i>i.e.</i> to sixtieths of a degree.</p>
+
+<p>For certain purposes connected with the proper motion of stars they are
+still appealed to, and they served as the certain and trustworthy data
+for succeeding generations of theorists to work upon. It was long,
+indeed, after Tycho's death before observations approaching in accuracy
+to his were again made.</p>
+
+<p>In every sense, therefore, he was a pioneer: let us proceed to trace his
+history.</p>
+
+<p>Born the eldest son of a noble family&mdash;"as noble and ignorant as sixteen
+undisputed quarterings could make them," as one of his biographers
+says&mdash;in a period when, even more than at present, killing and hunting
+were the only natural aristocratic pursuits, when all study was regarded
+as something only fit for monks, and when science was looked at askance
+as something unsavoury, useless, and semi-diabolic, there was little in
+his introduction to the world urging him in the direction where his
+genius lay. Of course he was destined for a soldier; but fortunately his
+uncle, George Brah&eacute;, a more educated man than his father, having no son
+of his own, was anxious to adopt him, and though not permitted to do so
+for a time, succeeded in getting his way on the birth of a second son,
+Steno&mdash;who, by the way, ultimately became Privy Councillor to the King
+of Denmark.</p>
+
+<p>Tycho's uncle gave him what he would never have got at home&mdash;a good
+education; and ultimately put him to study law. At the age of thirteen
+he entered the University of Copenhagen, and while there occurred the
+determining influence of his life.</p>
+
+<p>An eclipse of the sun in those days was not regarded with the
+cold-blooded inquisitiveness or matter-of-fact apathy, according as
+there is or is not anything to be learnt from it, with which such an
+event is now regarded. Every<span class='pagenum'><a name="Page_40" id="Page_40">[Pg 40]</a></span> occurrence in the heavens was then
+believed to carry with it the destiny of nations and the fate of
+individuals, and accordingly was of surpassing interest. Ever since the
+time of Hipparchus it had been possible for some capable man here and
+there to predict the occurrence of eclipses pretty closely. The thing is
+not difficult. The prediction was not, indeed, to the minute and second,
+as it is now; but the day could usually be hit upon pretty accurately
+some time ahead, much as we now manage to hit upon the return of a
+comet&mdash;barring accidents; and the hour could be predicted as the event
+approached.</p>
+
+<p>Well, the boy Tycho, among others, watched for this eclipse on August
+21st, 1560; and when it appeared at its appointed time, every instinct
+for the marvellous, dormant in his strong nature, awoke to strenuous
+life, and he determined to understand for himself a science permitting
+such wonderful possibilities of prediction. He was sent to Leipzig with
+a tutor to go on with his study of law, but he seems to have done as
+little law as possible: he spent all his money on books and instruments,
+and sat up half the night studying and watching the stars.</p>
+
+<p>In 1563 he observed a conjunction of Jupiter and Saturn, the precursor,
+and <i>cause</i> as he thought it, of the great plague. He found that the old
+planetary tables were as much as a month in error in fixing this event,
+and even the Copernican tables were several days out; so he formed the
+resolve to devote his life to improving astronomical tables. This
+resolve he executed with a vengeance. His first instrument was a jointed
+ruler with sights for fixing the position of planets with respect to the
+stars, and observing their stations and retrogressions. By thus
+measuring the angles between a planet and two fixed stars, its position
+can be plotted down on a celestial map or globe.</p>
+
+<p><span class='pagenum'><a name="Page_41" id="Page_41">[Pg 41]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_17" id="Fig_17"></a>
+<img src="images/fig17.jpg" width="400" height="527" alt="Fig. 17." title="" />
+<span class="caption"><span class="smcap">Fig. 17.</span>&mdash;Portrait of Tycho.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_42" id="Page_42">[Pg 42]</a></span></p><p>In 1565 his uncle George died, and made Tycho his heir. He returned to
+Denmark, but met with nothing but ridicule and contempt for his absurd
+drivelling away of time over useless pursuits. So he went back to
+Germany&mdash;first to Wittenberg, thence, driven by the plague, to Rostock.</p>
+
+<p>Here his fiery nature led him into an absurd though somewhat dangerous
+adventure. A quarrel at some feast, on a mathematical point, with a
+countryman, Manderupius, led to the fixing of a duel, and it was fought
+with swords at 7 p.m. at the end of December, when, if there was any
+light at all, it must have been of a flickering and unsatisfactory
+nature. The result of this insane performance was that Tycho got his
+nose cut clean off.</p>
+
+<p>He managed however to construct an artificial one, some say of gold and
+silver, some say of putty and brass; but whatever it was made of there
+is no doubt that he wore it for the rest of his life, and it is a most
+famous feature. It excited generally far more interest than his
+astronomical researches. It is said, moreover, to have very fairly
+resembled the original, but whether this remark was made by a friend or
+by an enemy I cannot say. One account says that he used to carry about
+with him a box of cement to apply whenever his nose came off, which it
+periodically did.</p>
+
+<p>About this time he visited Augsburg, met with some kindred and
+enlightened spirits in that town, and with much enthusiasm and spirit
+constructed a great quadrant. These early instruments were tremendous
+affairs. A great number of workmen were employed upon this quadrant, and
+it took twenty men to carry it to its place and erect it. It stood in
+the open air for five years, and then was destroyed by a storm. With it
+he made many observations.</p>
+
+<p><span class='pagenum'><a name="Page_43" id="Page_43">[Pg 43]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_18" id="Fig_18"></a>
+<img src="images/fig18.jpg" width="400" height="592" alt="Fig. 18." title="" />
+<span class="caption"><span class="smcap">Fig. 18.</span>&mdash;Early out-door quadrant of Tycho; for
+observing altitudes by help of the sights <i>D</i>, <i>L</i> and the plumb line.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_44" id="Page_44">[Pg 44]</a></span></p><p>On his return to Denmark in 1571, his fame preceded him, and he was
+much better received; and in order to increase his power of constructing
+instruments he took up the study of alchemy, and like the rest of the
+persuasion tried to make gold. The precious metals were by many old
+philosophers considered to be related in some way to the heavenly
+bodies: silver to the moon, for instance&mdash;as we still see by the name
+lunar caustic applied to nitrate of silver; gold to the sun, copper to
+Mars, lead to Saturn. Hence astronomy and alchemy often went together.
+Tycho all his life combined a little alchemy with his astronomical
+labours, and he constructed a wonderful patent medicine to cure all
+disorders, which had as wide a circulation in Europe in its time as
+Holloway's pills; he gives a tremendous receipt for it, with liquid gold
+and all manner of ingredients in it; among them, however, occurs a
+little antimony&mdash;a well-known sudorific&mdash;and to this, no doubt, whatever
+efficacy the medicine possessed was due.</p>
+
+<p>So he might have gone on wasting his time, were it not that in November,
+1572, a new star made its appearance, as they have done occasionally
+before and since. On the average one may say that about every fifty
+years a new star of fair magnitude makes its temporary appearance. They
+are now known to be the result of some catastrophe or collision, whereby
+immense masses of incandescent gas are produced. This one seen by Tycho
+became as bright as Jupiter, and then died away in about a year and a
+half. Tycho observed all its changes, and endeavoured to measure its
+distance from the earth, with the result that it was proved to belong to
+the region of the fixed stars, at an immeasurable distance, and was not
+some nearer and more trivial phenomenon.</p>
+
+<p>He was asked by the University of Copenhagen to give a course of
+lectures on astronomy; but this was a step he felt some aristocratic
+aversion to, until a little friendly pressure was brought to bear upon
+him by a request from the king, and delivered they were.</p>
+
+<p>He now seems to have finally thrown off his aristocratic prejudices, and
+to have indulged himself in treading on the corns of nearly all the high
+and mighty people he came into contact with. In short, he became what we
+might now call a violent Radical; but he was a good-hearted man,
+nevertheless, and many are the tales told of his visits to<span class='pagenum'><a name="Page_45" id="Page_45">[Pg 45]</a></span> sick
+peasants, of his consulting the stars as to their fate&mdash;all in perfect
+good faith&mdash;and of the medicines which he concocted and prescribed for
+them.</p>
+
+<p>The daughter of one of these peasants he married, and very happy the
+marriage seems to have been.</p>
+
+<div class="figcenter" style="width: 450px;"><a name="Fig_19" id="Fig_19"></a>
+<img src="images/fig19.jpg" width="350" height="351" alt="Fig. 19." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 19.</span>&mdash;Map of Denmark, showing the island of Huen.<br />
+<i>Walker &amp; Boutallse.</i></span>
+</div>
+
+<p>Now comes the crowning episode in Tycho's life. Frederick II., realizing
+how eminent a man they had among them, and how much he could do if only
+he had the means&mdash;for we must understand that Tycho, though of good
+family and well off, was by no means what we would call a wealthy
+man&mdash;Frederick II. made him a splendid and enlightened offer. The offer
+was this: that if Tycho would agree to settle down and make his
+astronomical observations in Denmark, he should have an estate in Norway
+settled upon him, a pension of &pound;400 a year for life, a site for a large
+observatory, and &pound;20,000 to build it with.</p>
+
+<p><span class='pagenum'><a name="Page_46" id="Page_46">[Pg 46]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_20" id="Fig_20"></a>
+<img src="images/fig20.jpg" width="400" height="253" alt="Fig. 20." title="" />
+<span class="caption"><span class="smcap">Fig. 20.</span>&mdash;Uraniburg.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_47" id="Page_47">[Pg 47]</a></span></p>
+<div class="figcenter" style="width: 450px;"><br /><a name="Fig_21" id="Fig_21"></a>
+<img src="images/fig21.jpg" width="350" height="568" alt="Fig. 21." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 21.</span>&mdash;Astrolabe. An old instrument with sights for
+marking the positions of the celestial bodies roughly. A sort of
+skeleton celestial globe.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_48" id="Page_48">[Pg 48]</a></span></p>
+<div class="figcenter" style="width: 350px;"><br /><a name="Fig_22" id="Fig_22"></a>
+<img src="images/fig22.jpg" width="350" height="552" alt="Fig. 22." title="" />
+<span class="caption"><span class="smcap">Fig. 22.</span>&mdash;Tycho's large sextant; for measuring the angular distance
+between two bodies by direct sighting.</span>
+</div>
+
+<p>Well, if ever money was well spent, this was. By its means Denmark
+before long headed the nations of Europe in the matter of science&mdash;a
+thing it has not done before or since. The site granted was the island
+of Huen, between Copenhagen and Elsinore; and here the most magnificent
+observatory ever built was raised, and called Uraniburg&mdash;the castle of
+the heavens. It was built on a hill in the centre of the island, and
+included gardens, printing shops, laboratory, dwelling-houses, and four
+observatories&mdash;all furnished with the most splendid instruments that
+Tycho could devise, and that could then be constructed. It was decorated
+with pictures and sculptures of eminent men,<span class='pagenum'><a name="Page_49" id="Page_49">[Pg 49]</a></span> and altogether was a most
+gorgeous place. &pound;20,000 no doubt went far in those days, but the
+original grant was supplemented by Tycho himself, who is said to have
+spent another equal sum out of his own pocket on the place.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_23" id="Fig_23"></a>
+<img src="images/fig23.jpg" width="350" height="541" alt="Fig. 23." title="" />
+<span class="caption"><span class="smcap">Fig. 23.</span>&mdash;The Quadrant in Uraniburg; or altitude and azimuth
+instrument.</span>
+</div>
+
+<p>For twenty years this great temple of science was continually worked in
+by him, and he soon became the foremost scientific man in Europe.
+Philosophers, statesmen, and occasionally kings, came to visit the great
+astronomer, and to inspect his curiosities.</p>
+
+<p><span class='pagenum'><a name="Page_50" id="Page_50">[Pg 50]</a></span></p>
+
+<div class="figcenter" style="width: 550px;"><a name="Fig_24" id="Fig_24"></a>
+<img src="images/fig24.jpg" width="400" height="632" alt="Fig. 24." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 24.</span>&mdash;Tycho&#39;s form of transit circle.</span>
+<p><small>The method of utilising the extremely uniform rotation of the earth by
+watching the planets and stars as they cross the meridian, and recording
+their times of transit; observing also at the same time their meridian
+altitudes (see observer <i>F</i>), was the invention of Tycho, and
+constitutes his greatest achievement. His method is followed to this day
+in all observatories.</small></p>
+</div>
+
+<p><span class='pagenum'><a name="Page_51" id="Page_51">[Pg 51]</a></span></p>
+<div class="figcenter" style="width: 400px;"><br /><a name="Fig_25" id="Fig_25"></a>
+<img src="images/fig25.jpg" width="400" height="391" alt="Fig. 25." title="" />
+<div class="caption1"><span class="smcap">Fig. 25.</span>&mdash;A modern transit circle, showing essentially
+the same parts as in Tycho's instrument, viz. the observer watching the
+transit, the clock, the recorder of the observation, and the graduated
+circle; the latter to be read by a second observer.</div>
+</div>
+
+<p>And very wholesome for some of these great personages must have been the
+treatment they met with. For Tycho was no respecter of persons. His
+humbly-born wife sat at the head of the table, whoever was there; and he
+would snub and contradict a chancellor just as soon as he would a serf.
+Whatever form his pride may have taken when a youth, in his maturity it
+impelled him to ignore differences of rank not substantially justified,
+and he seemed to take a delight in exposing the ignorance of shallow
+titled persons, to whom contradiction and exposure were most unusual
+experiences.</p>
+
+<p><span class='pagenum'><a name="Page_52" id="Page_52">[Pg 52]</a></span></p><p>For sick peasants he would take no end of trouble, and went about
+doctoring them for nothing, till he set all the professional doctors
+against him; so that when his day of misfortune came, as come it did,
+their influence was not wanting to help to ruin one who spoilt their
+practice, and whom they derided as a quack.</p>
+
+<p>But some of the great ignorant folk who came to visit his temple of
+science, and to inspect its curiosities, felt themselves insulted&mdash;not
+always without reason. He kept a tame maniac in the house, named Lep,
+and he used to regard the sayings of this personage as oracular,
+presaging future events, and far better worth listening to than ordinary
+conversation. Consequently he used to have him at his banquets and feed
+him himself; and whenever Lep opened his mouth to speak, every one else
+was peremptorily ordered to hold his tongue, so that Lep's words might
+be written down. In fact it was something like an exaggerated edition of
+Betsy Trotwood and Mr. Dick.</p>
+
+<p>"It must have been an odd dinner party" (says Prof. Stuart), "with this
+strange, wild, terribly clever man, with his red hair and brazen nose,
+sometimes flashing with wit and knowledge, sometimes making the whole
+company, princes and servants alike, hold their peace and listen humbly
+to the ravings of a poor imbecile."</p>
+
+<p>To people he despised he did not show his serious instruments. He had
+other attractions, in the shape of a lot of toy machinery, little
+windmills, and queer doors, and golden globes, and all manner of
+ingenious tricks and automata, many of which he had made himself, and
+these he used to show them instead; and no doubt they were well enough
+pleased with them. Those of the visitors, however, who really cared to
+see and understand his instruments, went away enchanted with his genius
+and hospitality.</p>
+
+<p>I may, perhaps, be producing an unfair impression of imperiousness and
+insolence. Tycho was fiery, no doubt, but<span class='pagenum'><a name="Page_53" id="Page_53">[Pg 53]</a></span> I think we should wrong him
+if we considered him insolent. Most of the nobles of his day were
+haughty persons, accustomed to deal with serfs, and very likely to sneer
+at and trample on any meek man of science whom they could easily
+despise. So Tycho was not meek; he stood up for the honour of his
+science, and paid them back in their own coin, with perhaps a little
+interest. That this behaviour was not worldly-wise is true enough, but I
+know of no commandment enjoining us to be worldly-wise.</p>
+
+<p>If we knew more about his so-called imbecile <i>prot&eacute;g&eacute;</i> we should
+probably find some reason for the interest which Tycho took in him.
+Whether he was what is now called a "clairvoyant" or not, Tycho
+evidently regarded his utterances as oracular, and of course when one is
+receiving what may be a revelation from heaven it is natural to suppress
+ordinary conversation.</p>
+
+<p>Among the noble visitors whom he received and entertained, it is
+interesting to notice James I. of England, who spent eight days at
+Uraniburg on the occasion of his marriage with Anne of Denmark in 1590,
+and seems to have been deeply impressed by his visit.</p>
+
+<p>Among other gifts, James presented Tycho with a dog (depicted in <a href="#Fig_24">Fig.
+24</a>), and this same animal was subsequently the cause of trouble. For it
+seems that one day the Chancellor of Denmark, Walchendorf, brutally
+kicked the poor beast; and Tycho, who was very fond of animals, gave him
+a piece of his mind in no measured language. Walchendorf went home
+determined to ruin him. King Frederick, however, remained his true
+friend, doubtless partly influenced thereto by his Queen Sophia, an
+enlightened woman who paid many visits to Uraniburg, and knew Tycho
+well. But unfortunately Frederick died; and his son, a mere boy, came to
+the throne.</p>
+
+<p>Now was the time for the people whom Tycho had offended, for those who
+were jealous of his great fame and importance, as well as for those who
+cast longing eyes<span class='pagenum'><a name="Page_54" id="Page_54">[Pg 54]</a></span> on his estate and endowments. The boy-king, too,
+unfortunately paid a visit to Tycho, and, venturing upon a decided
+opinion on some recondite subject, received a quiet setting down which
+he ill relished.</p>
+
+<p>Letters written by Tycho about this time are full of foreboding. He
+greatly dreads having to leave Uraniburg, with which his whole life has
+for twenty years been bound up. He tries to comfort himself with the
+thought that, wherever he is sent, he will have the same heavens and the
+same stars over his head.</p>
+
+<p>Gradually his Norwegian estate and his pension were taken away, and in
+five years poverty compelled him to abandon his magnificent temple, and
+to take a small house in Copenhagen.</p>
+
+<p>Not content with this, Walchendorf got a Royal Commission appointed to
+inquire into the value of his astronomical labours. This sapient body
+reported that his work was not only useless, but noxious; and soon after
+he was attacked by the populace in the public street.</p>
+
+<p>Nothing was left for him now but to leave the country, and he went into
+Germany, leaving his wife and instruments to follow him whenever he
+could find a home for them.</p>
+
+<p>His wanderings in this dark time&mdash;some two years&mdash;are not quite clear;
+but at last the enlightened Emperor of Bohemia, Rudolph II., invited him
+to settle in Prague. Thither he repaired, a castle was given him as an
+observatory, a house in the city, and 3000 crowns a year for life. So
+his instruments were set up once more, students flocked to hear him and
+to receive work at his hands&mdash;among them a poor youth, John Kepler, to
+whom he was very kind, and who became, as you know, a still greater man
+than his master.</p>
+
+<p>But the spirit of Tycho was broken, and though some good work was done
+at Prague&mdash;more observations made, and the Rudolphine tables begun&mdash;yet
+the hand of death was upon him. A painful disease seized him, attended
+with<span class='pagenum'><a name="Page_55" id="Page_55">[Pg 55]</a></span> sleeplessness and temporary delirium, during the paroxysms of
+which he frequently exclaimed, <i>Ne frustra vixisse videar</i>. ("Oh that it
+may not appear that I have lived in vain!")</p>
+
+<p>Quietly, however, at last, and surrounded by his friends and relatives,
+this fierce, passionate soul passed away, on the 24th of October, 1601.</p>
+
+<p>His beloved instruments, which were almost a part of himself, were
+stored by Rudolph in a museum with scrupulous care, until the taking of
+Prague by the Elector Palatine's troops. In this disturbed time they got
+smashed, dispersed, and converted to other purposes. One thing only was
+saved&mdash;the great brass globe, which some thirty years after was
+recognized by a later king of Denmark as having belonged to Tycho, and
+deposited in the Library of the Academy of Sciences at Copenhagen, where
+I believe it is to this day.</p>
+
+<p>The island of Huen was overrun by the Danish nobility, and nothing now
+remains of Uraniburg but a mound of earth and two pits.</p>
+
+<p>As to the real work of Tycho, that has become immortal enough,&mdash;chiefly
+through the labours of his friend and scholar whose life we shall
+consider in the next lecture.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_56" id="Page_56">[Pg 56]</a></span></p>
+<h4><a name="SUMMARY_OF_FACTS_FOR_LECTURE_III" id="SUMMARY_OF_FACTS_FOR_LECTURE_III"></a>SUMMARY OF FACTS FOR LECTURE III</h4>
+
+
+<p><i>Life and work of Kepler.</i> Kepler was born in December, 1571, at Weil in
+W&uuml;rtemberg. Father an officer in the duke's army, mother something of a
+virago, both very poor. Kepler was utilized as a tavern pot-boy, but
+ultimately sent to a charity school, and thence to the University of
+T&uuml;bingen. Health extremely delicate; he was liable to violent attacks
+all his life. Studied mathematics, and accepted an astronomical
+lectureship at Graz as the first post which offered. Endeavoured to
+discover some connection between the number of the planets, their times
+of revolution, and their distances from the sun. Ultimately hit upon his
+fanciful regular-solid hypothesis, and published his first book in 1597.
+In 1599 was invited by Tycho to Prague, and there appointed Imperial
+mathematician, at a handsome but seldom paid salary. Observed the new
+star of 1604. Endeavoured to find the law of refraction of light from
+Vitellio's measurements, but failed. Analyzed Tycho's observations to
+find the true law of motion of Mars. After incredible labour, through
+innumerable wrong guesses, and six years of almost incessant
+calculation, he at length emerged in his two "laws"&mdash;discoveries which
+swept away all epicycles, deferents, equants, and other remnants of the
+Greek system, and ushered in the dawn of modern astronomy.</p>
+
+<p><span class="smcap">Law I.</span> <i>Planets move in ellipses, with the Sun in one focus.</i></p>
+
+<p><span class="smcap">Law II.</span> <i>The radius vector (or line joining sun and planet) sweeps out
+equal areas in equal times.</i></p>
+
+<p>Published his second book containing these laws in 1609. Death of
+Rudolph in 1612, and subsequent increased misery and misfortune of
+Kepler. Ultimately discovered the connection between the times and
+distances of the planets for which he had been groping all his mature
+life, and announced it in 1618:&mdash;</p>
+
+<p><span class="smcap">Law III.</span> <i>The square of the time of revolution (or year) of each planet
+is proportional to the cube of its mean distance from the sun.</i></p>
+
+<p>The book in which this law was published ("On Celestial Harmonies") was
+dedicated to James of England. In 1620 had to intervene to protect his
+mother from being tortured for witchcraft. Accepted a professorship at
+Linz. Published the Rudolphine tables in 1627, embodying Tycho's
+observations and his own theory. Made a last effort to overcome his<span class='pagenum'><a name="Page_57" id="Page_57">[Pg 57]</a></span>
+poverty by getting the arrears of his salary paid at Prague, but was
+unsuccessful, and, contracting brain fever on the journey, died in
+November, 1630, aged 59.</p>
+
+<p>A man of keen imagination, indomitable perseverance, and uncompromising
+love of truth, Kepler overcame ill-health, poverty, and misfortune, and
+placed himself in the very highest rank of scientific men. His laws, so
+extraordinarily discovered, introduced order and simplicity into what
+else would have been a chaos of detailed observations; and they served
+as a secure basis for the splendid erection made on them by Newton.</p>
+
+<p class="poem">
+<i>Seven planets of the Ptolemaic system&mdash;</i><br />
+<span style="margin-left: 2em;">Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.</span><br />
+
+<i>Six planets of the Copernican system&mdash;</i><br />
+<span style="margin-left: 2em;">Mercury, Venus, Earth, Mars, Jupiter, Saturn.</span><br />
+
+<i>The five regular solids, in appropriate order&mdash;</i><br />
+<span style="margin-left: 2em;">Octahedron, Icosahedron, Dodecahedron, Tetrahedron, Cube.</span><br />
+</p>
+
+
+<div class='center'><br />
+<table border="0" cellpadding="4" cellspacing="0" summary="Kepler's Third Law">
+<tr>
+ <td align='center' colspan='5'><i>Table illustrating Kepler's third law.</i></td>
+</tr>
+<tr class='tr4'>
+ <td class='tdcbrbl'><small>Planet.</small></td>
+ <td class='tdcbr'><small>Mean distance<br />from Sun.<br />D</small></td>
+ <td class='tdcbr'><small>Length<br />of Year.<br />T</small></td>
+ <td class='tdcbr'><small>Cube of the<br />Distance.<br />D<sup>3</sup></small></td>
+ <td class='tdcbr'><small>Square of the<br />Time.<br />T<sup>2</sup></small></td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>Mercury</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&middot;3871</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;&middot;24084</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&middot;05801</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&middot;05801</td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>Venus</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&middot;7233</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;&middot;61519</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&middot;37845</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&middot;37846</td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>Earth</td>
+ <td class='tdlbrpl1'>1&middot;0000</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;1&middot;0000</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;1&middot;0000</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;1&middot;0000</td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>Mars</td>
+ <td class='tdlbrpl1'>1&middot;5237</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;1&middot;8808</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;3&middot;5375</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;&nbsp;&nbsp;3&middot;5375</td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>Jupiter</td>
+ <td class='tdlbrpl1'>5&middot;2028</td>
+ <td class='tdlbrpl1'>11&middot;862</td>
+ <td class='tdlbrpl1'>140&middot;83</td>
+ <td class='tdlbrpl1'>140&middot;70</td>
+</tr>
+<tr class='tr5'>
+ <td class='tdlbrblpl1'>Saturn</td>
+ <td class='tdlbrpl1'>9&middot;5388</td>
+ <td class='tdlbrpl1'>29&middot;457</td>
+ <td class='tdlbrpl1'>867&middot;92</td>
+ <td class='tdlbrpl1'>867&middot;70</td>
+</tr>
+</table></div>
+
+<p>The length of the earth's year is 365&middot;256 days; its mean distance from
+the sun, taken above as unity, is 92,000,000 miles.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_58" id="Page_58">[Pg 58]</a></span></p>
+<h3><a name="LECTURE_III" id="LECTURE_III"></a>LECTURE III</h3>
+
+<h5>KEPLER AND THE LAWS OF PLANETARY MOTION</h5>
+
+
+<p><span class="smcap">It</span> is difficult to imagine a stronger contrast between two men engaged
+in the same branch of science than exists between Tycho Brah&eacute;, the
+subject of last lecture, and Kepler, our subject on the present
+occasion.</p>
+
+<p>The one, rich, noble, vigorous, passionate, strong in mechanical
+ingenuity and experimental skill, but not above the average in
+theoretical and mathematical power.</p>
+
+<p>The other, poor, sickly, devoid of experimental gifts, and unfitted by
+nature for accurate observation, but strong almost beyond competition in
+speculative subtlety and innate mathematical perception.</p>
+
+<p>The one is the complement of the other; and from the fact of their
+following each other so closely arose the most surprising benefits to
+science.</p>
+
+<p>The outward life of Kepler is to a large extent a mere record of poverty
+and misfortune. I shall only sketch in its broad features, so that we
+may have more time to attend to his work.</p>
+
+<p>He was born (so his biographer assures us) in longitude 29&deg; 7', latitude
+48&deg; 54', on the 21st of December, 1571. His parents seem to have been of
+fair condition, but by reason, it is said, of his becoming surety for a
+friend, the father lost all his slender income, and was reduced to
+keeping a tavern. Young John Kepler was thereupon taken from school,
+and<span class='pagenum'><a name="Page_59" id="Page_59">[Pg 59]</a></span> employed as pot-boy between the ages of nine and twelve. He was a
+sickly lad, subject to violent illnesses from the cradle, so that his
+life was frequently despaired of. Ultimately he was sent to a monastic
+school and thence to the University of T&uuml;bingen, where he graduated
+second on the list. Meanwhile home affairs had gone to rack and ruin.
+His father abandoned the home, and later died abroad. The mother
+quarrelled with all her relations, including her son John; who was
+therefore glad to get away as soon as possible.</p>
+
+<p>All his connection with astronomy up to this time had been the hearing
+the Copernican theory expounded in University lectures, and defending it
+in a college debating society.</p>
+
+<p>An astronomical lectureship at Graz happening to offer itself, he was
+urged to take it, and agreed to do so, though stipulating that it should
+not debar him from some more brilliant profession when there was a
+chance.</p>
+
+<p>For astronomy in those days seems to have ranked as a minor science,
+like mineralogy or meteorology now. It had little of the special dignity
+with which the labours of Kepler himself were destined so greatly to aid
+in endowing it.</p>
+
+<p>Well, he speedily became a thorough Copernican, and as he had a most
+singularly restless and inquisitive mind, full of appreciation of
+everything relating to number and magnitude&mdash;was a born speculator and
+thinker just as Mozart was a born musician, or Bidder a born
+calculator&mdash;he was agitated by questions such as these: Why are there
+exactly six planets? Is there any connection between their orbital
+distances, or between their orbits and the times of describing them?
+These things tormented him, and he thought about them day and night. It
+is characteristic of the spirit of the times&mdash;this questioning why there
+should be six planets. Nowadays, we should simply record the fact and
+look out for a seventh. Then, some occult property of the number six was
+groped for, such as that it was equal to 1 + 2 + 3 and likewise equal to
+1 &times; 2 &times; 3, and so on. Many fine reasons had been given for the seven
+planets of the Ptolemaic<span class='pagenum'><a name="Page_60" id="Page_60">[Pg 60]</a></span> system (see, for instance, <a href="#Page_106">p. 106</a>), but for
+the six planets of the Copernican system the reasons were not so cogent.</p>
+
+<p>Again, with respect to their successive distances from the sun, some law
+would seem to regulate their distance, but it was not known.
+(Parenthetically I may remark that it is not known even now: a crude
+empirical statement known as Bode's law&mdash;see page 294&mdash;is all that has
+been discovered.)</p>
+
+<p>Once more, the further the planet the slower it moved; there seemed to
+be some law connecting speed and distance. This also Kepler made
+continual attempts to discover.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_26" id="Fig_26"></a>
+<img src="images/fig26.jpg" width="350" height="269" alt="Fig. 26." title="" />
+<span class="caption"><span class="smcap">Fig. 26.</span>&mdash;Orbits of some of the planets drawn to scale:
+showing the gap between Mars and Jupiter.</span>
+</div>
+
+<p>One of his ideas concerning the law of the successive distances was
+based on the inscription of a triangle in a circle. If you inscribe in a
+circle a large number of equilateral triangles, they envelop another
+circle bearing a definite ratio to the first: these might do for the
+orbits of two planets (<a href="#Fig_27">see Fig. 27</a>). Then try inscribing and
+circumscribing squares, hexagons, and other figures, and see if the
+circles thus defined would correspond to the several planetary orbits.
+But they would not give any satisfactory result. Brooding over this
+disappointment, the idea of trying solid figures suddenly strikes him.
+"What have plane figures to do with the celestial orbits?" he cries out;
+"inscribe the regular solids." And then&mdash;brilliant idea&mdash;he remembers
+that there are but five. Euclid had shown that there could be only five
+regular solids.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a> The number evidently corresponds to the gaps between
+the six planets. The reason of there being only six seems to be
+attained. This coincidence assures him he is on the right track, and
+with great enthusiasm and hope he "represents the earth's orbit by a
+sphere as the norm and measure of all"; round it he circumscribes a
+dodecahedron, and puts another sphere round that, which is approximately
+the orbit of Mars; round that, again, a tetrahedron, the corners of
+which mark the sphere of the orbit of Jupiter; round that sphere, again,
+he places a cube, which roughly gives the orbit of Saturn.</p>
+
+<p><span class='pagenum'><a name="Page_61" id="Page_61">[Pg 61]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_27" id="Fig_27"></a>
+<img src="images/fig27.jpg" width="400" height="400" alt="Fig. 27." title="" />
+<span class="caption"><span class="smcap">Fig. 27.</span>&mdash;Many-sided polygon or approximate circle
+enveloped by straight lines, as for instance by a number of equilateral
+triangles.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_62" id="Page_62">[Pg 62]</a></span></p><p>On the other hand, he inscribes in the sphere of the earth's orbit an
+icosahedron; and inside the sphere determined by that, an octahedron;
+which figures he takes to inclose the spheres of Venus and of Mercury
+respectively.</p>
+
+<p>The imagined discovery is purely fictitious and accidental. First of
+all, eight planets are now known; and secondly, their real distances
+agree only very approximately with Kepler's hypothesis.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_28" id="Fig_28"></a>
+<img src="images/fig28.jpg" width="400" height="371" alt="Fig. 28." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 28.</span>&mdash;Frameworks with inscribed and circumscribed
+spheres, representing the five regular solids distributed as Kepler
+supposed them to be among the planetary orbits. (See &quot;Summary&quot; at
+beginning of this lecture, <a href="#Page_57">p. 57</a>.)</div>
+</div>
+
+<p>Nevertheless, the idea gave him great delight. He says:&mdash;"The intense
+pleasure I have received from this discovery can never be told in words.
+I regretted no more the time wasted; I tired of no labour; I shunned no
+toil of reckoning, days and nights spent in calculation, until I could
+see whether my hypothesis would agree with the orbits of Copernicus, or
+whether my joy was to vanish into air."</p>
+
+<p>He then went on to speculate as to the cause of the<span class='pagenum'><a name="Page_63" id="Page_63">[Pg 63]</a></span> planets' motion.
+The old idea was that they were carried round by angels or celestial
+intelligences. Kepler tried to establish some propelling force emanating
+from the sun, like the spokes of a windmill.</p>
+
+<p>This first book of his brought him into notice, and served as an
+introduction to Tycho and to Galileo.</p>
+
+<p>Tycho Brah&eacute; was at this time at Prague under the patronage of the
+Emperor Rudolph; and as he was known to have by far the best planetary
+observations of any man living, Kepler wrote to him to know if he might
+come and examine them so as to perfect his theory.</p>
+
+<p>Tycho immediately replied, "Come, not as a stranger, but as a very
+welcome friend; come and share in my observations with such instruments
+as I have with me, and as a dearly beloved associate." After this visit,
+Tycho wrote again, offering him the post of mathematical assistant,
+which after hesitation was accepted. Part of the hesitation Kepler
+expresses by saying that "for observations his sight was dull, and for
+mechanical operations his hand was awkward. He suffered much from weak
+eyes, and dare not expose himself to night air." In all this he was, of
+course, the antipodes of Tycho, but in mathematical skill he was greatly
+his superior.</p>
+
+<p>On his way to Prague he was seized with one of his periodical illnesses,
+and all his means were exhausted by the time he could set forward again,
+so that he had to apply for help to Tycho.</p>
+
+<p>It is clear, indeed, that for some time now he subsisted entirely on the
+bounty of Tycho, and he expresses himself most deeply grateful for all
+the kindness he received from that noble and distinguished man, the head
+of the scientific world at that date.</p>
+
+<p>To illustrate Tycho's kindness and generosity, I must read you a letter
+written to him by Kepler. It seems that Kepler, on one of his absences
+from Prague, driven half mad with poverty and trouble, fell foul of
+Tycho, whom he<span class='pagenum'><a name="Page_64" id="Page_64">[Pg 64]</a></span> thought to be behaving badly in money matters to him and
+his family, and wrote him a violent letter full of reproaches and
+insults. Tycho's secretary replied quietly enough, pointing out the
+groundlessness and ingratitude of the accusation.</p>
+
+<p>Kepler repents instantly, and replies:&mdash;</p>
+
+<div class="blockquot"><p>"<span class="smcap">Most Noble Tycho</span>," (these are the words of his letter), "how shall
+I enumerate or rightly estimate your benefits conferred on me? For
+two months you have liberally and gratuitously maintained me, and
+my whole family; you have provided for all my wishes; you have done
+me every possible kindness; you have communicated to me everything
+you hold most dear; no one, by word or deed, has intentionally
+injured me in anything; in short, not to your children, your wife,
+or yourself have you shown more indulgence than to me. This being
+so, as I am anxious to put on record, I cannot reflect without
+consternation that I should have been so given up by God to my own
+intemperance as to shut my eyes on all these benefits; that,
+instead of modest and respectful gratitude, I should indulge for
+three weeks in continual moroseness towards all your family, in
+headlong passion and the utmost insolence towards yourself, who
+possess so many claims on my veneration, from your noble family,
+your extraordinary learning, and distinguished reputation. Whatever
+I have said or written against the person, the fame, the honour,
+and the learning of your excellency; or whatever, in any other way,
+I have injuriously spoken or written (if they admit no other more
+favourable interpretation), as, to my grief, I have spoken and
+written many things, and more than I can remember; all and
+everything I recant, and freely and honestly declare and profess to
+be groundless, false, and incapable of proof."</p></div>
+
+<p>Tycho accepted the apology thus heartily rendered, and the temporary
+breach was permanently healed.</p>
+
+<p>In 1601, Kepler was appointed "Imperial mathematician," to assist Tycho
+in his calculations.</p>
+
+<p>The Emperor Rudolph did a good piece of work in thus maintaining these
+two eminent men, but it is quite clear that it was as astrologers that
+he valued them; and all he<span class='pagenum'><a name="Page_65" id="Page_65">[Pg 65]</a></span> cared for in the planetary motions was
+limited to their supposed effect on his own and his kingdom's destiny.
+He seems to have been politically a weak and superstitious prince, who
+was letting his kingdom get into hopeless confusion, and entangling
+himself in all manner of political complications. While Bohemia
+suffered, however, the world has benefited at his hands; and the tables
+upon which Tycho was now engaged are well called the Rudolphine tables.</p>
+
+<p>These tables of planetary motion Tycho had always regarded as the main
+work of his life; but he died before they were finished, and on his
+death-bed he intrusted the completion of them to Kepler, who loyally
+undertook their charge.</p>
+
+<p>The Imperial funds were by this time, however, so taxed by wars and
+other difficulties that the tables could only be proceeded with very
+slowly, a staff of calculators being out of the question. In fact,
+Kepler could not get even his own salary paid: he got orders, and
+promises, and drafts on estates for it; but when the time came for them
+to be honoured they were worthless, and he had no power to enforce his
+claims.</p>
+
+<p>So everything but brooding had to be abandoned as too expensive, and he
+proceeded to study optics. He gave a very accurate explanation of the
+action of the human eye, and made many hypotheses, some of them shrewd
+and close to the mark, concerning the law of refraction of light in
+dense media: but though several minor points of interest turned up,
+nothing of the first magnitude came out of this long research.</p>
+
+<p>The true law of refraction was discovered some years after by a Dutch
+professor, Willebrod Snell.</p>
+
+<p>We must now devote a little time to the main work of Kepler's life. All
+the time he had been at Prague he had been making a severe study of the
+motion of the planet Mars, analyzing minutely Tycho's books of
+observations, in order to find out, if possible, the true theory of his
+motion.<span class='pagenum'><a name="Page_66" id="Page_66">[Pg 66]</a></span> Aristotle had taught that circular motion was the only perfect
+and natural motion, and that the heavenly bodies therefore necessarily
+moved in circles.</p>
+
+<p>So firmly had this idea become rooted in men's minds, that no one ever
+seems to have contemplated the possibility of its being false or
+meaningless.</p>
+
+<p>When Hipparchus and others found that, as a matter of fact, the planets
+did <i>not</i> revolve in simple circles, they did not try other curves, as
+we should at once do now, but they tried combinations of circles, as we
+saw in <a href="#DATES_AND_SUMMARY_OF_FACTS_FOR_LECTURE_I">Lecture I</a>. The small circle carried by a bigger one was called an
+Epicycle. The carrying circle was called the Deferent. If for any reason
+the earth had to be placed out of the centre, the main planetary orbit
+was called an Excentric, and so on.</p>
+
+<p>But although the planetary paths might be roughly represented by a
+combination of circles, their speeds could not, on the hypothesis of
+uniform motion in each circle round the earth as a fixed body. Hence was
+introduced the idea of an Equant, <i>i.e.</i> an arbitrary point, not the
+earth, about which the speed might be uniform. Copernicus, by making the
+sun the centre, had been able to simplify a good deal of this, and to
+abolish the equant.</p>
+
+<p>But now that Kepler had the accurate observations of Tycho to refer to,
+he found immense difficulty in obtaining the true positions of the
+planets for long together on any such theory.</p>
+
+<p>He specially attacked the motion of the planet Mars, because that was
+sufficiently rapid in its changes for a considerable collection of data
+to have accumulated with respect to it. He tried all manner of circular
+orbits for the earth and for Mars, placing them in all sorts of aspects
+with respect to the sun. The problem to be solved was to choose such an
+orbit and such a law of speed, for both the earth and Mars, that a line
+joining them, produced out to the stars, should always mark correctly
+the apparent position of Mars as seen from the earth. He had to arrange
+the size<span class='pagenum'><a name="Page_67" id="Page_67">[Pg 67]</a></span> of the orbits that suited best, then the positions of their
+centres, both being supposed excentric with respect to the sun; but he
+could not get any such arrangement to work with uniform motion about the
+sun. So he reintroduced the equant, and thus had another variable at his
+disposal&mdash;in fact, two, for he had an equant for the earth and another
+for Mars, getting a pattern of the kind suggested in <a href="#Fig_29">Fig. 29</a>.</p>
+
+<p>The equants might divide the line in any arbitrary ratio. All sorts of
+combinations had to be tried, the relative positions of the earth and
+Mars to be worked out for each, and compared with Tycho's recorded
+observations. It was easy to get them to agree for a short time, but
+sooner or later a discrepancy showed itself.</p>
+
+<div class="figcenter" style="width: 550px;"><a name="Fig_29" id="Fig_29"></a>
+<img src="images/fig29.jpg" width="400" height="205" alt="Fig. 29." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 29.</span>&mdash;<i>S</i> represents the sun; <i>EC</i>, the centre of the
+earth's orbit, to be placed as best suited; <i>MC</i>, the same for Mars;
+<i>EE</i>, the earth's equant, or point about which the earth uniformly
+revolved (<i>i.e.</i> the point determining the law of speed about the sun),
+likewise to be placed anywhere, but supposed to be in the line joining
+<i>S</i> to <i>EC</i>; <i>ME</i>, the same thing for Mars; with <i>?ME</i> for an
+alternative hypothesis that perhaps Mars' equant was on line joining
+<i>EC</i> with <i>MC</i>.</div>
+</div>
+
+<p>I need not say that all these attempts and gropings, thus briefly
+summarized, entailed enormous labour, and required not only great
+pertinacity, but a most singularly constituted mind, that could thus
+continue groping in the dark without a possible ray of theory to
+illuminate its search. Grope he did, however, with unexampled diligence.</p>
+
+<p>At length he hit upon a point that seemed nearly right. He thought he
+had found the truth; but no, before long the position of the planet, as
+calculated, and as recorded by Tycho, differed by eight minutes of arc,
+or about one-eighth of a degree. Could the observation be wrong by this
+small<span class='pagenum'><a name="Page_68" id="Page_68">[Pg 68]</a></span> amount? No, he had known Tycho, and knew that he was never wrong
+eight minutes in an observation.</p>
+
+<p>So he set out the whole weary way again, and said that with those eight
+minutes he would yet find out the law of the universe. He proceeded to
+see if by making the planet librate, or the plane of its orbit tilt up
+and down, anything could be done. He was rewarded by finding that at any
+rate the plane of the orbit did not tilt up and down: it was fixed, and
+this was a simplification on Copernicus's theory. It is not an absolute
+fixture, but the changes are very small (see Laplace, <a href="#Page_266">page 266</a>).</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_30" id="Fig_30"></a>
+<img src="images/fig30.jpg" width="350" height="349" alt="Fig. 30." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 30.</span>&mdash;Excentric circle supposed to be divided into
+equal areas. The sun, <i>S</i>, being placed at a selected point, it was
+possible to represent the varying speed of a planet by saying that it
+moved from <i>A</i> to <i>B</i>, from <i>B</i> to <i>C</i>, and so on, in equal times.</div>
+</div>
+
+<p>At last he thought of giving up the idea of <i>uniform</i> circular motion,
+and of trying <i>varying</i> circular motion, say inversely as its distance
+from the sun. To simplify calculation, he divided the orbit into
+triangles, and tried if making the triangles equal would do. A great
+piece of luck, they did beautifully: the rate of description of areas
+(not arcs) is uniform. Over this discovery he greatly rejoices. He feels
+as though he had been carrying on a war against the planet and had
+triumphed; but his gratulation was<span class='pagenum'><a name="Page_69" id="Page_69">[Pg 69]</a></span> premature. Before long fresh little
+errors appeared, and grew in importance. Thus he announces it himself:&mdash;</p>
+
+<p>"While thus triumphing over Mars, and preparing for him, as for one
+already vanquished, tabular prisons and equated excentric fetters, it is
+buzzed here and there that the victory is vain, and that the war is
+raging anew as violently as before. For the enemy left at home a
+despised captive has burst all the chains of the equations, and broken
+forth from the prisons of the tables."</p>
+
+<p>Still, a part of the truth had been gained, and was not to be abandoned
+any more. The law of speed was fixed: that which is now known as his
+second law. But what about the shape of the orbit&mdash;Was it after all
+possible that Aristotle, and every philosopher since Aristotle, had been
+wrong? that circular motion was not the perfect and natural motion, but
+that planets might move in some other closed curve?</p>
+
+<p>Suppose he tried an oval. Well, there are a great variety of ovals, and
+several were tried: with the result that they could be made to answer
+better than a circle, but still were not right.</p>
+
+<p>Now, however, the geometrical and mathematical difficulties of
+calculation, which before had been tedious and oppressive, threatened to
+become overwhelming; and it is with a rising sense of despondency that
+Kepler sees his six years' unremitting labour leading deeper and deeper
+into complication.</p>
+
+<p>One most disheartening circumstance appeared, viz. that when he made the
+circuit oval his law of equable description of areas broke down. That
+seemed to require the circular orbit, and yet no circular orbit was
+quite accurate.</p>
+
+<p>While thinking and pondering for weeks and months over this new dilemma
+and complication of difficulties, till his brain reeled, an accidental
+ray of light broke upon him in a way not now intelligible, or barely
+intelligible. Half the extreme breadth intercepted between the circle
+and oval<span class='pagenum'><a name="Page_70" id="Page_70">[Pg 70]</a></span> was <span class="above">429</span>&#8260;<span class="below">100,000</span> of the radius, and he remembered that the
+"optical inequality" of Mars was also about <span class="above">429</span>&#8260;<span class="below">100,000</span>. This
+coincidence, in his own words, woke him out of sleep; and for some
+reason or other impelled him instantly to try making the planet
+oscillate in the diameter of its epicycle instead of revolve round it&mdash;a
+singular idea, but Copernicus had had a similar one to explain the
+motions of Mercury.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_31" id="Fig_31"></a>
+<img src="images/fig31.jpg" width="350" height="225" alt="Fig. 31." title="" />
+<span class="caption"><span class="smcap">Fig. 31.</span>&mdash;Mode of drawing an ellipse. The two pins <i>F</i>
+are the foci.</span>
+</div>
+
+<p>Away he started through his calculations again. A long course of work
+night and day was rewarded by finding that he was now able to hit off
+the motions better than before; but what a singularly complicated motion
+it was. Could it be expressed no more simply? Yes, the curve so
+described by the planet is a comparatively simple one: it is a special
+kind of oval&mdash;the ellipse. Strange that he had not thought of it before.
+It was a famous curve, for the Greek geometers had studied it as one of
+the sections of a cone, but it was not so well known in Kepler's time.
+The fact that the planets move in it has raised it to the first
+importance, and it is familiar enough to us now. But did it satisfy the
+law<span class='pagenum'><a name="Page_71" id="Page_71">[Pg 71]</a></span> of speed? Could the rate of description of areas be uniform with
+it? Well, he tried the ellipse, and to his inexpressible delight he
+found that it did satisfy the condition of equable description of areas,
+if the sun was in one focus. So, moving the planet in a selected
+ellipse, with the sun in one focus, at a speed given by the equable area
+description, its position agreed with Tycho's observations within the
+limits of the error of experiment. Mars was finally conquered, and
+remains in his prison-house to this day. The orbit was found.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_32" id="Fig_32"></a>
+<img src="images/fig32.jpg" width="400" height="366" alt="Fig. 32." title="" />
+<span class="caption"><span class="smcap">Fig. 32.</span></span>
+</div>
+
+<p>In a paroxysm of delight Kepler celebrates his victory by a triumphant
+figure, sketched actually on his geometrical diagram&mdash;the diagram which
+proves that the law of equable description of areas can hold good with
+an ellipse. The above is a tracing of it.</p>
+
+<p>Such is a crude and bald sketch of the steps by which Kepler rose to his
+great generalizations&mdash;the two laws which have immortalized his name.</p>
+
+<p>All the complications of epicycle, equant, deferent, excentric, and the
+like, were swept at once away, and an orbit<span class='pagenum'><a name="Page_72" id="Page_72">[Pg 72]</a></span> of striking and beautiful
+properties substituted. Well might he be called, as he was, "the
+legislator," or law interpreter, "of the heavens."</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_33" id="Fig_33"></a>
+<img src="images/fig33.jpg" width="400" height="297" alt="Fig. 33." title="" />
+<span class="caption"><span class="smcap">Fig. 33.</span>&mdash;If <i>S</i> is the sun, a planet or comet moves from
+<i>P</i> to <i>P<sub>1</sub></i>, from <i>P<sub>2</sub></i> to <i>P<sub>3</sub></i>, and from <i>P<sub>4</sub></i> to <i>P<sub>5</sub></i> in
+the same time; if the shaded areas are equal.</span>
+</div>
+
+<p>He concludes his book on the motions of Mars with a half comic appeal to
+the Emperor to provide him with the sinews of war for an attack on
+Mars's relations&mdash;father Jupiter, brother Mercury, and the rest&mdash;but the
+death of his unhappy patron in 1612 put an end to all these schemes, and
+reduced Kepler to the utmost misery. While at Prague his salary was in
+continual arrear, and it was with difficulty that he could provide
+sustenance for his family. He had been there eleven years, but they had
+been hard years of poverty, and he could leave without regret were it
+not that he should have to leave Tycho's instruments and observations
+behind him. While he was hesitating what best to do, and reduced to the
+verge of despair, his wife, who had long been suffering from low spirits
+and despondency, and his three children, were taken ill; one of the sons
+died of small-pox, and the wife eleven days after of low fever and
+epilepsy. No money could be got at Prague, so after a<span class='pagenum'><a name="Page_73" id="Page_73">[Pg 73]</a></span> short time he
+accepted a professorship at Linz, and withdrew with his two quite young
+remaining children.</p>
+
+<p>He provided for himself now partly by publishing a prophesying almanack,
+a sort of Zadkiel arrangement&mdash;a thing which he despised, but the
+support of which he could not afford to do without. He is continually
+attacking and throwing sarcasm at astrology, but it was the only thing
+for which people would pay him, and on it after a fashion he lived. We
+do not find that his circumstances were ever prosperous, and though
+8,000 crowns were due to him from Bohemia he could not manage to get
+them paid.</p>
+
+<p>About this time occurred a singular interruption to his work. His old
+mother, of whose fierce temper something has already been indicated, had
+been engaged in a law-suit for some years near their old home in
+W&uuml;rtemberg. A change of judge having in process of time occurred, the
+defendant saw his way to turn the tables on the old lady by accusing her
+of sorcery. She was sent to prison, and condemned to the torture, with
+the usual intelligent idea of extracting a "voluntary" confession.
+Kepler had to hurry from Linz to interpose. He succeeded in saving her
+from the torture, but she remained in prison for a year or so. Her
+spirit, however, was unbroken, for no sooner was she released than she
+commenced a fresh action against her accuser. But fresh trouble was
+averted by the death of the poor old dame at the age of nearly eighty.</p>
+
+<p>This narration renders the unflagging energy shown by her son in his
+mathematical wrestlings less surprising.</p>
+
+<p>Interspersed with these domestic troubles, and with harassing and
+unsuccessful attempts to get his rights, he still brooded over his old
+problem of some possible connection between the distances of the planets
+from the sun and their times of revolution, <i>i.e.</i> the length of their
+years.</p>
+
+<p>It might well have been that there was no connection, that it was purely
+imaginary, like his old idea of the law of the successive distances of
+the planets, and like so<span class='pagenum'><a name="Page_74" id="Page_74">[Pg 74]</a></span> many others of the guesses and fancies which
+he entertained and spent his energies in probing. But fortunately this
+time there was a connection, and he lived to have the joy of discovering
+it.</p>
+
+<p>The connection is this, that if one compares the distance of the
+different planets from the sun with the length of time they take to go
+round him, the cube of the respective distances is proportional to the
+square of the corresponding times. In other words, the ratio of <i>r</i><sup>3</sup>
+to <i>T</i><sup>2</sup> for every planet is the same. Or, again, the length of a
+planet's year depends on the <span class="above">3</span>&#8260;<span class="below">2</span>th power of its distance from the sun.
+Or, once more, the speed of each planet in its orbit is as the inverse
+square-root of its distance from the sun. The product of the distance
+into the square of the speed is the same for each planet.</p>
+
+<p>This (however stated) is called Kepler's third law. It welds the planets
+together, and shows them to be one system. His rapture on detecting the
+law was unbounded, and he breaks out into an exulting rhapsody:&mdash;</p>
+
+<p>"What I prophesied two-and-twenty years ago, as soon as I discovered the
+five solids among the heavenly orbits&mdash;what I firmly believed long
+before I had seen Ptolemy's <i>Harmonies</i>&mdash;what I had promised my friends
+in the title of this book, which I named before I was sure of my
+discovery&mdash;what sixteen years ago, I urged as a thing to be sought&mdash;that
+for which I joined Tycho Brah&eacute;, for which I settled in Prague, for which
+I have devoted the best part of my life to astronomical contemplations,
+at length I have brought to light, and recognized its truth beyond my
+most sanguine expectations. It is not eighteen months since I got the
+first glimpse of light, three months since the dawn, very few days since
+the unveiled sun, most admirable to gaze upon, burst upon me. Nothing
+holds me; I will indulge my sacred fury; I will triumph over mankind by
+the honest confession that I have stolen the golden vases of the
+Egyptians to build up a tabernacle for my God far away<span class='pagenum'><a name="Page_75" id="Page_75">[Pg 75]</a></span> from the
+confines of Egypt. If you forgive me, I rejoice; if you are angry, I can
+bear it; the die is cast, the book is written, to be read either now or
+by posterity, I care not which; it may well wait a century for a reader,
+as God has waited six thousand years for an observer."</p>
+
+<p>Soon after this great work his third book appeared: it was an epitome of
+the Copernican theory, a clear and fairly popular exposition of it,
+which had the honour of being at once suppressed and placed on the list
+of books prohibited by the Church, side by side with the work of
+Copernicus himself, <i>De Revolutionibus Orbium C&#339;lestium</i>.</p>
+
+<p>This honour, however, gave Kepler no satisfaction&mdash;it rather occasioned
+him dismay, especially as it deprived him of all pecuniary benefit, and
+made it almost impossible for him to get a publisher to undertake
+another book.</p>
+
+<p>Still he worked on at the Rudolphine tables of Tycho, and ultimately,
+with some small help from Vienna, completed them; but he could not get
+the means to print them. He applied to the Court till he was sick of
+applying: they lay idle four years. At last he determined to pay for the
+type himself. What he paid it with, God knows, but he did pay it, and he
+did bring out the tables, and so was faithful to the behest of his
+friend.</p>
+
+<p>This great publication marks an era in astronomy. They were the first
+really accurate tables which navigators ever possessed; they were the
+precursors of our present <i>Nautical Almanack</i>.</p>
+
+<p>After this, the Grand Duke of Tuscany sent Kepler a golden chain, which
+is interesting inasmuch as it must really have come from Galileo, who
+was in high favour at the Italian Court at this time.</p>
+
+<p>Once more Kepler made a determined attempt to get his arrears of salary
+paid, and rescue himself and family from their bitter poverty. He
+travelled to Prague on purpose, attended the imperial meeting, and
+pleaded his own cause, but it was all fruitless; and exhausted by the
+journey, weakened by over-study, and disheartened by the failure, he
+caught a fever, and died in his fifty-ninth year. His body was buried at
+Ratisbon, and a century ago a proposal was made to erect a marble
+monument to his memory, but nothing was done. It matters little one way
+or the other whether Germany, having almost refused him bread during his
+life, should, a century and a half after his death, offer him a stone.</p>
+
+<p><span class='pagenum'><a name="Page_76" id="Page_76">[Pg 76]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_34" id="Fig_34"></a>
+<img src="images/fig34.jpg" width="400" height="492" alt="Fig. 34." title="" />
+<span class="caption"><span class="smcap">Fig. 34.</span>&mdash;Portrait of Kepler, older.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_77" id="Page_77">[Pg 77]</a></span></p><p>The contiguity of the lives of Kepler and Tycho furnishes a moral too
+obvious to need pointing out. What Kepler might have achieved had he
+been relieved of those ghastly struggles for subsistence one cannot
+tell, but this much is clear, that had Tycho been subjected to the same
+misfortune, instead of being born rich and being assisted by generous
+and enlightened patrons, he could have accomplished very little. His
+instruments, his observatory&mdash;the tools by which he did his work&mdash;would
+have been impossible for him. Frederick and Sophia of Denmark, and
+Rudolph of Bohemia, are therefore to be remembered as co-workers with
+him.</p>
+
+<p>Kepler, with his ill-health and inferior physical energy, was unable to
+command the like advantages. Much, nevertheless, he did; more one cannot
+but feel he might have done had he been properly helped. Besides, the
+world would have been free from the reproach of accepting the fruits of
+his bright genius while condemning the worker to a life of misery,
+relieved only by the beauty of his own thoughts and the ecstasy awakened
+in him by the harmony and precision of Nature.</p>
+
+<p>Concerning the method of Kepler, the mode by which he made his
+discoveries, we must remember that he gives us an account of all the
+steps, unsuccessful as well as successful, by which he travelled. He
+maps out his route like a traveller. In fact he compares himself to
+Columbus or Magellan, voyaging into unknown lands, and recording his
+wandering route. This being remembered, it will be found that his
+methods do not differ so utterly from those used by other philosophers
+in like case. His imagination was perhaps more luxuriant and was allowed
+freer play than most men's, but it was nevertheless always controlled by
+rigid examination and comparison of hypotheses with fact.</p>
+
+<p><span class='pagenum'><a name="Page_78" id="Page_78">[Pg 78]</a></span></p><p>Brewster says of him:&mdash;"Ardent, restless, burning to distinguish
+himself by discovery, he attempted everything; and once having obtained
+a glimpse of a clue, no labour was too hard in following or verifying
+it. A few of his attempts succeeded&mdash;a multitude failed. Those which
+failed seem to us now fanciful, those which succeeded appear to us
+sublime. But his methods were the same. When in search of what really
+existed he sometimes found it; when in pursuit of a chim&aelig;ra he could not
+but fail; but in either case he displayed the same great qualities, and
+that obstinate perseverance which must conquer all difficulties except
+those really insurmountable."</p>
+
+<p>To realize what he did for astronomy, it is necessary for us now to
+consider some science still in its infancy. Astronomy is so clear and so
+thoroughly explored now, that it is difficult to put oneself into a
+contemporary attitude. But take some other science still barely
+developed: meteorology, for instance. The science of the weather, the
+succession of winds and rain, sunshine and frost, clouds and fog, is now
+very much in the condition of astronomy before Kepler.</p>
+
+<p>We have passed through the stage of ascribing atmospheric
+disturbances&mdash;thunderstorms, cyclones, earthquakes, and the like&mdash;to
+supernatural agency; we have had our Copernican era: not perhaps brought
+about by a single individual, but still achieved. Something of the laws
+of cyclone and anticyclone are known, and rude weather predictions
+across the Atlantic are roughly possible. Barometers and thermometers
+and anemometers, and all their tribe, represent the astronomical
+instruments in the island of Huen; and our numerous meteorological
+observatories, with their continual record of events, represent the work
+of Tycho Brah&eacute;.</p>
+
+<p>Observation is heaped on observation; tables are compiled; volumes are
+filled with data; the hours of sunshine are recorded, the fall of rain,
+the moisture in the air, the kind of clouds, the temperature&mdash;millions
+of facts; but where is the<span class='pagenum'><a name="Page_79" id="Page_79">[Pg 79]</a></span> Kepler to study and brood over them? Where
+is the man to spend his life in evolving the beginnings of law and order
+from the midst of all this chaos?</p>
+
+<p>Perhaps as a man he may not come, but his era will come. Through this
+stage the science must pass, ere it is ready for the commanding
+intellect of a Newton.</p>
+
+<p>But what a work it will be for the man, whoever he be that undertakes
+it&mdash;a fearful monotonous grind of calculation, hypothesis, hypothesis,
+calculation, a desperate and groping endeavour to reconcile theories
+with facts.</p>
+
+<p>A life of such labour, crowned by three brilliant discoveries, the world
+owes (and too late recognizes its obligation) to the harshly treated
+German genius, Kepler.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_80" id="Page_80">[Pg 80]</a></span></p>
+<h4><a name="SUMMARY_OF_FACTS_FOR_LECTURES_IV_AND_V" id="SUMMARY_OF_FACTS_FOR_LECTURES_IV_AND_V"></a>SUMMARY OF FACTS FOR LECTURES IV AND V</h4>
+
+
+<p>In 1564, Michael Angelo died and Galileo was born; in 1642, Galileo died
+and Newton was born. Milton lived from 1608 to 1674.</p>
+
+<p>For teaching the plurality of worlds, with other heterodox doctrines,
+and refusing to recant, Bruno, after six years' imprisonment in Rome,
+was burnt at the stake on the 16th of February, 1600 <span class="ampm">A.D.</span> A "natural"
+death in the dungeons of the Inquisition saved Antonio de Dominis, the
+explainer of the rainbow, from the same fate, but his body and books
+were publicly burned at Rome in 1624.</p>
+
+<p>The persecution of Galileo began in 1615, became intense in 1632, and so
+lasted till his death and after.</p>
+
+<hr style='width: 10%;' />
+
+<p>Galileo Galilei, eldest son of Vincenzo de Bonajuti de Galilei, a noble
+Florentine, was born at Pisa, 18th of February, 1564. At the age of 17
+was sent to the University of Pisa to study medicine. Observed the swing
+of a pendulum and applied it to count pulse-beats. Read Euclid and
+Archimedes, and could be kept at medicine no more. At 26 was appointed
+Lecturer in Mathematics at Pisa. Read Bruno and became smitten with the
+Copernican theory. Controverted the Aristotelians concerning falling
+bodies, at Pisa. Hence became unpopular and accepted a chair at Padua,
+1592. Invented a thermometer. Wrote on astronomy, adopting the Ptolemaic
+system provisionally, and so opened up a correspondence with Kepler,
+with whom he formed a friendship. Lectured on the new star of 1604, and
+publicly renounced the old systems of astronomy. Invented a calculating
+compass or "Gunter's scale." In 1609 invented a telescope, after hearing
+of a Dutch optician's discovery. Invented the microscope soon after.
+Rapidly completed a better telescope and began a survey of the heavens.
+On the 8th of January, 1610, discovered Jupiter's satellites. Observed
+the mountains in the moon, and roughly measured their height. Explained
+the visibility of the new moon by <i>earth-shine</i>. Was invited to the
+Grand Ducal Court of Tuscany by Cosmo de Medici, and appointed
+philosopher to that personage. Discovered innumerable new stars, and the
+nebul&aelig;. Observed a triple appearance of Saturn. Discovered the<span class='pagenum'><a name="Page_81" id="Page_81">[Pg 81]</a></span> phases
+of Venus predicted by Copernicus, and spots on the sun. Wrote on
+floating bodies. Tried to get his satellites utilized for determining
+longitude at sea.</p>
+
+<p>Went to Rome to defend the Copernican system, then under official
+discussion, and as a result was formally forbidden ever to teach it. On
+the accession of Pope Urban VIII. in 1623, Galileo again visited Rome to
+pay his respects, and was well received. In 1632 appeared his
+"Dialogues" on the Ptolemaic and Copernican systems. Summoned to Rome,
+practically imprisoned, and "rigorously questioned." Was made to recant
+22nd of June, 1633. Forbidden evermore to publish anything, or to teach,
+or receive friends. Retired to Arcetri in broken down health. Death of
+his favourite daughter, Sister Maria Celeste. Wrote and meditated on the
+laws of motion. Discovered the moon's libration. In 1637 he became
+blind. The rigour was then slightly relaxed and many visited him: among
+them John Milton. Died 8th of January, 1642, aged 78. As a prisoner of
+the Inquisition his right to make a will or to be buried in consecrated
+ground was disputed. Many of his manuscripts were destroyed.</p>
+
+<p>Galileo, besides being a singularly clear-headed thinker and
+experimental genius, was also something of a musician, a poet, and an
+artist. He was full of humour as well as of solid common-sense, and his
+literary style is brilliant. Of his scientific achievements those now
+reckoned most weighty, are the discovery of the Laws of Motion, and the
+laying of the foundations of Mechanics.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Jupiters Satellite">
+<tr>
+ <td align='center' colspan='7'><i>Particulars of Jupiter's Satellites,<br />
+   Illustrating their obedience to Kepler's third law.</i></td>
+</tr>
+<tr class='tr4'>
+ <td class='tdcbrbl'><small>Satellite.</small></td>
+ <td class='tdcbr'><small>Diameter<br />in miles.</small></td>
+ <td class='tdcbr'><small>Time of<br />revolution<br />in hours.<br />(<i>T</i>)</small></td>
+ <td class='tdcbr'><small>Distance<br />from<br />Jupiter, in<br />Jovian<br />radii.<br />(<i>d</i>)</small></td>
+ <td class='tdcbr'><small><i>T</i><sup>2</sup></small></td>
+ <td class='tdcbr'><small><i>d</i><sup>3</sup></small></td>
+ <td class='tdcbr'><small><i>T</i><sup>2</sup><br /><i>d</i><sup>3</sup><br />which is<br />practically<br />constant.</small></td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>No. 1.</td>
+ <td class='tdlbrpl1'>2437</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;42&middot;47</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;6&middot;049</td>
+ <td class='tdrbrpl1'>1803&middot;7</td>
+ <td class='tdrbrpl1'>221&middot;44</td>
+ <td class='tdlbrpl1'>8&middot;149</td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>No. 2.</td>
+ <td class='tdlbrpl1'>2188</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;85&middot;23</td>
+ <td class='tdlbrpl1'>&nbsp;&nbsp;9&middot;623</td>
+ <td class='tdrbrpl1'>7264&middot;1</td>
+ <td class='tdrbrpl1'>891&middot;11</td>
+ <td class='tdlbrpl1'>8&middot;152</td>
+</tr>
+<tr>
+ <td class='tdlbrblpl1'>No. 3.</td>
+ <td class='tdlbrpl1'>3575</td>
+ <td class='tdlbrpl1'>177&middot;72</td>
+ <td class='tdlbrpl1'>15&middot;350</td>
+ <td class='tdrbrpl1'>29488&middot;&nbsp;&nbsp;</td>
+ <td class='tdrbrpl1'>3916&middot;8&nbsp;&nbsp;</td>
+ <td class='tdlbrpl1'>8&middot;153</td>
+</tr>
+<tr class='tr5'>
+ <td class='tdlbrblpl1'>No. 4.</td>
+ <td class='tdlbrpl1'>3059</td>
+ <td class='tdlbrpl1'>400&middot;53</td>
+ <td class='tdlbrpl1'>26&middot;998</td>
+ <td class='tdrbrpl1'>160426&middot;&nbsp;&nbsp;</td>
+ <td class='tdrbrpl1'>19679&middot;&nbsp;&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdlbrpl1'>8&middot;152</td>
+</tr>
+<tr>
+ <td align='center' colspan='7'>The diameter of Jupiter is 85,823 miles.</td>
+</tr>
+</table></div>
+
+
+<p class="center"><i>Falling Bodies.</i></p>
+
+
+<p>Since all bodies fall at the same rate, except for the disturbing effect
+of the resistance of the air, a statement of their rates of fall is of
+interest. In one second a freely falling body near the earth is found to
+drop 16 feet.<span class='pagenum'><a name="Page_82" id="Page_82">[Pg 82]</a></span> In two seconds it drops 64 feet altogether, viz. 16 feet
+in the first, and 48 feet in the next second; because at the beginning
+of every second after the first it has the accumulated velocity of
+preceding seconds. The height fallen by a dropped body is not
+proportional to the time simply, but to what is rather absurdly called
+the square of the time, <i>i.e.</i> the time multiplied by itself.</p>
+
+<p>For instance, in 3 seconds it drops 9 &times; 16 = 144 feet; in 4 seconds 16 &times;
+16, or 256 feet, and so on. The distances travelled in 1, 2, 3, 4, &amp;c.,
+seconds by a body dropped from rest and not appreciably resisted by the
+air, are 1, 4, 9, 16, 25, &amp;c., respectively, each multiplied by the
+constant 16 feet.</p>
+
+<p>Another way of stating the law is to say that the heights travelled in
+successive seconds proceed in the proportion 1, 3, 5, 7, 9, &amp;c.; again
+multiplied by 16 feet in each case.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_35" id="Fig_35"></a>
+<img src="images/fig35.jpg" width="400" height="200" alt="Fig. 35." title="" />
+<div class="caption1"><span class="smcap">Fig. 35.</span>&mdash;Curve described by a projectile, showing how it
+drops from the line of fire, <i>O D</i>, in successive seconds, the same
+distances <i>AP</i>, <i>BQ</i>, <i>CR</i>, &amp;c., as are stated above for a dropped
+body.</div>
+</div>
+
+<p>All this was experimentally established by Galileo.</p>
+
+<p>A body takes half a second to drop 4 feet; and a quarter of a second to
+drop 1 foot. The easiest way of estimating a quarter of a second with
+some accuracy is to drop a bullet one foot.</p>
+
+<p>A bullet thrown or shot in any direction falls just as much as if merely
+dropped; but instead of falling from the starting-point it drops
+vertically from the line of fire. (See fig. 35).</p>
+
+<p>The rate of fall depends on the intensity of gravity; if it could be
+doubled, a body would fall twice as far in the same time; but to make it
+fall a given distance in half the time the intensity of gravity would
+have to be quadrupled. At a place where the intensity of gravity is
+<span class="above">1</span>&#8260;<span class="below">3600</span> of what it is here, a body would fall as far in a minute as it
+now falls in a second. Such a place occurs at about the distance of the
+moon (<i>cf.</i> page 177).</p>
+
+<p>The fact that the height fallen through is proportional to the square
+of<span class='pagenum'><a name="Page_83" id="Page_83">[Pg 83]</a></span> the time proves that the attraction of the earth or the intensity of
+gravity is sensibly constant throughout ordinary small ranges. Over
+great distances of fall, gravity cannot be considered constant; so for
+things falling through great spaces the Galilean law of the square of
+the time does not hold.</p>
+
+<p>The fact that things near the earth fall 16 feet in the first second
+proves that the intensity of ordinary terrestrial gravity is 32 British
+units of force per pound of matter.</p>
+
+<p>The fact that all bodies fall at the same rate (when the resistance of
+the air is eliminated), proves that weight is proportional to mass; or
+more explicitly, that the gravitative attraction of the earth on matter
+near its surface depends on the amount of that matter, as estimated by
+its inertia, and on nothing else.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_84" id="Page_84">[Pg 84]</a></span></p>
+<h3><a name="LECTURE_IV" id="LECTURE_IV"></a>LECTURE IV</h3>
+
+<h5>GALILEO AND THE INVENTION OF THE TELESCOPE</h5>
+
+
+<p><span class="smcap">Contemporary</span> with the life of Kepler, but overlapping it at both ends,
+comes the great and eventful life of Galileo Galilei,<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a> a man whose
+influence on the development of human thought has been greater than that
+of any man whom we have yet considered, and upon whom, therefore, it is
+necessary for us, in order to carry out the plan of these lectures, to
+bestow much time. A man of great and wide culture, a so-called universal
+genius, it is as an experimental philosopher that he takes the first
+rank. In this capacity he must be placed alongside of Archimedes, and it
+is pretty certain that between the two there was no man of magnitude
+equal to either in experimental philosophy. It is perhaps too bold a
+speculation, but I venture to doubt whether in succeeding generations we
+find his equal in the domain of purely experimental science until we
+come to Faraday. Faraday was no doubt his superior, but I know of no
+other of whom the like can unhesitatingly be said. In mathematical and
+deductive science, of course, it is quite otherwise. Kepler, for
+instance, and many men before and since, have far excelled Galileo in
+mathematical skill and power, though at the same time his achievements
+in this department are by no means to be despised.</p>
+
+<p><span class='pagenum'><a name="Page_85" id="Page_85">[Pg 85]</a></span></p><p>Born at Pisa three centuries ago, on the very day that Michael Angelo
+lay dying in Rome, he inherited from his father a noble name, cultivated
+tastes, a keen love of truth, and an impoverished patrimony. Vincenzo de
+Galilei, a descendant of the important Bonajuti family, was himself a
+mathematician and a musician, and in a book of his still extant he
+declares himself in favour of free and open inquiry into scientific
+matters, unrestrained by the weight of authority and tradition.</p>
+
+<p>In all probability the son imbibed these precepts: certainly he acted on
+them.</p>
+
+<p>Vincenzo, having himself experienced the unremunerative character of
+scientific work, had a horror of his son's taking to it, especially as
+in his boyhood he was always constructing ingenious mechanical toys, and
+exhibiting other marks of precocity. So the son was destined for
+business&mdash;to be, in fact, a cloth-dealer. But he was to receive a good
+education first, and was sent to an excellent convent school.</p>
+
+<p>Here he made rapid progress, and soon excelled in all branches of
+classics and literature. He delighted in poetry, and in later years
+wrote several essays on Dante, Tasso, and Ariosto, besides composing
+some tolerable poems himself. He played skilfully on several musical
+instruments, especially on the lute, of which indeed he became a master,
+and on which he solaced himself when quite an old man. Besides this he
+seems to have had some skill as an artist, which was useful afterwards
+in illustrating his discoveries, and to have had a fine sensibility as
+an art critic, for we find several eminent painters of that day
+acknowledging the value of the opinion of the young Galileo.</p>
+
+<p>Perceiving all this display of ability, the father wisely came to the
+conclusion that the selling of woollen stuffs would hardly satisfy his
+aspirations for long, and that it was worth a sacrifice to send him to
+the University. So to the University of his native town he went, with
+the avowed object of studying medicine, that career seeming the most<span class='pagenum'><a name="Page_86" id="Page_86">[Pg 86]</a></span>
+likely to be profitable. Old Vincenzo's horror of mathematics or science
+as a means of obtaining a livelihood is justified by the fact that while
+the University Professor of Medicine received 2,000 scudi a year, the
+Professor of Mathematics had only 60, that is &pound;13 a year, or 7&frac12;<i>d.</i> a
+day.</p>
+
+<p>So the son had been kept properly ignorant of such poverty-stricken
+subjects, and to study medicine he went.</p>
+
+<p>But his natural bent showed itself even here. For praying one day in the
+Cathedral, like a good Catholic as he was all his life, his attention
+was arrested by the great lamp which, after lighting it, the verger had
+left swinging to and fro. Galileo proceeded to time its swings by the
+only watch he possessed&mdash;viz., his own pulse. He noticed that the time
+of swing remained as near as he could tell the same, notwithstanding the
+fact that the swings were getting smaller and smaller.</p>
+
+<p>By subsequent experiment he verified the law, and the isochronism of the
+pendulum was discovered. An immensely important practical discovery
+this, for upon it all modern clocks are based; and Huyghens soon applied
+it to the astronomical clock, which up to that time had been a crude and
+quite untrustworthy instrument.</p>
+
+<p>The best clock which Tycho Brah&eacute; could get for his observatory was
+inferior to one that may now be purchased for a few shillings; and this
+change is owing to the discovery of the pendulum by Galileo. Not that he
+applied it to clocks; he was not thinking of astronomy, he was thinking
+of medicine, and wanted to count people's pulses. The pendulum served;
+and "pulsilogies," as they were called, were thus introduced to and used
+by medical practitioners.</p>
+
+<p>The Tuscan Court came to Pisa for the summer months, for it was then a
+seaside place, and among the suite was Ostillio Ricci, a distinguished
+mathematician and old friend of the Galileo family. The youth visited
+him, and one day, it is said, heard a lesson in Euclid being given by
+Ricci to the pages while he stood outside the door entranced. Anyhow<span class='pagenum'><a name="Page_87" id="Page_87">[Pg 87]</a></span> he
+implored Ricci to help him into some knowledge of mathematics, and the
+old man willingly consented. So he mastered Euclid and passed on to
+Archimedes, for whom he acquired a great veneration.</p>
+
+<p>His father soon heard of this obnoxious proclivity, and did what he
+could to divert him back to medicine again. But it was no use.
+Underneath his Galen and Hippocrates were secreted copies of Euclid and
+Archimedes, to be studied at every available opportunity. Old Vincenzo
+perceived the bent of genius to be too strong for him, and at last gave
+way.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_36" id="Fig_36"></a>
+<img src="images/fig36.jpg" width="400" height="446" alt="Fig. 36." title="" />
+<div class="caption1"><span class="smcap">Fig. 36.</span>&mdash;Two forms of pulsilogy. The string is wound up
+till the swinging weight keeps time with the pulse, and the position of
+a bead or of an index connected with the string is then read on a scale
+or dial.</div>
+</div>
+
+<p>With prodigious rapidity the released philosopher now assimilated the
+elements of mathematics and physics, and at twenty-six we find him
+appointed for three years to<span class='pagenum'><a name="Page_88" id="Page_88">[Pg 88]</a></span> the University Chair of Mathematics, and
+enjoying the paternally dreaded stipend of 7&frac12;<i>d.</i> a day.</p>
+
+<p>Now it was that he pondered over the laws of falling bodies. He
+verified, by experiment, the fact that the velocity acquired by falling
+down any slope of given height was independent of the angle of slope.
+Also, that the height fallen through was proportional to the square of
+the time.</p>
+
+<p>Another thing he found experimentally was that all bodies, heavy and
+light, fell at the same rate, striking the ground at the same time.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a></p>
+
+<p>Now this was clean contrary to what he had been taught. The physics of
+those days were a simple reproduction of statements in old books.
+Aristotle had asserted certain things to be true, and these were
+universally believed. No one thought of trying the thing to see if it
+really were so. The idea of making an experiment would have savoured of
+impiety, because it seemed to tend towards scepticism, and cast a doubt
+on a reverend authority.</p>
+
+<p>Young Galileo, with all the energy and imprudence of youth (what a
+blessing that youth has a little imprudence and disregard of
+consequences in pursuing a high ideal!), as soon as he perceived that
+his instructors were wrong on the subject of falling bodies, instantly
+informed them of the fact. Whether he expected them to be pleased or not
+is a question. Anyhow, they were not pleased, but were much annoyed by
+his impertinent arrogance.</p>
+
+<p>It is, perhaps, difficult for us now to appreciate precisely their
+position. These doctrines of antiquity, which had come down hoary with
+age, and the discovery of which had<span class='pagenum'><a name="Page_89" id="Page_89">[Pg 89]</a></span> reawakened learning and quickened
+intellectual life, were accepted less as a science or a philosophy, than
+as a religion. Had they regarded Aristotle as a verbally inspired
+writer, they could not have received his statements with more
+unhesitating conviction. In any dispute as to a question of fact, such
+as the one before us concerning the laws of falling bodies, their method
+was not to make an experiment, but to turn over the pages of Aristotle;
+and he who could quote chapter and verse of this great writer was held
+to settle the question and raise it above the reach of controversy.</p>
+
+<p>It is very necessary for us to realize this state of things clearly,
+because otherwise the attitude of the learned of those days towards
+every new discovery seems stupid and almost insane. They had a
+crystallized system of truth, perfect, symmetrical&mdash;it wanted no
+novelty, no additions; every addition or growth was an imperfection, an
+excrescence, a deformity. Progress was unnecessary and undesired. The
+Church had a rigid system of dogma, which must be accepted in its
+entirety on pain of being treated as a heretic. Philosophers had a
+cast-iron system of truth to match&mdash;a system founded upon Aristotle&mdash;and
+so interwoven with the great theological dogmas that to question one was
+almost equivalent to casting doubt upon the other.</p>
+
+<p>In such an atmosphere true science was impossible. The life-blood of
+science is growth, expansion, freedom, development. Before it could
+appear it must throw off these old shackles of centuries. It must burst
+its old skin, and emerge, worn with the struggle, weakly and
+unprotected, but free and able to grow and to expand. The conflict was
+inevitable, and it was severe. Is it over yet? I fear not quite, though
+so nearly as to disturb science hardly at all. Then it was different; it
+was terrible. Honour to the men who bore the first shock of the battle!</p>
+
+<p>Now Aristotle had said that bodies fell at rates depending on their
+weight.</p>
+
+<p><span class='pagenum'><a name="Page_90" id="Page_90">[Pg 90]</a></span></p><p>A 5 lb. weight would fall five times as quick as a 1 lb. weight; a 50
+lb. weight fifty times as quick, and so on.</p>
+
+<p>Why he said so nobody knows. He cannot have tried. He was not above
+trying experiments, like his smaller disciples; but probably it never
+occurred to him to doubt the fact. It seems so natural that a heavy body
+should fall quicker than a light one; and perhaps he thought of a stone
+and a feather, and was satisfied.</p>
+
+<p>Galileo, however, asserted that the weight did not matter a bit, that
+everything fell at the same rate (even a stone and a feather, but for
+the resistance of the air), and would reach the ground in the same time.</p>
+
+<p>And he was not content to be pooh-poohed and snubbed. He knew he was
+right, and he was determined to make every one see the facts as he saw
+them. So one morning, before the assembled University, he ascended the
+famous leaning tower, taking with him a 100 lb. shot and a 1 lb. shot.
+He balanced them on the edge of the tower, and let them drop together.
+Together they fell, and together they struck the ground.</p>
+
+<p>The simultaneous clang of those two weights sounded the death-knell of
+the old system of philosophy, and heralded the birth of the new.</p>
+
+<p>But was the change sudden? Were his opponents convinced? Not a jot.
+Though they had seen with their eyes, and heard with their ears, the
+full light of heaven shining upon them, they went back muttering and
+discontented to their musty old volumes and their garrets, there to
+invent occult reasons for denying the validity of the observation, and
+for referring it to some unknown disturbing cause.</p>
+
+<p>They saw that if they gave way on this one point they would be letting
+go their anchorage, and henceforward would be liable to drift along with
+the tide, not knowing whither. They dared not do this. No; they <i>must</i>
+cling to the old traditions; they could not cast away their rotting
+ropes and sail out on to the free ocean of God's truth in a spirit of
+fearless faith.</p>
+
+<p><span class='pagenum'><a name="Page_91" id="Page_91">[Pg 91]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_37" id="Fig_37"></a>
+<img src="images/fig37.jpg" width="400" height="623" alt="Fig. 37." title="" />
+<span class="caption"><span class="smcap">Fig. 37.</span>&mdash;Tower of Pisa.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_92" id="Page_92">[Pg 92]</a></span></p><p>Yet they had received a shock: as by a breath of fresh salt breeze and
+a dash of spray in their faces, they had been awakened out of their
+comfortable lethargy. They felt the approach of a new era.</p>
+
+<p>Yes, it was a shock; and they hated the young Galileo for giving it
+them&mdash;hated him with the sullen hatred of men who fight for a lost and
+dying cause.</p>
+
+<p>We need scarcely blame these men; at least we need not blame them
+overmuch. To say that they acted as they did is to say that they were
+human, were narrow-minded, and were the apostles of a lost cause. But
+<i>they</i> could not know this; <i>they</i> had no experience of the past to
+guide them; the conditions under which they found themselves were novel,
+and had to be met for the first time. Conduct which was excusable then
+would be unpardonable now, in the light of all this experience to guide
+us. Are there any now who practically repeat their error, and resist new
+truth? who cling to any old anchorage of dogma, and refuse to rise with
+the tide of advancing knowledge? There may be some even now.</p>
+
+<p>Well, the unpopularity of Galileo smouldered for a time, until, by
+another noble imprudence, he managed to offend a semi-royal personage,
+Giovanni de Medici, by giving his real opinion, when consulted, about a
+machine which de Medici had invented for cleaning out the harbour of
+Leghorn. He said it was as useless as it in fact turned out to be.
+Through the influence of the mortified inventor he lost favour at Court;
+and his enemies took advantage of the fact to render his chair
+untenable. He resigned before his three years were up, and retired to
+Florence.</p>
+
+<p>His father at this time died, and the family were left in narrow
+circumstances. He had a brother and three sisters to provide for.</p>
+
+<p><span class='pagenum'><a name="Page_93" id="Page_93">[Pg 93]</a></span></p><p>He was offered a professorship at Padua for six years by the Senate of
+Venice, and willingly accepted it.</p>
+
+<p>Now began a very successful career. His introductory address was marked
+by brilliant eloquence, and his lectures soon acquired fame. He wrote
+for his pupils on the laws of motion, on fortifications, on sundials, on
+mechanics, and on the celestial globe: some of these papers are now
+lost, others have been printed during the present century.</p>
+
+<p>Kepler sent him a copy of his new book, <i>Mysterium Cosmographicum</i>, and
+Galileo in thanking him for it writes him the following letter:&mdash;<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a></p>
+
+<div class="blockquot"><p>"I count myself happy, in the search after truth, to have so great
+an ally as yourself, and one who is so great a friend of the truth
+itself. It is really pitiful that there are so few who seek truth,
+and who do not pursue a perverse method of philosophising. But this
+is not the place to mourn over the miseries of our times, but to
+congratulate you on your splendid discoveries in confirmation of
+truth. I shall read your book to the end, sure of finding much that
+is excellent in it. I shall do so with the more pleasure, because
+<i>I have been for many years an adherent of the Copernican system</i>,
+and it explains to me the causes of many of the appearances of
+nature which are quite unintelligible on the commonly accepted
+hypothesis. <i>I have collected many arguments for the purpose of
+refuting the latter</i>; but I do not venture to bring them to the
+light of publicity, for fear of sharing the fate of our master,
+Copernicus, who, although he has earned immortal fame with some,
+yet with very many (so great is the number of fools) has become an
+object of ridicule and scorn. I should certainly venture to publish
+my speculations if there were more people like you. But this not
+being the case, I refrain from such an undertaking." </p></div>
+
+<p>Kepler urged him to publish his arguments in favour of the Copernican
+theory, but he hesitated for the present, knowing that his declaration
+would be received with ridicule and opposition, and thinking it wiser to
+get rather more<span class='pagenum'><a name="Page_94" id="Page_94">[Pg 94]</a></span> firmly seated in his chair before encountering the
+storm of controversy.</p>
+
+<p>The six years passed away, and the Venetian Senate, anxious not to lose
+so bright an ornament, renewed his appointment for another six years at
+a largely increased salary.</p>
+
+<p>Soon after this appeared a new star, the stella nova of 1604, not the
+one Tycho had seen&mdash;that was in 1572&mdash;but the same that Kepler was so
+much interested in.</p>
+
+<p>Galileo gave a course of three lectures upon it to a great audience. At
+the first the theatre was over-crowded, so he had to adjourn to a hall
+holding 1000 persons. At the next he had to lecture in the open air.</p>
+
+<p>He took occasion to rebuke his hearers for thronging to hear about an
+ephemeral novelty, while for the much more wonderful and important
+truths about the permanent stars and facts of nature they had but deaf
+ears.</p>
+
+<p>But the main point he brought out concerning the new star was that it
+upset the received Aristotelian doctrine of the immutability of the
+heavens. According to that doctrine the heavens were unchangeable,
+perfect, subject neither to growth nor to decay. Here was a body, not a
+meteor but a real distant star, which had not been visible and which
+would shortly fade away again, but which meanwhile was brighter than
+Jupiter.</p>
+
+<p>The staff of petrified professorial wisdom were annoyed at the
+appearance of the star, still more at Galileo's calling public attention
+to it; and controversy began at Padua. However, he accepted it; and now
+boldly threw down the gauntlet in favour of the Copernican theory,
+utterly repudiating the old Ptolemaic system which up to that time he
+had taught in the schools according to established custom.</p>
+
+<p>The earth no longer the only world to which all else in the firmament
+were obsequious attendants, but a mere insignificant speck among the
+host of heaven! Man no longer the centre and cynosure of creation, but,
+as<span class='pagenum'><a name="Page_95" id="Page_95">[Pg 95]</a></span> it were, an insect crawling on the surface of this little speck! All
+this not set down in crabbed Latin in dry folios for a few learned
+monks, as in Copernicus's time, but promulgated and argued in rich
+Italian, illustrated by analogy, by experiment, and with cultured wit;
+taught not to a few scholars here and there in musty libraries, but
+proclaimed in the vernacular to the whole populace with all the energy
+and enthusiasm of a recent convert and a master of language! Had a
+bombshell been exploded among the fossilized professors it had been less
+disturbing.</p>
+
+<p>But there was worse in store for them.</p>
+
+<p>A Dutch optician, Hans Lippershey by name, of Middleburg, had in his
+shop a curious toy, rigged up, it is said, by an apprentice, and made
+out of a couple of spectacle lenses, whereby, if one looked through it,
+the weather-cock of a neighbouring church spire was seen nearer and
+upside down.</p>
+
+<p>The tale goes that the Marquis Spinola, happening to call at the shop,
+was struck with the toy and bought it. He showed it to Prince Maurice of
+Nassau, who thought of using it for military reconnoitring. All this is
+trivial. What is important is that some faint and inaccurate echo of
+this news found its way to Padua, and into the ears of Galileo.</p>
+
+<p>The seed fell on good soil. All that night he sat up and pondered. He
+knew about lenses and magnifying glasses. He had read Kepler's theory of
+the eye, and had himself lectured on optics. Could he not hit on the
+device and make an instrument capable of bringing the heavenly bodies
+nearer? Who knew what marvels he might not so perceive! By morning he
+had some schemes ready to try, and one of them was successful.
+Singularly enough it was not the same plan as the Dutch optician's, it
+was another mode of achieving the same end.</p>
+
+<p>He took an old small organ pipe, jammed a suitably chosen spectacle
+glass into either end, one convex the other concave, and behold, he had
+the half of a wretchedly bad<span class='pagenum'><a name="Page_96" id="Page_96">[Pg 96]</a></span> opera glass capable of magnifying three
+times. It was better than the Dutchman's, however; it did not invert.</p>
+
+<div class="blockquot"><p>It is easy to understand the general principle of a telescope. A
+general knowledge of the common magnifying glass may be assumed.
+Roger Bacon knew about lenses; and the ancients often refer to
+them, though usually as burning glasses. The magnifying power of
+globes of water must have been noticed soon after the discovery of
+glass and the art of working it.</p>
+
+<p>A magnifying glass is most simply thought of as an additional lens
+to the eye. The eye has a lens by which ordinary vision is
+accomplished, an extra glass lens strengthens it and enables
+objects to be seen nearer and therefore apparently bigger. But to
+apply a magnifying glass to distant objects is impossible. In order
+to magnify distant objects, another function of lenses has also to
+be employed, viz., their power of forming real images, the power on
+which their use as burning-glasses depends: for the best focus is
+an image of the sun. Although the object itself is inaccessible,
+the image of it is by no means so, and to the image a magnifier can
+be applied. This is exactly what is done in the telescope; the
+object glass or large lens forms an image, which is then looked at
+through a magnifying glass or eye-piece.</p>
+
+<p>Of course the image is nothing like so big as the object. For
+astronomical objects it is almost infinitely less; still it is an
+exact representation at an accessible place, and no one expects a
+telescope to show distant bodies as big as they really are. All it
+does is to show them bigger than they could be seen without it.</p>
+
+<p>But if the objects are not distant, the same principle may still be
+applied, and two lenses may be used, one to form an image, the
+other to magnify it; only if the object can be put where we please,
+we can easily place it so that its image is already much bigger
+than the object even before magnification by the eye lens. This is
+the compound microscope, the invention of which soon followed the
+telescope. In fact the two instruments shade off into one another,
+so that the reading telescope or reading microscope of a laboratory
+(for reading thermometers, and small divisions generally) goes by
+either name at random.</p>
+
+<p>The arrangement so far described depicts things on the retina the
+unaccustomed way up. By using a concave glass<span class='pagenum'><a name="Page_97" id="Page_97">[Pg 97]</a></span> instead of a convex,
+and placing it so as to prevent any image being formed, except on
+the retina direct, this inconvenience is avoided.</p></div>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_38" id="Fig_38"></a>
+<img src="images/fig38.jpg" width="400" height="535" alt="Fig. 38." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 38.</span>&mdash;View of the half-moon in small telescope. The
+darker regions, or plains, used to be called &quot;seas.&quot;</span>
+</div>
+
+<p>Such a thing as Galileo made may now be bought at a toy-shop for I
+suppose half a crown, and yet what a potentiality lay in that "glazed
+optic tube," as Milton called it.<span class='pagenum'><a name="Page_98" id="Page_98">[Pg 98]</a></span> Away he went with it to Venice and
+showed it to the Signoria, to their great astonishment. "Many noblemen
+and senators," says Galileo, "though of advanced age, mounted to the top
+of one of the highest towers to watch the ships, which were visible
+through my glass two hours before they were seen entering the harbour,
+for it makes a thing fifty miles off as near and clear as if it were
+only five." Among the people too the instrument excited the greatest
+astonishment and interest, so that he was nearly mobbed. The Senate
+hinted to him that a present of the instrument would not be
+unacceptable, so Galileo took the hint and made another for them.</p>
+
+<div class="figcenter" style="width: 550px;"><a name="Fig_39" id="Fig_39"></a>
+<img src="images/fig39.jpg" width="400" height="354" alt="Fig. 39." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 39.</span>&mdash;Portion of the lunar surface more highly
+magnified, showing the shadows of a mountain range, deep pits, and other
+details.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_99" id="Page_99">[Pg 99]</a></span></p><p>They immediately doubled his salary at Padua, making it 1000 florins,
+and confirmed him in the enjoyment of it for life.</p>
+
+<p>He now eagerly began the construction of a larger and better instrument.
+Grinding the lenses with his own hands with consummate skill, he
+succeeded in making a telescope magnifying thirty times. Thus equipped
+he was ready to begin a survey of the heavens.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_40" id="Fig_40"></a>
+<img src="images/fig40.jpg" width="400" height="349" alt="Fig. 40." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 40.</span>&mdash;Another portion of the lunar surface, showing a
+so-called crater or vast lava pool and other evidences of ancient heat
+unmodified by water.</span>
+</div>
+
+<p>The first object he carefully examined was naturally the moon. He found
+there everything at first sight very like the earth, mountains and
+valleys, craters and plains, rocks, and apparently seas. You may imagine
+the hostility excited among the Aristotelian philosophers, especially no
+doubt<span class='pagenum'><a name="Page_100" id="Page_100">[Pg 100]</a></span> those he had left behind at Pisa, on the ground of his spoiling
+the pure, smooth, crystalline, celestial face of the moon as they had
+thought it, and making it harsh and rugged and like so vile and ignoble
+a body as the earth.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_41" id="Fig_41"></a>
+<img src="images/fig41.jpg" width="400" height="659" alt="Fig. 41." title="" />
+<div class="caption1"><span class="smcap">Fig. 41.</span>&mdash;Lunar landscape showing earth. The earth would
+be a stationary object in the moon&#39;s sky: its only apparent motion being
+a slow oscillation as of a pendulum (the result of the moon&#39;s
+libration).</div>
+</div>
+
+<p>He went further, however, into heterodoxy than this&mdash;he not only made
+the moon like the earth, but he made the earth shine like the moon. The
+visibility of "the old moon in the new moon's arms" he explained by
+earth-shine. Leonardo had given the same explanation a century before.
+Now one of the many stock arguments against Copernican<span class='pagenum'><a name="Page_101" id="Page_101">[Pg 101]</a></span> theory of the
+earth being a planet like the rest was that the earth was dull and dark
+and did not shine. Galileo argued that it shone just as much as the moon
+does, and in fact rather more&mdash;especially if it be covered with clouds.
+One reason of the peculiar brilliancy of Venus is that she is a very
+cloudy planet.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> Seen from the moon the earth would look exactly as the
+moon does to us, only a little brighter and sixteen times as big (four
+times the diameter).</p>
+
+<div class="figcenter" style="width: 650px;"><a name="Fig_42" id="Fig_42"></a>
+<img src="images/fig42.jpg" width="400" height="275" alt="Fig. 42." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 42.</span>&mdash;Galileo&#39;s method of estimating the height of
+lunar mountain.<br /></span>
+<div class="caption1"><i>AB'BC</i> is the illuminated half of the moon. <i>SA</i> is a solar ray just
+catching the peak of the mountain <i>M</i>. Then by geometry, as <i>MN</i> is to
+<i>MA</i>, so is <i>MA</i> to <i>MB'</i>; whence the height of the mountain, <i>MN</i>, can
+be determined. The earth and spectator are supposed to be somewhere in
+the direction <i>BA</i> produced, <i>i.e.</i> towards the top of the page.</div>
+</div>
+
+<div class="blockquot"><p>Galileo made a very good estimate of the height of lunar mountains,
+of which many are five miles high and some as much as seven. He did
+this simply by measuring from the half-moon's straight edge the
+distance at which their peaks caught the rising or setting sun. The
+above simple diagram shows that as this distance is to the diameter
+of the moon, so is the height of the sun-tipped mountain to the
+aforesaid distance. </p></div>
+
+<p>Wherever Galileo turned his telescope new stars appeared. The Milky Way,
+which had so puzzled the ancients, was found to be composed of stars.
+Stars that appeared single to the eye were some of them found to be
+double; and at intervals were found hazy nebulous wisps, some of which
+seemed to be star clusters, while others seemed only a fleecy cloud.</p>
+
+<p><span class='pagenum'><a name="Page_102" id="Page_102">[Pg 102]</a></span></p>
+<div class="figcenter" style="width: 500px;"><a name="Fig_43" id="Fig_43"></a>
+<img src="images/fig43.jpg" width="500" height="750" alt="Fig. 43." title="" />
+<span class="caption"><span class="smcap">Fig. 43.</span>&mdash;Some clusters and nebul&aelig;.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_103" id="Page_103">[Pg 103]</a></span></p>
+<div class="figcenter" style="width: 550px;"><br /><a name="Fig_44" id="Fig_44"></a>
+<img src="images/fig44.jpg" width="400" height="387" alt="Fig. 44." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 44.</span>&mdash;Jupiter&#39;s satellites, showing the stages of
+their discovery.</span>
+</div>
+
+<p>Now we come to his most brilliant, at least his most sensational,
+discovery. Examining Jupiter minutely on January 7, 1610, he noticed
+three little stars near it, which he noted down as fixing its then
+position. On the following night Jupiter had moved to the other side of
+the three stars. This was natural enough, but was it moving the right
+way? On examination it appeared not. Was it possible the tables were
+wrong? The next evening was cloudy, and he had to curb his feverish
+impatience. On the 10th there were only two, and those on the other
+side. On the 11th two again, but one bigger than the other. On the 12th
+the three re-appeared, and on the 13th there were four. No more
+appeared.</p>
+
+<p><span class='pagenum'><a name="Page_104" id="Page_104">[Pg 104]</a></span></p><p>Jupiter then had moons like the earth, four of them in fact, and they
+revolved round him in periods which were soon determined.</p>
+
+<div class="blockquot"><p>The reason why they were not all visible at first, and why their
+visibility so rapidly changes, is because they revolve round him
+almost in the plane of our vision, so that sometimes they are in
+front and sometimes behind him, while again at other times they
+plunge into his shadow and are thus eclipsed from the light of the
+sun which enables us to see them. A large modern telescope will
+show the moons when in front of Jupiter, but small telescopes will
+only show them when clear of the disk and shadow. Often all four
+can be thus seen, but three or two is a very common amount of
+visibility. Quite a small telescope, such as a ship's telescope, if
+held steadily, suffices to show the satellites of Jupiter, and very
+interesting objects they are. They are of habitable size, and may
+be important worlds for all we know to the contrary. </p></div>
+
+<p>The news of the discovery soon spread and excited the greatest interest
+and astonishment. Many of course refused to believe it. Some there were
+who having been shown them refused to believe their eyes, and asserted
+that although the telescope acted well enough for terrestrial objects,
+it was altogether false and illusory when applied to the heavens. Others
+took the safer ground of refusing to look through the glass. One of
+these who would not look at the satellites happened to die soon
+afterwards. "I hope," says Galileo, "that he saw them on his way to
+heaven."</p>
+
+<p>The way in which Kepler received the news is characteristic, though by
+adding four to the supposed number of planets it might have seemed to
+upset his notions about the five regular solids.</p>
+
+<div class="blockquot"><p>He says,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> "I was sitting idle at home thinking of you, most
+excellent Galileo, and your letters, when the news was brought me
+of the discovery of four planets by the help of the double
+eye-glass. Wachenfels stopped his carriage at<span class='pagenum'><a name="Page_105" id="Page_105">[Pg 105]</a></span> my door to tell me,
+when such a fit of wonder seized me at a report which seemed so
+very absurd, and I was thrown into such agitation at seeing an old
+dispute between us decided in this way, that between his joy, my
+colouring, and the laughter of us both, confounded as we were by
+such a novelty, we were hardly capable, he of speaking, or I of
+listening....</p>
+
+<p>"On our separating, I immediately fell to thinking how there could
+be any addition to the number of planets without overturning my
+<i>Mysterium Cosmographicon</i>, published thirteen years ago, according
+to which Euclid's five regular solids do not allow more than six
+planets round the sun.</p>
+
+<p>"But I am so far from disbelieving the existence of the four
+circumjovial planets that I long for a telescope to anticipate you
+if possible in discovering two round Mars (as the proportion seems
+to me to require) six or eight round Saturn, and one each round
+Mercury and Venus." </p></div>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_45" id="Fig_45"></a>
+<img src="images/fig45.jpg" width="400" height="423" alt="Fig. 45." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 45.</span>&mdash;Eclipses of Jupiter's satellites. The diagram
+shows the first (<i>i.e.</i> the nearest) moon in Jupiter's shadow, the
+second as passing between earth and Jupiter, and appearing to transit
+his disk, the third as on the verge of entering his shadow, and the
+fourth quite plainly and separately visible.</div>
+</div>
+
+<p>As an illustration of the opposite school, I will take<span class='pagenum'><a name="Page_106" id="Page_106">[Pg 106]</a></span> the following
+extract from Francesco Sizzi, a Florentine astronomer, who argues
+against the discovery thus:&mdash;</p>
+
+<div class="blockquot"><p>"There are seven windows in the head, two nostrils, two eyes, two
+ears, and a mouth; so in the heavens there are two favourable
+stars, two unpropitious, two luminaries, and Mercury alone
+undecided and indifferent. From which and many other similar
+phenomena of nature, such as the seven metals, &amp;c., which it were
+tedious to enumerate, we gather that the number of planets is
+necessarily seven.</p>
+
+<p>"Moreover, the satellites are invisible to the naked eye, and
+therefore can have no influence on the earth, and therefore would
+be useless, and therefore do not exist.</p>
+
+<p>"Besides, the Jews and other ancient nations as well as modern
+Europeans have adopted the division of the week into seven days,
+and have named them from the seven planets: now if we increase the
+number of the planets this whole system falls to the ground." </p></div>
+
+<p>To these arguments Galileo replied that whatever their force might be as
+a reason for believing beforehand that no more than seven planets would
+be discovered, they hardly seemed of sufficient weight to destroy the
+new ones when actually seen.</p>
+
+<p>Writing to Kepler at this time, Galileo ejaculates:</p>
+
+<div class="blockquot"><p>"Oh, my dear Kepler, how I wish that we could have one hearty laugh
+together! Here, at Padua, is the principal professor of philosophy
+whom I have repeatedly and urgently requested to look at the moon
+and planets through my glass, which he pertinaciously refuses to
+do. Why are you not here? What shouts of laughter we should have at
+this glorious folly! And to hear the professor of philosophy at
+Pisa labouring before the grand duke with logical arguments, as if
+with magical incantations, to charm the new planets out of the
+sky." </p></div>
+
+<p>A young German <i>prot&eacute;g&eacute;</i> of Kepler, Martin Horkey, was travelling in
+Italy, and meeting Galileo at Bologna was favoured with a view through
+his telescope. But supposing<span class='pagenum'><a name="Page_107" id="Page_107">[Pg 107]</a></span> that Kepler must necessarily be jealous of
+such great discoveries, and thinking to please him, he writes, "I cannot
+tell what to think about these observations. They are stupendous, they
+are wonderful, but whether they are true or false I cannot tell." He
+concludes, "I will never concede his four new planets to that Italian
+from Padua though I die for it." So he published a pamphlet asserting
+that reflected rays and optical illusions were the sole cause of the
+appearance, and that the only use of the imaginary planets was to
+gratify Galileo's thirst for gold and notoriety.</p>
+
+<p>When after this performance he paid a visit to his old instructor
+Kepler, he got a reception which astonished him. However, he pleaded so
+hard to be forgiven that Kepler restored him to partial favour, on this
+condition, that he was to look again at the satellites, and this time to
+see them and own that they were there.</p>
+
+<p>By degrees the enemies of Galileo were compelled to confess to the truth
+of the discovery, and the next step was to outdo him. Scheiner counted
+five, Rheiter nine, and others went as high as twelve. Some of these
+were imaginary, some were fixed stars, and four satellites only are
+known to this day.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p>
+
+<p>Here, close to the summit of his greatness, we must leave him for a
+time. A few steps more and he will be on the brow of the hill; a short
+piece of table-land, and then the descent begins.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_108" id="Page_108">[Pg 108]</a></span></p>
+<h3><a name="LECTURE_V" id="LECTURE_V"></a>LECTURE V</h3>
+
+<h5>GALILEO AND THE INQUISITION</h5>
+
+
+<p><span class="smcap">One</span> sinister event occurred while Galileo was at Padua, some time before
+the era we have now arrived at, before the invention of the
+telescope&mdash;two years indeed after he had first gone to Padua; an event
+not directly concerning Galileo, but which I must mention because it
+must have shadowed his life both at the time and long afterwards. It was
+the execution of Giordano Bruno for heresy. This eminent philosopher had
+travelled largely, had lived some time in England, had acquired new and
+heterodox views on a variety of subjects, and did not hesitate to
+propound them even after he had returned to Italy.</p>
+
+<p>The Copernican doctrine of the motion of the earth was one of his
+obnoxious heresies. Being persecuted to some extent by the Church, Bruno
+took refuge in Venice&mdash;a free republic almost independent of the
+Papacy&mdash;where he felt himself safe. Galileo was at Padua hard by: the
+University of Padua was under the government of the Senate of Venice:
+the two men must in all probability have met.</p>
+
+<p>Well, the Inquisition at Rome sent messengers to Venice with a demand
+for the extradition of Bruno&mdash;they wanted him at Rome to try him for
+heresy.</p>
+
+<p>In a moment of miserable weakness the Venetian republic gave him up, and
+Bruno was taken to Rome. There he was tried, and cast into the dungeons
+for six years, and because<span class='pagenum'><a name="Page_109" id="Page_109">[Pg 109]</a></span> he entirely refused to recant, was at length
+delivered over to the secular arm and burned at the stake on 16th
+February, Anno Domini 1600.</p>
+
+<p>This event could not but have cast a gloom over the mind of lovers and
+expounders of truth, and the lesson probably sank deep into Galileo's
+soul.</p>
+
+<p>In dealing with these historic events will you allow me to repudiate
+once for all the slightest sectarian bias or meaning. I have nothing to
+do with Catholic or Protestant as such. I have nothing to do with the
+Church of Rome as such. I am dealing with the history of science. But
+historically at one period science and the Church came into conflict. It
+was not specially one Church rather than another&mdash;it was the Church in
+general, the only one that then existed in those countries.
+Historically, I say, they came into conflict, and historically the
+Church was the conqueror. It got its way; and science, in the persons of
+Bruno, Galileo, and several others, was vanquished.</p>
+
+<p>Such being the facts, there is no help but to mention them in dealing
+with the history of science.</p>
+
+<p>Doubtless <i>now</i> the Church regards it as an unhappy victory, and gladly
+would ignore this painful struggle. This, however, is impossible. With
+their creed the Churchmen of that day could act in no other way. They
+were bound to prosecute heresy, and they were bound to conquer in the
+struggle or be themselves shattered.</p>
+
+<p>But let me insist on the fact that no one accuses the ecclesiastical
+courts of crime or evil motives. They attacked heresy after their
+manner, as the civil courts attacked witchcraft after <i>their</i> manner.
+Both erred grievously, but both acted with the best intentions.</p>
+
+<p>We must remember, moreover, that his doctrines were scientifically
+heterodox, and the University Professors of that day were probably quite
+as ready to condemn them as the Church was. To realise the position we
+must think of some subjects which <i>to-day</i> are scientifically
+heterodox,<span class='pagenum'><a name="Page_110" id="Page_110">[Pg 110]</a></span> and of the customary attitude adopted towards them by
+persons of widely differing creeds.</p>
+
+<p>If it be contended now, as it is, that the ecclesiastics treated Galileo
+well, I admit it freely: they treated him as well as they possibly
+could. They overcame him, and he recanted; but if he had not recanted,
+if he had persisted in his heresy, they would&mdash;well, they would still
+have treated his soul well, but they would have set fire to his body.
+Their mistake consisted not in cruelty, but in supposing themselves the
+arbiters of eternal truth; and by no amount of slurring and glossing
+over facts can they evade the responsibility assumed by them on account
+of this mistaken attitude.</p>
+
+<p>I am not here attacking the dogma of Papal Infallibility: it is
+historically, I believe, quite unaffected by the controversy respecting
+the motion of the earth, no Papal edict <i>ex cathedr&acirc;</i> having been
+promulgated on the subject.</p>
+
+<p>We left Galileo standing at his telescope and beginning his survey of
+the heavens. We followed him indeed through a few of his first great
+discoveries&mdash;the discovery of the mountains and other variety of surface
+in the moon, of the nebul&aelig; and a multitude of faint stars, and lastly of
+the four satellites of Jupiter.</p>
+
+<p>This latter discovery made an immense sensation, and contributed its
+share to his removal from Padua, which quickly followed it, as I shall
+shortly narrate; but first I think it will be best to continue our
+survey of his astronomical discoveries without regard to the place
+whence they were made.</p>
+
+<p>Before the end of the year Galileo had made another discovery&mdash;this time
+on Saturn. But to guard against the host of plagiarists and impostors,
+he published it in the form of an anagram, which, at the request of the
+Emperor Rudolph (a request probably inspired by Kepler), he interpreted;
+it ran thus: The furthest planet is triple.</p>
+
+<p>Very soon after he found that Venus was changing from a<span class='pagenum'><a name="Page_111" id="Page_111">[Pg 111]</a></span> full moon to a
+half moon appearance. He announced this also by an anagram, and waited
+till it should become a crescent, which it did.</p>
+
+<p>This was a dreadful blow to the anti-Copernicans, for it removed the
+last lingering difficulty to the reception of the Copernican doctrine.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_46" id="Fig_46"></a>
+<img src="images/fig46.jpg" width="400" height="403" alt="Fig. 46." title="" />
+<span class="caption"><span class="smcap">Fig. 46.</span>&mdash;Old drawings of Saturn by different observers,
+with the imperfect instruments of that day. The first is Galileo's idea
+of what he saw.</span>
+</div>
+
+<p>Copernicus had predicted, indeed, a hundred years before, that, if ever
+our powers of sight were sufficiently enhanced, Venus and Mercury would
+be seen to have phases like the<span class='pagenum'><a name="Page_112" id="Page_112">[Pg 112]</a></span> moon. And now Galileo with his
+telescope verifies the prediction to the letter.</p>
+
+<p>Here was a triumph for the grand old monk, and a bitter morsel for his
+opponents.</p>
+
+<div class="blockquot"><p>Castelli writes: "This must now convince the most obstinate." But
+Galileo, with more experience, replies:&mdash;"You almost make me laugh
+by saying that these clear observations are sufficient to convince
+the most obstinate; it seems you have yet to learn that long ago
+the observations were enough to convince those who are capable of
+reasoning, and those who wish to learn the truth; but that to
+convince the obstinate, and those who care for nothing beyond the
+vain applause of the senseless vulgar, not even the testimony of
+the stars would suffice, were they to descend on earth to speak for
+themselves. Let us, then, endeavour to procure some knowledge for
+ourselves, and rest contented with this sole satisfaction; but of
+advancing in popular opinion, or of gaining the assent of the
+book-philosophers, let us abandon both the hope and the desire."</p></div>
+
+<div class="figcenter" style="width: 550px;"><a name="Fig_47" id="Fig_47"></a>
+<img src="images/fig47.jpg" width="400" height="147" alt="Fig. 47." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 47.</span>&mdash;Phases of Venus. Showing also its apparent
+variations in size by reason of its varying distance from the earth.
+When fully illuminated it is necessarily most distant. It looks
+brightest to us when a broad crescent.</div>
+</div>
+
+<p>What a year's work it had been!</p>
+
+<p>In twelve months observational astronomy had made such a bound as it has
+never made before or since.</p>
+
+<p>Why did not others make any of these observations? Because no one could
+make telescopes like Galileo.</p>
+
+<p>He gathered pupils round him however, and taught them<span class='pagenum'><a name="Page_113" id="Page_113">[Pg 113]</a></span> how to work the
+lenses, so that gradually these instruments penetrated Europe, and
+astronomers everywhere verified his splendid discoveries.</p>
+
+<p>But still he worked on, and by March in the very next year, he saw
+something still more hateful to the Aristotelian philosophers, viz.
+spots on the sun.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_48" id="Fig_48"></a>
+<img src="images/fig48.jpg" width="400" height="361" alt="Fig. 48." title="" />
+<span class="caption"><span class="smcap">Fig. 48.</span></span>
+</div>
+
+<p>If anything was pure and perfect it was the sun, they said. Was this
+impostor going to blacken its face too?</p>
+
+<p>Well, there they were. They slowly formed and changed, and by moving all
+together showed him that the sun rotated about once a month.</p>
+
+<p><span class='pagenum'><a name="Page_114" id="Page_114">[Pg 114]</a></span></p><p>Before taking leave of Galileo's astronomical researches, I must
+mention an observation made at the end of 1612, that the apparent
+triplicity of Saturn (<a href="#Fig_46">Fig. 46</a>) had vanished.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_49" id="Fig_49"></a>
+<img src="images/fig49.jpg" width="400" height="258" alt="Fig. 49." title="" />
+<span class="caption"><span class="smcap">Fig. 49.</span>&mdash;A portion of the sun&#39;s disk as seen in a
+powerful modern telescope.</span>
+</div>
+
+<div class="blockquot"><p>"Looking on Saturn within these few days, I found it solitary,
+without the assistance of its accustomed stars, and in short
+perfectly round and defined, like Jupiter, and such it still
+remains. Now what can be said of so strange a metamorphosis? Are
+perhaps the two smaller stars consumed like spots on the sun? Have
+they suddenly vanished and fled? Or has Saturn devoured his own
+children? Or was the appearance indeed fraud and illusion, with
+which the glasses have so long time mocked me and so many others
+who have so often observed with me? Now perhaps the time is come to
+revive the withering hopes of those, who, guided by more profound
+contemplations, have fathomed all the fallacies of the new
+observations and recognized their impossibility! I cannot resolve
+what to<span class='pagenum'><a name="Page_115" id="Page_115">[Pg 115]</a></span> say in a chance so strange, so new, so unexpected. The
+shortness of time, the unexampled occurrence, the weakness of my
+intellect, the terror of being mistaken, have greatly confounded
+me." </p></div>
+
+<p>However, he plucked up courage, and conjectured that the two attendants
+would reappear, by revolving round the planet.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_50" id="Fig_50"></a>
+<img src="images/fig50.jpg" width="400" height="334" alt="Fig. 50." title="" />
+<span class="caption"><span class="smcap">Fig. 50.</span>&mdash;Saturn and his rings, as seen under the most
+favourable circumstances.</span>
+</div>
+
+<p>The real reason of their disappearance is well known to us now. The
+plane of Saturn's rings oscillates slowly about our line of sight, and
+so we sometimes see them edgeways and sometimes with a moderate amount
+of obliquity. The rings are so thin that, when turned precisely
+edgeways, they become invisible. The two imaginary attendants were the
+most conspicuous portions of the ring, subsequently called <i>ans&aelig;</i>.</p>
+
+<p>I have thought it better not to interrupt this catalogue of<span class='pagenum'><a name="Page_116" id="Page_116">[Pg 116]</a></span> brilliant
+discoveries by any biographical details; but we must now retrace our
+steps to the years 1609 and 1610, the era of the invention of the
+telescope.</p>
+
+<p>By this time Galileo had been eighteen years at Padua, and like many
+another man in like case, was getting rather tired of continual
+lecturing. Moreover, he felt so full of ideas that he longed to have a
+better opportunity of following them up, and more time for thinking them
+out.</p>
+
+<p>Now in the holidays he had been accustomed to return to his family home
+at Pisa, and there to come a good deal into contact with the Grand-Ducal
+House of Tuscany. Young Cosmo di Medici became in fact his pupil, and
+arrived at man's estate with the highest opinion of the philosopher.
+This young man had now come to the throne as Cosmo II., and to him
+Galileo wrote saying how much he should like more time and leisure, how
+full he was of discoveries if he only had the chance of a reasonable
+income without the necessity of consuming so large a portion of his time
+in elementary teaching, and practically asking to be removed to some
+position in the Court. Nothing was done for a time, but negotiations
+proceeded, and soon after the discovery of Jupiter's satellites Cosmo
+wrote making a generous offer, which Galileo gladly and enthusiastically
+accepted, and at once left Padua for Florence. All his subsequent
+discoveries date from Florence.</p>
+
+<p>Thus closed his brilliant and happy career as a professor at the
+University of Padua. He had been treated well: his pay had become larger
+than that of any Professor of Mathematics up to that time; and, as you
+know, immediately after his invention of the telescope the Venetian
+Senate, in a fit of enthusiasm, had doubled it and secured it to him for
+life wherever he was. To throw up his chair and leave the place the very
+next year scarcely seems a strictly honourable procedure. It was legal
+enough no doubt, and it is easy for small men to criticize a great one,
+but nevertheless I think we must admit that it is a step<span class='pagenum'><a name="Page_117" id="Page_117">[Pg 117]</a></span> such as a man
+with a keen sense of honour would hardly have taken.</p>
+
+<p>One quite feels and sympathizes with the temptation. Not emolument, but
+leisure; freedom from harassing engagements and constant teaching, and
+liberty to prosecute his studies day and night without interference:
+this was the golden prospect before him. He yielded, but one cannot help
+wishing he had not.</p>
+
+<p>As it turned out it was a false step&mdash;the first false step of his public
+career. When made it was irretrievable, and it led to great misery.</p>
+
+<p>At first it seemed brilliant enough. The great philosopher of the Tuscan
+Court was courted and flattered by princes and nobles, he enjoyed a
+world-wide reputation, lived as luxuriously as he cared for, had his
+time all to himself, and lectured but very seldom, on great occasions or
+to a few crowned heads.</p>
+
+<p>His position was in fact analogous to that of Tycho Brah&eacute; in his island
+of Huen.</p>
+
+<p>Misfortune overtook both. In Tycho's case it arose mainly from the death
+of his patron. In Galileo's it was due to a more insidious cause, to
+understand which cause aright we must remember the political divisions
+of Italy at that date.</p>
+
+<p>Tuscany was a Papal State, and thought there was by no means free.
+Venice was a free republic, and was even hostile to the Papacy. In 1606
+the Pope had placed it under an interdict. In reply it had ejected every
+Jesuit.</p>
+
+<p>Out of this atmosphere of comparative enlightenment and freedom into
+that hotbed of medi&aelig;valism and superstition went Galileo with his eyes
+open. Keen was the regret of his Paduan and Venetian friends; bitter
+were their remonstrances and exhortations. But he was determined to go,
+and, not without turning some of his old friends into enemies, he went.</p>
+
+<p>Seldom has such a man made so great a mistake: never, I suppose, has one
+been so cruelly punished for it.</p>
+
+<p><span class='pagenum'><a name="Page_118" id="Page_118">[Pg 118]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_51" id="Fig_51"></a>
+<img src="images/fig51.jpg" width="400" height="406" alt="Fig. 51." title="" />
+<span class="caption"><span class="smcap">Fig. 51.</span>&mdash;Map of Italy.</span>
+</div>
+
+<p>We must remember, however, that Galileo, though by no means a saint, was
+yet a really religious man, a devout Catholic and thorough adherent of
+the Church, so that he would have no dislike to place himself under her
+sway. Moreover, he had been born a Tuscan, his family had lived at
+Florence or Pisa, and it felt like going home. His theological attitude
+is worthy of notice, for he was not in the least a sceptic. He quite
+acquiesces in the authority of the Bible, especially in all matters
+concerning faith and conduct; as to its statements in scientific
+matters, he argues that we are so liable to misinterpret their meaning
+that it<span class='pagenum'><a name="Page_119" id="Page_119">[Pg 119]</a></span> is really easier to examine Nature for truth in scientific
+matters, and that when direct observation and Scripture seem to clash,
+it is because of our fallacious interpretation of one or both of them.
+He is, in fact, what one now calls a "reconciler."</p>
+
+<p>It is curious to find such a man prosecuted for heresy, when to-day his
+opinions are those of the orthodox among the orthodox. But so it ever
+is, and the heresy of one generation becomes the commonplace of the
+next.</p>
+
+<p>He accepts Joshua's miracle, for instance, not as a striking poem, but
+as a literal fact; and he points out how much more simply it could be
+done on the Copernican system by stopping the earth's rotation for a
+short time, than by stopping the sun and moon and all the host of heaven
+as on the old Ptolemaic system, or again by stopping only the sun and
+not any of the other bodies, and so throwing astronomy all wrong.</p>
+
+<p>This reads to us like satire, but no doubt it was his genuine opinion.</p>
+
+<p>These Scriptural reconciliations of his, however, angered the religious
+authorities still more. They said it was bad enough for this heretic to
+try and upset old <i>scientific</i> beliefs, and to spoil the face of
+<i>Nature</i> with his infidel discoveries, but at least he might leave the
+Bible alone; and they addressed an indignant remonstrance to Rome, to
+protect it from the hands of ignorant laymen.</p>
+
+<p>Thus, wherever he turned he encountered hostility. Of course he had many
+friends&mdash;some of them powerful like Cosmo, all of them faithful and
+sincere. But against the power of Rome what could they do? Cosmo dared
+no more than remonstrate, and ultimately his successor had to refrain
+from even this, so enchained and bound was the spirit of the rulers of
+those days; and so when his day of tribulation came he stood alone and
+helpless in the midst of his enemies.</p>
+
+<p>You may wonder, perhaps, why this man should excite<span class='pagenum'><a name="Page_120" id="Page_120">[Pg 120]</a></span> so much more
+hostility than many another man who was suffered to believe and teach
+much the same doctrines unmolested. But no other man had made such
+brilliant and exciting discoveries. No man stood so prominently forward
+in the eyes of all Christendom as the champion of the new doctrines. No
+other man stated them so clearly and forcibly, nor drove them home with
+such brilliant and telling illustrations.</p>
+
+<p>And again, there was the memory of his early conflict with the
+Aristotelians at Pisa, of his scornful and successful refutation of
+their absurdities. All this made him specially obnoxious to the
+Aristotelian Jesuits in their double capacity both of priests and of
+philosophers, and they singled him out for relentless official
+persecution.</p>
+
+<p>Not yet, however, is he much troubled by them. The chief men at Rome
+have not yet moved. Messages, however, keep going up from Tuscany to
+Rome respecting the teachings of this man, and of the harm he is doing
+by his pertinacious preaching of the Copernican doctrine that the earth
+moves.</p>
+
+<p>At length, in 1615, Pope Paul V. wrote requesting him to come to Rome to
+explain his views. He went, was well received, made a special friend of
+Cardinal Barberino&mdash;an accomplished man in high position, who became in
+fact the next Pope. Galileo showed cardinals and others his telescope,
+and to as many as would look through it he showed Jupiter's satellites
+and his other discoveries. He had a most successful visit. He talked, he
+harangued, he held forth in the midst of fifteen or twenty disputants at
+once, confounding his opponents and putting them to shame.</p>
+
+<p>His method was to let the opposite arguments be stated as fully and
+completely as possible, himself aiding, and often adducing the most
+forcible and plausible arguments against his own views; and then, all
+having been well stated, he would proceed to utterly undermine and
+demolish<span class='pagenum'><a name="Page_121" id="Page_121">[Pg 121]</a></span> the whole fabric, and bring out the truth in such a way as to
+convince all honest minds. It was this habit that made him such a
+formidable antagonist. He never shrank from meeting an opposing
+argument, never sought to ignore it, or cloak it in a cloud of words.
+Every hostile argument he seemed to delight in, as a foe to be crushed,
+and the better and stronger they sounded the more he liked them. He knew
+many of them well, he invented a number more, and had he chosen could
+have out-argued the stoutest Aristotelian on his own grounds. Thus did
+he lead his adversaries on, almost like Socrates, only to ultimately
+overwhelm them in a more hopeless rout. All this in Rome too, in the
+heart of the Catholic world. Had he been worldly-wise, he would
+certainly have kept silent and unobtrusive till he had leave to go away
+again. But he felt like an apostle of the new doctrines, whose mission
+it was to proclaim them even in this centre of the world and of the
+Church.</p>
+
+<p>Well, he had an audience with the Pope&mdash;a chat an hour long&mdash;and the two
+parted good friends, mutually pleased with each other.</p>
+
+<p>He writes that he is all right now, and might return home when he liked.
+But the question began to be agitated whether the whole system of
+Copernicus ought not to be condemned as impious and heretical. This view
+was persistently urged upon the Pope and College of Cardinals, and it
+was soon to be decided upon.</p>
+
+<p>Had Galileo been unfaithful to the Church he could have left them to
+stultify themselves in any way they thought proper, and himself have
+gone; but he felt supremely interested in the result, and he stayed. He
+writes:&mdash;</p>
+
+<div class="blockquot"><p>"So far as concerns the clearing of my own character, I might
+return home immediately; but although this new question regards me
+no more than all those who for the last eighty years have supported
+those opinions both in public and private, yet, as perhaps I may be
+of some<span class='pagenum'><a name="Page_122" id="Page_122">[Pg 122]</a></span> assistance in that part of the discussion which depends on
+the knowledge of truths ascertained by means of the sciences which
+I profess, I, as a zealous and Catholic Christian, neither can nor
+ought to withhold that assistance which my knowledge affords, and
+this business keeps me sufficiently employed." </p></div>
+
+<p>It is possible that his stay was the worst thing for the cause he had at
+heart. Anyhow, the result was that the system was condemned, and both
+the book of Copernicus and the epitome of it by Kepler were placed on
+the forbidden list,<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> and Galileo himself was formally ordered never
+to teach or to believe the motion of the earth.</p>
+
+<p>He quitted Rome in disgust, which before long broke out in satire. The
+only way in which he could safely speak of these views now was as if
+they were hypothetical and uncertain, and so we find him writing to the
+Archduke Leopold, with a presentation copy of his book on the tides, the
+following:&mdash;</p>
+
+<div class="blockquot"><p>"This theory occurred to me when in Rome whilst the theologians
+were debating on the prohibition of Copernicus's book, and of the
+opinion maintained in it of the motion of the earth, which I at
+that time believed: until it pleased those gentlemen to suspend the
+book, and declare the opinion false and repugnant to the Holy
+Scriptures. Now, as I know how well it becomes me to obey and
+believe the decisions of my superiors, which proceed out of more
+knowledge than the weakness of my intellect can attain to, this
+theory which I send you, which is founded on the motion of the
+earth, I now look upon as a fiction and a dream, and beg your
+highness to receive it as such. But as poets often learn to prize
+the creations of their fancy, so in like manner do I set some value
+on this absurdity of mine. It is true that when I sketched this
+little work I did hope that Copernicus would not, after eighty
+years, be convicted of error; and I had intended to develop and
+amplify it<span class='pagenum'><a name="Page_123" id="Page_123">[Pg 123]</a></span> further, but a voice from heaven suddenly awakened me,
+and at once annihilated all my confused and entangled fancies." </p></div>
+
+<p>This sarcasm, if it had been in print, would probably have been
+dangerous. It was safe in a private letter, but it shows us his real
+feelings.</p>
+
+<p>However, he was left comparatively quiet for a time. He was getting an
+old man now, and passed the time studiously enough, partly at his house
+in Florence, partly at his villa in Arcetri, a mile or so out of the
+town.</p>
+
+<p>Here was a convent, and in it his two daughters were nuns. One of them,
+who passed under the name of Sister Maria Celeste, seems to have been a
+woman of considerable capacity&mdash;certainly she was of a most affectionate
+disposition&mdash;and loved and honoured her father in the most dutiful way.</p>
+
+<p>This was a quiet period of his life, spoiled only by occasional fits of
+illness and severe rheumatic pains, to which the old man was always
+liable. Many little circumstances are known of this peaceful time. For
+instance, the convent clock won't go, and Galileo mends it for them. He
+is always doing little things for them, and sending presents to the Lady
+Superior and his two daughters.</p>
+
+<p>He was occupied now with problems in hydrostatics, and on other matters
+unconnected with astronomy: a large piece of work which I must pass
+over. Most interesting and acute it is, however.</p>
+
+<p>In 1623, when the old Pope died, there was elected to the Papal throne,
+as Urban VIII., Cardinal Barberino, a man of very considerable
+enlightenment, and a personal friend of Galileo's, so that both he and
+his daughters rejoice greatly, and hope that things will come all right,
+and the forbidding edict be withdrawn.</p>
+
+<p>The year after this election he manages to make another journey to Rome
+to compliment his friend on his elevation<span class='pagenum'><a name="Page_124" id="Page_124">[Pg 124]</a></span> to the Pontifical chair. He
+had many talks with Urban, and made himself very agreeable.</p>
+
+<p>Urban wrote to the Grand Duke Ferdinand, son of Cosmo:&mdash;</p>
+
+<div class="blockquot"><p>"For We find in him not only literary distinction but also love of
+piety, and he is strong in those qualities by which Pontifical good
+will is easily obtainable. And now, when he has been brought to
+this city to congratulate Us on Our elevation, We have very
+lovingly embraced him; nor can We suffer him to return to the
+country whither your liberality recalls him without an ample
+provision of Pontifical love. And that you may know how dear he is
+to Us, We have willed to give him this honourable testimonial of
+virtue and piety. And We further signify that every benefit which
+you shall confer upon him, imitating or even surpassing your
+father's liberality, will conduce to Our gratification."</p></div>
+
+<p>Encouraged, doubtless, by these marks of approbation, and reposing too
+much confidence in the individual good will of the Pope, without heeding
+the crowd of half-declared enemies who were seeking to undermine his
+reputation, he set about, after his return to Florence, his greatest
+literary and most popular work, <i>Dialogues on the Ptolemaic and
+Copernican Systems</i>. This purports to be a series of four conversations
+between three characters: Salviati, a Copernican philosopher; Sagredo, a
+wit and scholar, not specially learned, but keen and critical, and who
+lightens the talk with chaff; Simplicio, an Aristotelian philosopher,
+who propounds the stock absurdities which served instead of arguments to
+the majority of men.</p>
+
+<p>The conversations are something between Plato's <i>Dialogues</i> and Sir
+Arthur Helps's <i>Friends in Council</i>. The whole is conducted with great
+good temper and fairness; and, discreetly enough, no definite conclusion
+is arrived at, the whole being left in abeyance as if for a fifth and
+decisive dialogue, which, however, was never written, and perhaps was
+only intended in case the reception was favourable.</p>
+
+<p>The preface also sets forth that the object of the writer is<span class='pagenum'><a name="Page_125" id="Page_125">[Pg 125]</a></span> to show
+that the Roman edict forbidding the Copernican doctrine was not issued
+in ignorance of the facts of the case, as had been maliciously reported,
+and that he wishes to show how well and clearly it was all known
+beforehand. So he says the dialogue on the Copernican side takes up the
+question purely as a mathematical hypothesis or speculative figment, and
+gives it every artificial advantage of which the theory is capable.</p>
+
+<p>This piece of caution was insufficient to blind the eyes of the
+Cardinals; for in it the arguments in favour of the earth's motion are
+so cogent and unanswerable, and are so popularly stated, as to do more
+in a few years to undermine the old system than all that he had written
+and spoken before. He could not get it printed for two years after he
+had written it, and then only got consent through a piece of
+carelessness or laziness on the part of the ecclesiastical censor
+through whose hands the manuscript passed&mdash;for which he was afterwards
+dismissed.</p>
+
+<p>However, it did appear, and was eagerly read; the more, perhaps, as the
+Church at once sought to suppress it.</p>
+
+<p>The Aristotelians were furious, and represented to the Pope that he
+himself was the character intended by Simplicio, the philosopher whose
+opinions get alternately refuted and ridiculed by the other two, till he
+is reduced to an abject state of impotence.</p>
+
+<p>The idea that Galileo had thus cast ridicule upon his friend and patron
+is no doubt a gratuitous and insulting libel: there is no telling
+whether or not Urban believed it, but certainly his countenance changed
+to Galileo henceforward, and whether overruled by his Cardinals, or
+actuated by some other motive, his favour was completely withdrawn.</p>
+
+<p>The infirm old man was instantly summoned to Rome. His friends pleaded
+his age&mdash;he was now seventy&mdash;his ill-health, the time of year, the state
+of the roads, the quarantine existing on account of the plague. It was
+all of no avail,<span class='pagenum'><a name="Page_126" id="Page_126">[Pg 126]</a></span> to Rome he must go, and on the 14th of February he
+arrived.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_52" id="Fig_52"></a>
+<img src="images/fig52.jpg" width="400" height="522" alt="Fig. 52." title="" />
+<span class="caption"><span class="smcap">Fig. 52.</span>&mdash;Portrait of Galileo.</span>
+</div>
+
+<p>His daughter at Arcetri was in despair; and anxiety and fastings and
+penances self-inflicted on his account, dangerously reduced her health.</p>
+
+<p>At Rome he was not imprisoned, but he was told to keep indoors, and show
+himself as little as possible. He was<span class='pagenum'><a name="Page_127" id="Page_127">[Pg 127]</a></span> allowed, however, to stay at the
+house of the Tuscan Ambassador instead of in gaol.</p>
+
+<p>By April he was removed to the chambers of the Inquisition, and examined
+several times. Here, however, the anxiety was too much, and his health
+began to give way seriously; so, before long, he was allowed to return
+to the Ambassador's house; and, after application had been made, was
+allowed to drive in the public garden in a half-closed carriage. Thus in
+every way the Inquisition dealt with him as leniently as they could. He
+was now their prisoner, and they might have cast him into their
+dungeons, as many another had been cast. By whatever they were
+influenced&mdash;perhaps the Pope's old friendship, perhaps his advanced age
+and infirmities&mdash;he was not so cruelly used.</p>
+
+<p>Still, they had their rules; he <i>must</i> be made to recant and abjure his
+heresy; and, if necessary, torture must be applied. This he knew well
+enough, and his daughter knew it, and her distress may be imagined.
+Moreover, it is not as if they had really been heretics, as if they
+hated or despised the Church of Rome. On the contrary, they loved and
+honoured the Church. They were sincere and devout worshippers, and only
+on a few scientific matters did Galileo presume to differ from his
+ecclesiastical superiors: his disagreement with them occasioned him real
+sorrow; and his dearest hope was that they could be brought to his way
+of thinking and embrace the truth.</p>
+
+<p>Every time he was sent for by the Inquisition he was in danger of
+torture unless he recanted. All his friends urged him repeatedly to
+submit. They said resistance was hopeless and fatal. Within the memory
+of men still young, Giordano Bruno had been burnt alive for a similar
+heresy. This had happened while Galileo was at Padua. Venice was full of
+it. And since that, only eight years ago indeed, Antonio de Dominis,
+Archbishop of Salpetria, had been sentenced to the same fate: "to be
+handed over to the secular arm to be dealt with as mercifully as
+possible<span class='pagenum'><a name="Page_128" id="Page_128">[Pg 128]</a></span> without the shedding of blood." So ran the hideous formula
+condemning a man to the stake. After his sentence, this unfortunate man
+died in the dungeons in which he had been incarcerated six years&mdash;died
+what is called a "natural" death; but the sentence was carried out,
+notwithstanding, on his lifeless body and his writings. His writings for
+which he had been willing to die!</p>
+
+<p>These were the tender mercies of the Inquisition; and this was the kind
+of meaning lurking behind many of their well-sounding and merciful
+phrases. For instance, what they call "rigorous examination," we call
+"torture." Let us, however, remember in our horror at this mode of
+compelling a prisoner to say anything they wished, that they were a
+legally constituted tribunal; that they acted with well established
+rules, and not in passion; and that torture was a recognized mode of
+extracting evidence, not only in ecclesiastical but in civil courts, at
+that date.</p>
+
+<p>All this, however, was but poor solace to the pitiable old philosopher,
+thus ruthlessly haled up and down, questioned and threatened, threatened
+and questioned, receiving agonizing letters from his daughter week by
+week, and trying to keep up a little spirit to reply as happily and
+hopefully as he could.</p>
+
+<p>This condition of things could not go on. From February to June the
+suspense lasted. On the 20th of June he was summoned again, and told he
+would be wanted all next day for a rigorous examination. Early in the
+morning of the 21st he repaired thither, and the doors were shut. Out of
+those chambers of horror he did not reappear till the 24th. What went on
+all those three days no one knows. He himself was bound to secrecy. No
+outsider was present. The records of the Inquisition are jealously
+guarded. That he was technically tortured is certain; that he actually
+underwent the torment of the rack is doubtful. Much learning has been
+expended upon the question, especially in Germany. Several eminent
+scholars have held the fact of<span class='pagenum'><a name="Page_129" id="Page_129">[Pg 129]</a></span> actual torture to be indisputable
+(geometrically certain, one says), and they confirm it by the hernia
+from which he afterwards suffered, this being a well-known and frequent
+consequence.</p>
+
+<p>Other equally learned commentators, however, deny that the last stage
+was reached. For there are five stages all laid down in the rules of the
+Inquisition, and steadily adhered to in a rigorous examination, at each
+stage an opportunity being given for recantation, every utterance,
+groan, or sigh being strictly recorded. The recantation so given has to
+be confirmed a day or two later, under pain of a precisely similar
+ordeal.</p>
+
+<p>The five stages are:&mdash;1st. The official threat in the court. 2nd. The
+taking to the door of the torture chamber and renewing the official
+threat. 3rd. The taking inside and showing the instruments. 4th.
+Undressing and binding upon the rack. 5th. <i>Territio realis.</i></p>
+
+<p>Through how many of these ghastly acts Galileo passed I do not know. I
+hope and believe not the last.</p>
+
+<p>There are those who lament that he did not hold out, and accept the
+crown of martyrdom thus offered to him. Had he done so we know his
+fate&mdash;a few years' languishing in the dungeons, and then the flames.</p>
+
+<p>Whatever he ought to have done, he did not hold out&mdash;he gave way. At one
+stage or another of the dread ordeal he said: "I am in your hands. I
+will say whatever you wish." Then was he removed to a cell while his
+special form of perjury was drawn up.</p>
+
+<p>The next day, clothed as a penitent, the venerable old man was taken to
+the Convent of Minerva, where the Cardinals and prelates were assembled
+for the purpose of passing judgment upon him.</p>
+
+<p>The text of the judgment I have here, but it is too long to read. It
+sentences him&mdash;1st. To the abjuration. 2nd. To formal imprisonment for
+life. 3rd. To recite the seven penitential psalms every week.</p>
+
+<p><span class='pagenum'><a name="Page_130" id="Page_130">[Pg 130]</a></span></p><p>Ten Cardinals were present; but, to their honour be it said, three
+refused to sign; and this blasphemous record of intolerance and bigoted
+folly goes down the ages with the names of seven Cardinals immortalized
+upon it.</p>
+
+<p>This having been read, he next had to read word for word the abjuration
+which had been drawn up for him, and then sign it.</p>
+
+
+<p class="center"><span class="smcap">The Abjuration of Galileo.</span></p>
+
+<div class="blockquot"><p>"I, Galileo Galilei, son of the late Vincenzo Galilei, of Florence,
+aged seventy years, being brought personally to judgment, and
+kneeling before you Most Eminent and Most Reverend Lords Cardinals,
+General Inquisitors of the universal Christian republic against
+heretical depravity, having before my eyes the Holy Gospels, which
+I touch with my own hands, swear that I have always believed, and
+now believe, and with the help of God will in future believe, every
+article which the Holy Catholic and Apostolic Church of Rome holds,
+teaches, and preaches. But because I have been enjoined by this
+Holy Office altogether to abandon the false opinion which maintains
+that the sun is the centre and immovable, and forbidden to hold,
+defend, or teach the said false doctrine in any manner, and after
+it hath been signified to me that the said doctrine is repugnant
+with the Holy Scripture, I have written and printed a book, in
+which I treat of the same doctrine now condemned, and adduce
+reasons with great force in support of the same, without giving any
+solution, and therefore have been judged grievously suspected of
+heresy; that is to say, that I held and believed that the sun is
+the centre of the universe and is immovable, and that the earth is
+not the centre and is movable; willing, therefore, to remove from
+the minds of your Eminences, and of every Catholic Christian, this
+vehement suspicion rightfully entertained towards me, with a
+sincere heart and unfeigned faith, I abjure, curse, and detest the
+said errors and heresies, and generally every other error and sect
+contrary to Holy Church; and I swear that I will never more in
+future say or assert anything verbally, or in writing, which may
+give rise to a similar<span class='pagenum'><a name="Page_131" id="Page_131">[Pg 131]</a></span> suspicion of me; but if I shall know any
+heretic, or any one suspected of heresy, that I will denounce him
+to this Holy Office, or to the Inquisitor or Ordinary of the place
+where I may be; I swear, moreover, and promise, that I will fulfil
+and observe fully, all the penances which have been or shall be
+laid on me by this Holy Office. But if it shall happen that I
+violate any of my said promises, oaths, and protestations (which
+God avert!), I subject myself to all the pains and punishments
+which have been decreed and promulgated by the sacred canons, and
+other general and particular constitutions, against delinquents of
+this description. So may God help me, and his Holy Gospels which I
+touch with my own hands. I, the above-named Galileo Galilei, have
+abjured, sworn, promised, and bound myself as above, and in witness
+thereof with my own hand have subscribed this present writing of my
+abjuration, which I have recited word for word. At Rome, in the
+Convent of Minerva, 22nd June, 1633. I, Galileo Galilei, have
+abjured as above with my own hand."</p></div>
+
+<p>Those who believe the story about his muttering to a friend, as he rose
+from his knees, "e pur si muove," do not realize the scene.</p>
+
+<p>1st. There was no friend in the place.</p>
+
+<p>2nd. It would have been fatally dangerous to mutter anything before such
+an assemblage.</p>
+
+<p>3rd. He was by this time an utterly broken and disgraced old man;
+wishful, of all things, to get away and hide himself and his miseries
+from the public gaze; probably with his senses deadened and stupefied by
+the mental sufferings he had undergone, and no longer able to think or
+care about anything&mdash;except perhaps his daughter,&mdash;certainly not about
+any motion of this wretched earth.</p>
+
+<p>Far and wide the news of the recantation spread. Copies of the
+abjuration were immediately sent to all Universities, with instructions
+to the professors to read it publicly.</p>
+
+<p>At Florence, his home, it was read out in the Cathedral<span class='pagenum'><a name="Page_132" id="Page_132">[Pg 132]</a></span> church, all his
+friends and adherents being specially summoned to hear it.</p>
+
+<p>For a short time more he was imprisoned in Rome; but at length was
+permitted to depart, never more of his own will to return.</p>
+
+<p>He was allowed to go to Siena. Here his daughter wrote consolingly,
+rejoicing at his escape, and saying how joyfully she already recited the
+penitential psalms for him, and so relieved him of that part of his
+sentence.</p>
+
+<p>But the poor girl was herself, by this time, ill&mdash;thoroughly worn out
+with anxiety and terror; she lay, in fact, on what proved to be her
+death-bed. Her one wish was to see her dearest lord and father, so she
+calls him, once more. The wish was granted. His prison was changed, by
+orders from Rome, from Siena to Arcetri, and once more father and
+daughter embraced. Six days after this she died.</p>
+
+<p>The broken-hearted old man now asks for permission to go to live in
+Florence, but is met with the stern answer that he is to stay at
+Arcetri, is not to go out of the house, is not to receive visitors, and
+that if he asks for more favours, or transgresses the commands laid upon
+him, he is liable to be haled back to Rome and cast into a dungeon.
+These harsh measures were dictated, not by cruelty, but by the fear of
+his still spreading heresy by conversation, and so he was to be kept
+isolated.</p>
+
+<p>Idle, however, he was not and could not be. He often complains that his
+head is too busy for his body. In the enforced solitude of Arcetri he
+was composing those dialogues on motion which are now reckoned his
+greatest and most solid achievement. In these the true laws of motion
+are set forth for the first time (see page 167). One more astronomical
+discovery also he was to make&mdash;that of the moon's libration.</p>
+
+<p>And then there came one more crushing blow. His eyes became inflamed and
+painful&mdash;the sight of one of them<span class='pagenum'><a name="Page_133" id="Page_133">[Pg 133]</a></span> failed, the other soon went; he
+became totally blind. But this, being a heaven-sent infliction, he could
+bear with resignation, though it must have been keenly painful to a
+solitary man of his activity. "Alas!" says he, in one of his letters,
+"your dear friend and servant is totally blind. Henceforth this heaven,
+this universe, which by wonderful observations I had enlarged a hundred
+and a thousand times beyond the conception of former ages, is shrunk for
+me into the narrow space which I myself fill in it. So it pleases God;
+it shall therefore please me also."</p>
+
+<p>He was now allowed an amanuensis, and the help of his pupils Torricelli,
+Castelli, and Viviani, all devotedly attached to him, and Torricelli
+very famous after him. Visitors also were permitted, after approval by a
+Jesuit supervisor; and under these circumstances many visited him, among
+them a man as immortal as himself&mdash;John Milton, then only twenty-nine,
+travelling in Italy. Surely a pathetic incident, this meeting of these
+two great men&mdash;the one already blind, the other destined to become so.
+No wonder that, as in his old age he dictated his masterpiece, the
+thoughts of the English poet should run on the blind sage of Tuscany,
+and the reminiscence of their conversation should lend colour to the
+poem.</p>
+
+<p>Well, it were tedious to follow the petty annoyances and troubles to
+which Galileo was still subject&mdash;how his own son was set to see that no
+unauthorized procedure took place, and that no heretic visitors were
+admitted; how it was impossible to get his new book printed till long
+afterwards; and how one form of illness after another took possession of
+him. The merciful end came at last, and at the age of seventy-eight he
+was released from the Inquisition.</p>
+
+<p>They wanted to deny him burial&mdash;they did deny him a monument; they
+threatened to cart his bones away from Florence if his friends attempted
+one. And so they hoped that he and his work might be forgotten.</p>
+
+<p><span class='pagenum'><a name="Page_134" id="Page_134">[Pg 134]</a></span></p><p>Poor schemers! Before the year was out an infant was born in
+Lincolnshire, whose destiny it was to round and complete and carry
+forward the work of their victim, so that, until man shall cease from
+the planet, neither the work nor its author shall have need of a
+monument.</p>
+
+<hr style='width: 20%;' />
+
+<p>Here might I end, were it not that the same kind of struggle as went on
+fiercely in the seventeenth century is still smouldering even now. Not
+in astronomy indeed, as then; nor yet in geology, as some fifty years
+ago; but in biology mainly&mdash;perhaps in other subjects. I myself have
+heard Charles Darwin spoken of as an atheist and an infidel, the theory
+of evolution assailed as unscriptural, and the doctrine of the ascent of
+man from a lower state of being, as opposed to the fall of man from some
+higher condition, denied as impious and un-Christian.</p>
+
+<p>Men will not learn by the past; still they brandish their feeble weapons
+against the truths of Nature, as if assertions one way or another could
+alter fact, or make the thing other than it really is. As Galileo said
+before his spirit was broken, "In these and other positions certainly no
+man doubts but His Holiness the Pope hath always an absolute power of
+admitting or condemning them; but it is not in the power of any creature
+to make them to be true or false, or otherwise than of their own nature
+and in fact they are."</p>
+
+<p>I know nothing of the views of any here present; but I have met educated
+persons who, while they might laugh at the men who refused to look
+through a telescope lest they should learn something they did not like,
+yet also themselves commit the very same folly. I have met<span class='pagenum'><a name="Page_135" id="Page_135">[Pg 135]</a></span> persons who
+utterly refuse to listen to any view concerning the origin of man other
+than that of a perfect prim&aelig;val pair in a garden, and I am constrained
+to say this much: Take heed lest some prophet, after having excited your
+indignation at the follies and bigotry of a bygone generation, does not
+turn upon you with the sentence, "Thou art the man."</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_136" id="Page_136">[Pg 136]</a></span></p>
+<h4><a name="SUMMARY_OF_FACTS_FOR_LECTURE_VI" id="SUMMARY_OF_FACTS_FOR_LECTURE_VI"></a>SUMMARY OF FACTS FOR LECTURE VI</h4>
+
+<p class="center"><i>Science before Newton</i></p>
+
+
+<p><i>Dr. Gilbert</i>, of Colchester, Physician to Queen Elizabeth, was an
+excellent experimenter, and made many discoveries in magnetism and
+electricity. He was contemporary with Tycho Brah&eacute;, and lived from 1540
+to 1603.</p>
+
+<p><i>Francis Bacon</i>, Lord Verulam, 1561&ndash;1626, though a brilliant writer, is
+not specially important as regards science. He was not a scientific man,
+and his rules for making discoveries, or methods of induction, have
+never been consciously, nor often indeed unconsciously, followed by
+discoverers. They are not in fact practical rules at all, though they
+were so intended. His really strong doctrines are that phenomena must be
+studied direct, and that variations in the ordinary course of nature
+must be induced by aid of experiment; but he lacked the scientific
+instinct for pursuing these great truths into detail and special cases.
+He sneered at the work and methods of both Gilbert and Galileo, and
+rejected the Copernican theory as absurd. His literary gifts have
+conferred on him an artificially high scientific reputation, especially
+in England; at the same time his writings undoubtedly helped to make
+popular the idea of there being new methods for investigating Nature,
+and, by insisting on the necessity for freedom from preconceived ideas
+and opinions, they did much to release men from the bondage of
+Aristotelian authority and scholastic tradition.</p>
+
+<p>The greatest name between Galileo and Newton is that of Descartes.</p>
+
+<p><i>Ren&eacute; Descartes</i> was born at La Haye in Touraine, 1596, and died at
+Stockholm in 1650. He did important work in mathematics, physics,
+anatomy, and philosophy. Was greatest as a philosopher and
+mathematician. At the age of twenty-one he served as a volunteer under
+Prince Maurice of Nassau, but spent most of his later life in Holland.
+His famous <i>Discourse on Method</i> appeared at Leyden in 1637, and his
+<i>Principia</i> at Amsterdam in 1644; great pains being taken to avoid the
+condemnation of the Church.</p>
+
+<p>Descartes's main scientific achievement was the application of algebra
+to geometry; his most famous speculation was the "theory of vortices,"
+invented to account for the motion of planets. He also made many
+discoveries in optics and physiology. His best known immediate pupils
+were the Princess Elizabeth of Bohemia, and Christina, Queen of Sweden.</p>
+
+<p>He founded a distinct school of thought (the Cartesian), and was the
+precursor of the modern mathematical method of investigating science,
+just as Galileo and Gilbert were the originators of the modern
+experimental method.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_137" id="Page_137">[Pg 137]</a></span></p>
+<h3><a name="LECTURE_VI" id="LECTURE_VI"></a>LECTURE VI</h3>
+
+<h5>DESCARTES AND HIS THEORY OF VORTICES</h5>
+
+
+<p><span class="smcap">After</span> the dramatic life we have been considering in the last two
+lectures, it is well to have a breathing space, to look round on what
+has been accomplished, and to review the state of scientific thought,
+before proceeding to the next great era. For we are still in the early
+morning of scientific discovery: the dawn of the modern period, faintly
+heralded by Copernicus, brought nearer by the work of Tycho and Kepler,
+and introduced by the discoveries of Galileo&mdash;the dawn has occurred, but
+the sun is not yet visible. It is hidden by the clouds and mists of the
+long night of ignorance and prejudice. The light is sufficient, indeed,
+to render these earth-born vapours more visible: it is not sufficient to
+dispel them. A generation of slow and doubtful progress must pass,
+before the first ray of sunlight can break through the eastern clouds
+and the full orb of day itself appear.</p>
+
+<p>It is this period of hesitating progress and slow leavening of men's
+ideas that we have to pass through in this week's lecture. It always
+happens thus: the assimilation of great and new ideas is always a slow
+and gradual process: there is no haste either here or in any other
+department of Nature. <i>Die Zeit ist unendlich lang.</i> Steadily the forces
+work, sometimes seeming to accomplish<span class='pagenum'><a name="Page_138" id="Page_138">[Pg 138]</a></span> nothing; sometimes even the
+motion appears retrograde; but in the long run the destined end is
+reached, and the course, whether of a planet or of men's thoughts about
+the universe, is permanently altered. Then, the controversy was about
+the <i>earth's</i> place in the universe; now, if there be any controversy of
+the same kind, it is about <i>man's</i> place in the universe; but the
+process is the same: a startling statement by a great genius or prophet,
+general disbelief, and, it may be, an attitude of hostility, gradual
+acceptance by a few, slow spreading among the many, ending in universal
+acceptance and faith often as unquestioning and unreasoning as the old
+state of unfaith had been. Now the process is comparatively speedy:
+twenty years accomplishes a great deal: then it was tediously slow, and
+a century seemed to accomplish very little. Periodical literature may be
+responsible for some waste of time, but it certainly assists the rapid
+spread of ideas. The rate with which ideas are assimilated by the
+general public cannot even now be considered excessive, but how much
+faster it is than it was a few centuries ago may be illustrated by the
+attitude of the public to Darwinism now, twenty-five years after <i>The
+Origin of Species</i>, as compared with their attitude to the Copernican
+system a century after <i>De Revolutionibus</i>. By the way, it is, I know,
+presumptuous for me to have an opinion, but I cannot hear Darwin
+compared to or mentioned along with Newton without a shudder. The stage
+in which he found biology seems to me far more comparable with the
+Ptolemaic era in astronomy, and he himself to be quite fairly comparable
+to Copernicus.</p>
+
+<p>Let us proceed to summarize the stage at which the human race had
+arrived at the epoch with which we are now dealing.</p>
+
+<p>The Copernican view of the solar system had been stated, restated,
+fought, and insisted on; a chain of brilliant telescopic discoveries had
+made it popular and<span class='pagenum'><a name="Page_139" id="Page_139">[Pg 139]</a></span> accessible to all men of any intelligence:
+henceforth it must be left to slowly percolate and sink into the minds
+of the people. For the nations were waking up now, and were accessible
+to new ideas. England especially was, in some sort, at the zenith of its
+glory; or, if not at the zenith, was in that full flush of youth and
+expectation and hope which is stronger and more prolific of great deeds
+and thoughts than a maturer period.</p>
+
+<p>A common cause against a common and detested enemy had roused in the
+hearts of Englishmen a passion of enthusiasm and patriotism; so that the
+mean elements of trade, their cheating yard-wands, were forgotten for a
+time; the Armada was defeated, and the nation's true and conscious adult
+life began. Commerce was now no mere struggle for profit and hard
+bargains; it was full of the spirit of adventure and discovery; a new
+world had been opened up; who could tell what more remained unexplored?
+Men awoke to the splendour of their inheritance, and away sailed Drake
+and Frobisher and Raleigh into the lands of the West.</p>
+
+<p>For literature, you know what a time it was. The author of <i>Hamlet</i> and
+<i>Othello</i> was alive: it is needless to say more. And what about science?
+The atmosphere of science is a more quiet and less stirring one; it
+thrives best when the fever of excitement is allayed; it is necessarily
+a later growth than literature. Already, however, our second great man
+of science was at work in a quiet country town&mdash;second in point of time,
+I mean, Roger Bacon being the first. Dr. Gilbert, of Colchester, was the
+second in point of time, and the age was ripening for the time when
+England was to be honoured with such a galaxy of scientific
+luminaries&mdash;Hooke and Boyle and Newton&mdash;as the world had not yet known.</p>
+
+<p>Yes, the nations were awake. "In all directions," as Draper says,
+"Nature was investigated: in all directions<span class='pagenum'><a name="Page_140" id="Page_140">[Pg 140]</a></span> new methods of examination
+were yielding unexpected and beautiful results. On the ruins of its
+ivy-grown cathedrals Ecclesiasticism [or Scholasticism], surprised and
+blinded by the breaking day, sat solemnly blinking at the light and life
+about it, absorbed in the recollection of the night that had passed,
+dreaming of new phantoms and delusions in its wished-for return, and
+vindictively striking its talons at any derisive assailant who
+incautiously approached too near."</p>
+
+<p>Of the work of Gilbert there is much to say; so there is also of Roger
+Bacon, whose life I am by no means sure I did right in omitting. But
+neither of them had much to do with astronomy, and since it is in
+astronomy that the most startling progress was during these centuries
+being made, I have judged it wiser to adhere mainly to the pioneers in
+this particular department.</p>
+
+<p>Only for this reason do I pass Gilbert with but slight mention. He knew
+of the Copernican theory and thoroughly accepted it (it is convenient to
+speak of it as the Copernican theory, though you know that it had been
+considerably improved in detail since the first crude statement by
+Copernicus), but he made in it no changes. He was a cultivated
+scientific man, and an acute experimental philosopher; his main work lay
+in the domain of magnetism and electricity. The phenomena connected with
+the mariner's compass had been studied somewhat by Roger Bacon; and they
+were now examined still more thoroughly by Gilbert, whose treatise <i>De
+Magnete</i>, marks the beginning of the science of magnetism.</p>
+
+<p>As an appendix to that work he studied the phenomenon of amber, which
+had been mentioned by Thales. He resuscitated this little fact after its
+burial of 2,200 years, and greatly extended it. He it was who invented
+the name electricity&mdash;I wish it had been a shorter one. Mankind invents
+names much better than do philosophers. What can be better than "heat,"
+"light," "sound"?<span class='pagenum'><a name="Page_141" id="Page_141">[Pg 141]</a></span> How favourably they compare with electricity,
+magnetism, galvanism, electro-magnetism, and magneto-electricity! The
+only long-established monosyllabic name I know invented by a philosopher
+is "gas"&mdash;an excellent attempt, which ought to be imitated.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a></p>
+
+<p>Of Lord Bacon, who flourished about the same time (a little later), it
+is necessary to say something, because many persons are under the
+impression that to him and his <i>Novum Organon</i> the reawakening of the
+world, and the overthrow of Aristotelian tradition, are mainly due. His
+influence, however, has been exaggerated. I am not going to enter into a
+discussion of the <i>Novum Organon</i>, and the mechanical methods which he
+propounded as certain to evolve truth if patiently pursued; for this is
+what he thought he was doing&mdash;giving to the world an infallible recipe
+for discovering truth, with which any ordinarily industrious man could
+make discoveries by means of collection and discrimination of instances.
+You will take my statement for what it is worth, but I assert this: that
+many of the methods which Bacon lays down are not those which the
+experience of mankind has found to be serviceable; nor are they such as
+a scientific man would have thought of devising.</p>
+
+<p>True it is that a real love and faculty for science are born in a man,
+and that to the man of scientific capacity rules of procedure are
+unnecessary; his own intuition is sufficient, or he has mistaken his
+vocation,&mdash;but that is not my point. It is not that Bacon's methods are
+useless because the best men do not need them; if they had been founded
+on a careful study of the methods actually employed, though it might be
+unconsciously employed, by scientific men&mdash;as the methods of induction,
+stated long after by John Stuart<span class='pagenum'><a name="Page_142" id="Page_142">[Pg 142]</a></span> Mill, were founded&mdash;then, no doubt,
+their statement would have been a valuable service and a great thing to
+accomplish. But they were not this. They are the ideas of a brilliant
+man of letters, writing in an age when scientific research was almost
+unknown, about a subject in which he was an amateur. I confess I do not
+see how he, or John Stuart Mill, or any one else, writing in that age,
+could have formulated the true rules of philosophizing; because the
+materials and information were scarcely to hand. Science and its methods
+were only beginning to grow. No doubt it was a brilliant attempt. No
+doubt also there are many good and true points in the statement,
+especially in his insistence on the attitude of free and open candour
+with which the investigation of Nature should be approached. No doubt
+there was much beauty in his allegories of the errors into which men
+were apt to fall&mdash;the <i>idola</i> of the market-place, of the tribe, of the
+theatre, and of the den; but all this is literature, and on the solid
+progress of science may be said to have had little or no effect.
+Descartes's <i>Discourse on Method</i> was a much more solid production.</p>
+
+<p>You will understand that I speak of Bacon purely as a scientific man. As
+a man of letters, as a lawyer, a man of the world, and a statesman, he
+is beyond any criticism of mine. I speak only of the purely scientific
+aspect of the <i>Novum Organon</i>. <i>The Essays</i> and <i>The Advancement of
+Learning</i> are masterly productions; and as a literary man he takes high
+rank.</p>
+
+<p>The over-praise which, in the British Isles, has been lavished upon his
+scientific importance is being followed abroad by what may be an
+unnecessary amount of detraction. This is always the worst of setting up
+a man on too high a pinnacle; some one has to undertake the ungrateful
+task of pulling him down again. Justus von Liebig addressed himself to
+this task with some vigour in his <i>Reden und Abhandlung</i> (Leipzig,
+1874), where he quotes from<span class='pagenum'><a name="Page_143" id="Page_143">[Pg 143]</a></span> Bacon a number of suggestions for absurd
+experimentation.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a></p>
+
+<p>The next paragraph I read, not because I endorse it, but because it is
+always well to hear both sides of a question. You have probably been
+long accustomed to read over-estimates of Bacon's importance, and
+extravagant laudation of his writings as making an epoch in science;
+hear what Draper says on the opposite side:&mdash;<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a></p>
+
+<div class="blockquot"><p>"The more closely we examine the writings of Lord Bacon, the more
+unworthy does he seem to have been of the great reputation which
+has been awarded to him. The popular delusion to which he owes so
+much originated at a time when the history of science was unknown.
+They who first brought him into notice knew nothing of the old
+school of Alexandria. This boasted founder of a new philosophy
+could not comprehend, and would not accept, the greatest of all
+scientific doctrines when it was plainly set before his eyes.</p>
+
+<p>"It has been represented that the invention of the true method of
+physical science was an amusement of Bacon's hours of relaxation
+from the more laborious studies of law, and duties of a Court.</p>
+
+<p>"His chief admirers have been persons of a literary turn, who have
+an idea that scientific discoveries are accomplished by a
+mechanico-mental operation. Bacon never produced any great
+practical result himself, no great physicist has ever made any use
+of his method. He has had the same to do with the development of
+modern science that the inventor of the orrery has had to do with
+the discovery of the mechanism of the world. Of all the important<span class='pagenum'><a name="Page_144" id="Page_144">[Pg 144]</a></span>
+physical discoveries, there is not one which shows that its author
+made it by the Baconian instrument.</p>
+
+<p>"Newton never seems to have been aware that he was under any
+obligation to Bacon. Archimedes, and the Alexandrians, and the
+Arabians, and Leonardo da Vinci did very well before he was born;
+the discovery of America by Columbus and the circumnavigation by
+Magellan can hardly be attributed to him, yet they were the
+consequences of a truly philosophical reasoning. But the
+investigation of Nature is an affair of genius, not of rules. No
+man can invent an <i>organon</i> for writing tragedies and epic poems.
+Bacon's system is, in its own terms, an idol of the theatre. It
+would scarcely guide a man to a solution of the riddle of &AElig;lia
+L&aelig;lia Crispis, or to that of the charade of Sir Hilary.</p>
+
+<p>"Few scientific pretenders have made more mistakes than Lord Bacon.
+He rejected the Copernican system, and spoke insolently of its
+great author; he undertook to criticize adversely Gilbert's
+treatise <i>De Magnete</i>; he was occupied in the condemnation of any
+investigation of final causes, while Harvey was deducing the
+circulation of the blood from Aquapendente's discovery of the
+valves in the veins; he was doubtful whether instruments were of
+any advantage, while Galileo was investigating the heavens with the
+telescope. Ignorant himself of every branch of mathematics, he
+presumed that they were useless in science but a few years before
+Newton achieved by their aid his immortal discoveries.</p>
+
+<p>"It is time that the sacred name of philosophy should be severed
+from its long connection with that of one who was a pretender in
+science, a time-serving politician, an insidious lawyer, a corrupt
+judge, a treacherous friend, a bad man."</p></div>
+
+<p>This seems to me a depreciation as excessive as are the eulogies
+commonly current. The truth probably lies somewhere between the two
+extremes. It is unfair to judge Bacon's methods by thinking of physical
+science in its present stage. To realise his position we must think of a
+subject still in its very early infancy, one in which the advisability
+of applying experimental methods is still doubted; one which has been
+studied by means of books<span class='pagenum'><a name="Page_145" id="Page_145">[Pg 145]</a></span> and words and discussion of normal instances,
+instead of by collection and observation of the unusual and irregular,
+and by experimental production of variety. If we think of a subject
+still in this infantile and almost pre-scientific stage, Bacon's words
+and formul&aelig; are far from inapplicable; they are, within their
+limitations, quite necessary and wholesome. A subject in this stage,
+strange to say, exists,&mdash;psychology; now hesitatingly beginning to
+assume its experimental weapons amid a stifling atmosphere of distrust
+and suspicion. Bacon's lack of the modern scientific instinct must be
+admitted, but he rendered humanity a powerful service in directing it
+from books to nature herself, and his genius is indubitable. A judicious
+account of his life and work is given by Prof. Adamson, in the
+<i>Encyclop&aelig;dia Britannica</i>, and to this article I now refer you.</p>
+
+<hr style='width: 10%;' />
+
+<p>Who, then, was the man of first magnitude filling up the gap in
+scientific history between the death of Galileo and the maturity of
+Newton? Unknown and mysterious are the laws regulating the appearance of
+genius. We have passed in review a Pole, a Dane, a German, and an
+Italian,&mdash;the great man is now a Frenchman, Ren&eacute; Descartes, born in
+Touraine, on the 31st of March, 1596.</p>
+
+<p>His mother died at his birth; the father was of no importance, save as
+the owner of some landed property. The boy was reared luxuriously, and
+inherited a fair fortune. Nearly all the men of first rank, you notice,
+were born well off. Genius born to poverty might, indeed, even then
+achieve name and fame&mdash;as we see in the case of Kepler&mdash;but it was
+terribly handicapped. Handicapped it is still, but far less than of old;
+and we may hope it will become gradually still less so as enlightenment
+proceeds, and the tremendous moment of great men to a nation is more
+clearly and actively perceived.</p>
+
+<p>It is possible for genius, when combined with strong character, to
+overcome all obstacles, and reach the highest<span class='pagenum'><a name="Page_146" id="Page_146">[Pg 146]</a></span> eminence, but the
+struggle must be severe; and the absence of early training and
+refinement during the receptive years of youth must be a lifelong
+drawback.</p>
+
+<p>Descartes had none of these drawbacks; life came easily to him, and, as
+a consequence perhaps, he never seems to have taken it quite seriously.
+Great movements and stirring events were to him opportunities for the
+study of men and manners; he was not the man to court persecution, nor
+to show enthusiasm for a losing or struggling cause.</p>
+
+<p>In this, as in many other things, he was imbued with a very modern
+spirit, a cynical and sceptical spirit, which, to an outside and
+superficial observer like myself, seems rather rife just now.</p>
+
+<p>He was also imbued with a phase of scientific spirit which you sometimes
+still meet with, though I believe it is passing away, viz. an uncultured
+absorption in his own pursuits, and some feeling of contempt for
+classical and literary and &aelig;sthetic studies.</p>
+
+<p>In politics, art, and history he seems to have had no interest. He was a
+spectator rather than an actor on the stage of the world; and though he
+joined the army of that great military commander Prince Maurice of
+Nassau, he did it not as a man with a cause at heart worth fighting for,
+but precisely in the spirit in which one of our own gilded youths would
+volunteer in a similar case, as a good opportunity for frolic and for
+seeing life.</p>
+
+<p>He soon tired of it and withdrew&mdash;at first to gay society in Paris. Here
+he might naturally have sunk into the gutter with his companions, but
+for a great mental shock which became the main epoch and turning-point
+of his life, the crisis which diverted him from frivolity to
+seriousness. It was a purely intellectual emotion, not excited by
+anything in the visible or tangible world; nor could it be called
+conversion in the common acceptation of that term. He tells us that on
+the 10th of November, 1619, at the age of twenty-four, a brilliant idea
+flashed upon him&mdash;the first idea, namely, of<span class='pagenum'><a name="Page_147" id="Page_147">[Pg 147]</a></span> his great and powerful
+mathematical method, of which I will speak directly; and in the flush of
+it he foresaw that just as geometers, starting with a few simple and
+evident propositions or axioms, ascend by a long and intricate ladder of
+reasoning to propositions more and more abstruse, so it might be
+possible to ascend from a few data, to all the secrets and facts of the
+universe, by a process of mathematical reasoning.</p>
+
+<p>"Comparing the mysteries of Nature with the laws of mathematics, he
+dared to hope that the secrets of both could be unlocked with the same
+key."</p>
+
+<p>That night he lapsed gradually into a state of enthusiasm, in which he
+saw three dreams or visions, which he interpreted at the time, even
+before waking, to be revelations from the Spirit of Truth to direct his
+future course, as well as to warn him from the sins he had already
+committed.</p>
+
+<p>His account of the dreams is on record, but is not very easy to follow;
+nor is it likely that a man should be able to convey to others any
+adequate idea of the deepest spiritual or mental agitation which has
+shaken him to his foundations.</p>
+
+<p>His associates in Paris were now abandoned, and he withdrew, after some
+wanderings, to Holland, where he abode the best part of his life and did
+his real work.</p>
+
+<p>Even now, however, he took life easily. He recommends idleness as
+necessary to the production of good mental work. He worked and meditated
+but a few hours a day: and most of those in bed. He used to think best
+in bed, he said. The afternoon he devoted to society and recreation.
+After supper he wrote letters to various persons, all plainly intended
+for publication, and scrupulously preserved. He kept himself free from
+care, and was most cautious about his health, regarding himself, no
+doubt, as a subject of experiment, and wishful to see how long he could
+prolong his life. At one time he writes to a friend that he shall be
+seriously disappointed if he does not manage to see 100 years.</p>
+
+<p><span class='pagenum'><a name="Page_148" id="Page_148">[Pg 148]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_53" id="Fig_53"></a>
+<img src="images/fig53.jpg" width="400" height="543" alt="Fig. 53." title="" />
+<span class="caption"><span class="smcap">Fig. 53.</span>&mdash;Descartes.</span>
+</div>
+
+<p>This plan of not over-working himself, and limiting the hours devoted to
+serious thought, is one that might perhaps advantageously be followed by
+some over-laborious students of the present day. At any rate it conveys
+a lesson; for the amount of ground covered by Descartes, in a life not
+very long, is extraordinary. He must, however, have had a<span class='pagenum'><a name="Page_149" id="Page_149">[Pg 149]</a></span> singular
+aptitude for scientific work; and the judicious leaven of selfishness
+whereby he was able to keep himself free from care and embarrassments
+must have been a great help to him.</p>
+
+<p>And what did his versatile genius accomplish during his fifty-four years
+of life?</p>
+
+<p>In philosophy, using the term as meaning mental or moral philosophy and
+metaphysics, as opposed to natural philosophy or physics, he takes a
+very high rank, and it is on this that perhaps his greatest fame rests.
+(He is the author, you may remember, of the famous aphorism, "<i>Cogito,
+ergo sum</i>.")</p>
+
+<p>In biology I believe he may be considered almost equally great:
+certainly he spent a great deal of time in dissecting, and he made out a
+good deal of what is now known of the structure of the body, and of the
+theory of vision. He eagerly accepted the doctrine of the circulation of
+the blood, then being taught by Harvey, and was an excellent anatomist.</p>
+
+<p>You doubtless know Professor Huxley's article on Descartes in the <i>Lay
+Sermons</i>, and you perceive in what high estimation he is there held.</p>
+
+<p>He originated the hypothesis that animals are automata, for which indeed
+there is much to be said from some points of view; but he unfortunately
+believed that they were unconscious and non-sentient automata, and this
+belief led his disciples into acts of abominable cruelty. Professor
+Huxley lectured on this hypothesis and partially upheld it not many
+years since. The article is included in his volume called <i>Science and
+Culture</i>.</p>
+
+<p>Concerning his work in mathematics and physics I can speak with more
+confidence. He is the author of the Cartesian system of algebraic or
+analytic geometry, which has been so powerful an engine of research, far
+easier to wield than the old synthetic geometry. Without it Newton could
+never have written the <i>Principia</i>, or made his greatest<span class='pagenum'><a name="Page_150" id="Page_150">[Pg 150]</a></span> discoveries.
+He might indeed have invented it for himself, but it would have consumed
+some of his life to have brought it to the necessary perfection.</p>
+
+<div class="blockquot"><p>The principle of it is the specification of the position of a point
+in a plane by two numbers, indicating say its distance from two
+lines of reference in the plane; like the latitude and longitude of
+a place on the globe. For instance, the two lines of reference
+might be the bottom edge and the left-hand vertical edge of a wall;
+then a point on the wall, stated as being for instance 6 feet along
+and 2 feet up, is precisely determined. These two distances are
+called co-ordinates; horizontal ones are usually denoted by <i>x</i>,
+and vertical ones by <i>y</i>.</p>
+
+<p>If, instead of specifying two things, only one statement is made,
+such as <i>y</i> = 2, it is satisfied by a whole row of points, all the
+points in a horizontal line 2 feet above the ground. Hence <i>y</i> = 2
+may be said to represent that straight line, and is called the
+equation to that straight line. Similarly <i>x</i> = 6 represents a
+vertical straight line 6 feet (or inches or some other unit) from
+the left-hand edge. If it is asserted that <i>x</i> = 6 and <i>y</i> = 2,
+only one point can be found to satisfy both conditions, viz. the
+crossing point of the above two straight lines.</p>
+
+<p>Suppose an equation such as <i>x</i> = <i>y</i> to be given. This also is
+satisfied by a row of points, viz. by all those that are
+equidistant from bottom and left-hand edges. In other words, <i>x</i> =
+<i>y</i> represents a straight line slanting upwards at 45&deg;. The
+equation <i>x</i> = 2<i>y</i> represents another straight line with a
+different angle of slope, and so on. The equation <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup>
+= 36 represents a circle of radius 6. The equation 3<i>x</i><sup>2</sup> +
+4<i>y</i><sup>2</sup> = 25 represents an ellipse; and in general every algebraic
+equation that can be written down, provided it involve only two
+variables, <i>x</i> and <i>y</i>, represents some curve in a plane; a curve
+moreover that can be drawn, or its properties completely
+investigated without drawing, from the equation. Thus algebra is
+wedded to geometry, and the investigation of geometric relations by
+means of algebraic equations is called analytical geometry, as
+opposed to the old Euclidian or synthetic mode of treating the
+subject by reasoning consciously directed to the subject by help of
+figures.</p>
+
+<p>If there be three variables&mdash;<i>x</i>, <i>y</i>, and <i>z</i>,&mdash;instead of only
+two, an equation among them represents not a curve in a plane but a
+surface in space; the three variables corresponding to the three
+dimensions of space: length, breadth, and thickness.</p>
+
+<p>An equation with four variables usually requires space of four
+dimensions for its geometrical interpretation, and so on.</p>
+
+<p><span class='pagenum'><a name="Page_151" id="Page_151">[Pg 151]</a></span></p><p>Thus geometry can not only be reasoned about in a more mechanical
+and therefore much easier, manner, but it can be extended into
+regions of which we have and can have no direct conception, because
+we are deficient in sense organs for accumulating any kind of
+experience in connexion with such ideas. </p></div>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_54" id="Fig_54"></a>
+<img src="images/fig54.jpg" width="350" height="622" alt="Fig. 54." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 54.</span>&mdash;The eye diagram. [From Descartes' <i>Principia</i>.]
+Three external points are shown depicted on the retina: the image being
+appreciated by a representation of the brain.</div>
+</div>
+
+<p>In physics proper Descartes' tract on optics is of considerable
+historical interest. He treats all the subjects he takes up in an able
+and original manner.</p>
+
+<p>In Astronomy he is the author of that famous and long upheld theory, the
+doctrine of vortices.</p>
+
+<p><span class='pagenum'><a name="Page_152" id="Page_152">[Pg 152]</a></span></p><p>He regarded space as a plenum full of an all-pervading fluid. Certain
+portions of this fluid were in a state of whirling motion, as in a
+whirlpool or eddy of water; and each planet had its own eddy, in which
+it was whirled round and round, as a straw is caught and whirled in a
+common whirlpool. This idea he works out and elaborates very fully,
+applying it to the system of the world, and to the explanation of all
+the motions of the planets.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_55" id="Fig_55"></a>
+<img src="images/fig55.jpg" width="350" height="572" alt="Fig. 55." title="" />
+<span class="caption"><span class="smcap">Fig. 55.</span>&mdash;Descartes's diagram of vortices, from his
+<i>Principia</i>.</span>
+</div>
+
+<p>This system evidently supplied a void in men's minds, left vacant by the
+overthrow of the Ptolemaic system, and<span class='pagenum'><a name="Page_153" id="Page_153">[Pg 153]</a></span> it was rapidly accepted. In the
+English Universities it held for a long time almost undisputed sway; it
+was in this faith that Newton was brought up.</p>
+
+<p>Something was felt to be necessary to keep the planets moving on their
+endless round; the <i>primum mobile</i> of Ptolemy had been stopped; an angel
+was sometimes assigned to each planet to carry it round, but though a
+widely diffused belief, this was a fantastic and not a serious
+scientific one. Descartes's vortices seemed to do exactly what was
+wanted.</p>
+
+<p>It is true they had no connexion with the laws of Kepler. I doubt
+whether he knew about the laws of Kepler; he had not much opinion of
+other people's work; he read very little&mdash;found it easier to think. (He
+travelled through Florence once when Galileo was at the height of his
+renown without calling upon or seeing him.) In so far as the motion of a
+planet was not circular, it had to be accounted for by the jostling and
+crowding and distortion of the vortices.</p>
+
+<p>Gravitation he explained by a settling down of bodies toward the centre
+of each vortex; and cohesion by an absence of relative motion tending to
+separate particles of matter. He "can imagine no stronger cement."</p>
+
+<p>The vortices, as Descartes imagined them, are not now believed in. Are
+we then to regard the system as absurd and wholly false? I do not see
+how we can do this, when to this day philosophers are agreed in
+believing space to be completely full of fluid, which fluid is certainly
+capable of vortex motion, and perhaps everywhere does possess that
+motion. True, the now imagined vortices are not the large whirls of
+planetary size, they are rather infinitesimal whirls of less than atomic
+dimensions; still a whirling fluid is believed in to this day, and many
+are seeking to deduce all the properties of matter (rigidity,
+elasticity, cohesion gravitation, and the rest) from it.</p>
+
+<p>Further, although we talk glibly about gravitation and magnetism, and so
+on, we do not really know what they are.<span class='pagenum'><a name="Page_154" id="Page_154">[Pg 154]</a></span> Progress is being made, but we
+do not yet properly know. Much, overwhelmingly much, remains to be
+discovered, and it ill-behoves us to reject any well-founded and
+long-held theory as utterly and intrinsically false and absurd. The more
+one gets to know, the more one perceives a kernel of truth even in the
+most singular statements; and scientific men have learned by experience
+to be very careful how they lop off any branch of the tree of knowledge,
+lest as they cut away the dead wood they lose also some green shoot,
+some healthy bud of unperceived truth.</p>
+
+<p>However, it may be admitted that the idea of a Cartesian vortex in
+connexion with the solar system applies, if at all, rather to an
+earlier&mdash;its nebulous&mdash;stage, when the whole thing was one great whirl,
+ready to split or shrink off planetary rings at their appropriate
+distances.</p>
+
+<p>Soon after he had written his great work, the <i>Principia Mathematica</i>,
+and before he printed it, news reached him of the persecution and
+recantation of Galileo. "He seems to have been quite thunderstruck at
+the tidings," says Mr. Mahaffy, in his <i>Life of Descartes</i>.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> "He had
+started on his scientific journeys with the firm determination to enter
+into no conflict with the Church, and to carry out his system of pure
+mathematics and physics without ever meddling with matters of faith. He
+was rudely disillusioned as to the possibility of this severance. He
+wrote at once&mdash;apparently, November 20th, 1633&mdash;to Mersenne to say he
+would on no account publish his work&mdash;nay, that he had at first resolved
+to burn all his papers, for that he would never prosecute philosophy at
+the risk of being censured by his Church. 'I could hardly have
+believed,' he says, 'that an Italian, and in favour with the Pope as I
+hear, could be considered criminal for nothing else than for seeking to
+establish the earth's motion; though I know it has formerly been
+censured by some Cardinals. But I thought I had heard that since then it
+was constantly<span class='pagenum'><a name="Page_155" id="Page_155">[Pg 155]</a></span> being taught, even at Rome; and I confess that if the
+opinion of the earth's movement is false, all the foundations of my
+philosophy are so also, because it is demonstrated clearly by them. It
+is so bound up with every part of my treatise that I could not sever it
+without making the remainder faulty; and although I consider all my
+conclusions based on very certain and clear demonstrations, I would not
+for all the world sustain them against the authority of the Church.'"</p>
+
+<p>Ten years later, however, he did publish the book, for he had by this
+time hit on an ingenious compromise. He formally denied that the earth
+moved, and only asserted that it was carried along with its water and
+air in one of those larger motions of the celestial ether which produce
+the diurnal and annual revolutions of the solar system. So, just as a
+passenger on the deck of a ship might be called stationary, so was the
+earth. He gives himself out therefore as a follower of Tycho rather than
+of Copernicus, and says if the Church won't accept this compromise he
+must return to the Ptolemaic system; but he hopes they won't compel him
+to do that, seeing that it is manifestly untrue.</p>
+
+<p>This elaborate deference to the powers that be did not indeed save the
+work from being ultimately placed upon the forbidden list by the Church,
+but it saved himself, at any rate, from annoying persecution. He was
+not, indeed, at all willing to be persecuted, and would no doubt have at
+once withdrawn anything they wished. I should be sorry to call him a
+time-server, but he certainly had plenty of that worldly wisdom in which
+some of his predecessors had been so lamentably deficient. Moreover, he
+was really a sceptic, and cared nothing at all about the Church or its
+dogmas. He knew the Church's power, however, and the advisability of
+standing well with it: he therefore professed himself a Catholic, and
+studiously kept his science and his Christianity distinct.</p>
+
+<p><span class='pagenum'><a name="Page_156" id="Page_156">[Pg 156]</a></span></p><p>In saying that he was a sceptic you must not understand that he was in
+the least an atheist. Very few men are; certainly Descartes never
+thought of being one. The term is indeed ludicrously inapplicable to
+him, for a great part of his philosophy is occupied with what he
+considers a rigorous proof of the existence of the Deity.</p>
+
+<p>At the age of fifty-three he was sent for to Stockholm by Christina,
+Queen of Sweden, a young lady enthusiastically devoted to study of all
+kinds and determined to surround her Court with all that was most famous
+in literature and science. Thither, after hesitation, Descartes went. He
+greatly liked royalty, but he dreaded the cold climate. Born in
+Touraine, a Swedish winter was peculiarly trying to him, especially as
+the energetic Queen would have lessons given her at five o'clock in the
+morning. She intended to treat him well, and was immensely taken with
+him; but this getting up at five o'clock on a November morning, to a man
+accustomed all his life to lie in bed till eleven, was a cruel hardship.
+He was too much of a courtier, however, to murmur, and the early morning
+audience continued. His health began to break down: he thought of
+retreating, but suddenly he gave way and became delirious. The Queen's
+physician attended him, and of course wanted to bleed him. This, knowing
+all he knew of physiology, sent him furious, and they could do nothing
+with him. After some days he became quiet, was bled twice, and gradually
+sank, discoursing with great calmness on his approaching death, and duly
+fortified with all the rites of the Catholic Church.</p>
+
+<p>His general method of research was as nearly as possible a purely
+deductive one:&mdash;<i>i.e.</i>, after the manner of Euclid he starts with a few
+simple principles, and then, by a chain of reasoning, endeavours to
+deduce from them their consequences, and so to build up bit by bit an
+edifice<span class='pagenum'><a name="Page_157" id="Page_157">[Pg 157]</a></span> of connected knowledge. In this he was the precursor of Newton.
+This method, when rigorously pursued, is the most powerful and
+satisfactory of all, and results in an ordered province of science far
+superior to the fragmentary conquests of experiment. But few indeed are
+the men who can handle it safely and satisfactorily: and none without
+continual appeals to experiment for verification. It was through not
+perceiving the necessity for verification that he erred. His importance
+to science lies not so much in what he actually discovered as in his
+anticipation of the right conditions for the solution of problems in
+physical science. He in fact made the discovery that Nature could after
+all be interrogated mathematically&mdash;a fact that was in great danger of
+remaining unknown. For, observe, that the mathematical study of Nature,
+the discovery of truth with a piece of paper and a pen, has a perilous
+similarity at first sight to the straw-thrashing subtleties of the
+Greeks, whose methods of investigating nature by discussing the meaning
+of words and the usage of language and the necessities of thought, had
+proved to be so futile and unproductive.</p>
+
+<p>A reaction had set in, led by Galileo, Gilbert, and the whole modern
+school of experimental philosophers, lasting down to the present
+day:&mdash;men who teach that the only right way of investigating Nature is
+by experiment and observation.</p>
+
+<p>It is indeed a very right and an absolutely necessary way; but it is not
+the only way. A foundation of experimental fact there must be; but upon
+this a great structure of theoretical deduction can be based, all
+rigidly connected together by pure reasoning, and all necessarily as
+true as the premises, provided no mistake is made. To guard against the
+possibility of mistake and oversight, especially oversight, all
+conclusions must sooner or later be brought to the test of experiment;
+and if disagreeing therewith, the theory itself must be re-examined,<span class='pagenum'><a name="Page_158" id="Page_158">[Pg 158]</a></span>
+and the flaw discovered, or else the theory must be abandoned.</p>
+
+<p>Of this grand method, quite different from the gropings in the dark of
+Kepler&mdash;this method, which, in combination with experiment, has made
+science what it now is&mdash;this which in the hands of Newton was to lead to
+such stupendous results, we owe the beginning and early stages to Ren&eacute;
+Descartes.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_159" id="Page_159">[Pg 159]</a></span></p>
+<h4><a name="SUMMARY_OF_FACTS_FOR_LECTURES_VII_AND_VIII" id="SUMMARY_OF_FACTS_FOR_LECTURES_VII_AND_VIII"></a>SUMMARY OF FACTS FOR LECTURES VII AND VIII</h4>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Newton's Contemporaries">
+<tr>
+ <td align='left'>Otto Guericke</td>
+ <td align='left'>1602&ndash;1686</td>
+</tr>
+<tr>
+ <td align='left'>Hon. Robert Boyle</td>
+ <td align='left'>1626&ndash;1691</td>
+</tr>
+<tr>
+ <td align='left'>Huyghens</td>
+ <td align='left'>1629&ndash;1695</td>
+</tr>
+<tr>
+ <td align='left'>Christopher Wren</td>
+ <td align='left'>1632&ndash;1723</td>
+</tr>
+<tr>
+ <td align='left'>Robert Hooke</td>
+ <td align='left'>1635&ndash;1702</td>
+</tr>
+<tr>
+ <td align='left'><span class="smcap">Newton</span></td>
+ <td align='left'>1642&ndash;1727</td>
+</tr>
+<tr>
+ <td align='left'>Edmund Halley</td>
+ <td align='left'>1656&ndash;1742</td>
+</tr>
+<tr>
+ <td align='left'>James Bradley</td>
+ <td align='left'>1692&ndash;1762</td></tr>
+</table></div>
+
+<p class="center"><i>Chronology of Newton's Life.</i></p>
+
+
+<p>Isaac Newton was born at Woolsthorpe, near Grantham, Lincolnshire, on
+Christmas Day, 1642. His father, a small freehold farmer, also named
+Isaac, died before his birth. His mother, <i>n&eacute;e</i> Hannah Ayscough, in two
+years married a Mr. Smith, rector of North Witham, but was again left a
+widow in 1656. His uncle, W. Ayscough, was rector of a near parish and a
+graduate of Trinity College, Cambridge. At the age of fifteen Isaac was
+removed from school at Grantham to be made a farmer of, but as it seemed
+he would not make a good one his uncle arranged for him to return to
+school and thence to Cambridge, where he entered Trinity College as a
+sub-sizar in 1661. Studied Descartes's geometry. Found out a method of
+infinite series in 1665, and began the invention of Fluxions. In the
+same year and the next he was driven from Cambridge by the plague. In
+1666, at Woolsthorpe, the apple fell. In 1667 he was elected a fellow of
+his college, and in 1669 was specially noted as possessing an
+unparalleled genius by Dr. Barrow, first Lucasian Professor of
+Mathematics. The same year Dr. Barrow retired from his chair in favour
+of Newton, who was thus elected at the age of twenty-six. He lectured
+first on optics with great success. Early in 1672 he was elected a
+Fellow of the Royal Society, and communicated his researches in optics,
+his reflecting telescope, and his discovery of the compound nature of
+white light. Annoying controversies arose; but he nevertheless
+contributed a good many other most important papers in optics, including
+observations in diffraction, and colours of thin plates. He also
+invented the modern sextant. In 1672 a letter from Paris was read at the
+Royal Society concerning a new and accurate determination of the size of
+the earth by Picard. When Newton heard of it he began the <i>Principia</i>,
+working in silence. In 1684 arose a<span class='pagenum'><a name="Page_160" id="Page_160">[Pg 160]</a></span> discussion between Wren, Hooke, and
+Halley concerning the law of inverse square as applied to gravity and
+the path it would cause the planets to describe. Hooke asserted that he
+had a solution, but he would not produce it. After waiting some time for
+it Halley went to Cambridge to consult Newton on the subject, and thus
+discovered the existence of the first part of the <i>Principia</i>, wherein
+all this and much more was thoroughly worked out. On his representations
+to the Royal Society the manuscript was asked for, and when complete was
+printed and published in 1687 at Halley's expense. While it was being
+completed Newton and seven others were sent to uphold the dignity of the
+University, before the Court of High Commission and Judge Jeffreys,
+against a high-handed action of James II. In 1682 he was sent to
+Parliament, and was present at the coronation of William and Mary. Made
+friends with Locke. In 1694 Montague, Lord Halifax, made him Warden, and
+in 1697 Master, of the Mint. Whiston succeeded him as Lucasian
+Professor. In 1693 the method of fluxions was published. In 1703 Newton
+was made President of the Royal Society, and held the office to the end
+of his life. In 1705 he was knighted by Anne. In 1713 Cotes helped him
+to bring out a new edition of the <i>Principia</i>, completed as we now have
+it. On the 20th of March 1727, he died: having lived from Charles I. to
+George II.</p>
+
+
+<p class="center"><br /><span class="smcap">The Laws of Motion, discovered by Galileo, stated by Newton.</span></p>
+
+<p><i>Law 1.</i>&mdash;If no force acts on a body in motion, it continues to move
+uniformly in a straight line.</p>
+
+<p><i>Law 2.</i>&mdash;If force acts on a body, it produces a change of motion
+proportional to the force and in the same direction.</p>
+
+<p><i>Law 3.</i>&mdash;When one body exerts force on another, that other reacts with
+equal force upon the one.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_161" id="Page_161">[Pg 161]</a></span></p>
+<h3><a name="LECTURE_VII" id="LECTURE_VII"></a>LECTURE VII</h3>
+
+<h5>SIR ISAAC NEWTON</h5>
+
+
+<p><span class="smcap">The</span> little hamlet of Woolsthorpe lies close to the village of
+Colsterworth, about six miles south of Grantham, in the county of
+Lincoln. In the manor house of Woolsthorpe, on Christmas Day, 1642, was
+born to a widowed mother a sickly infant who seemed not long for this
+world. Two women who were sent to North Witham to get some medicine for
+him scarcely expected to find him alive on their return. However, the
+child lived, became fairly robust, and was named Isaac, after his
+father. What sort of a man this father was we do not know. He was what
+we may call a yeoman, that most wholesome and natural of all classes. He
+owned the soil he tilled, and his little estate had already been in the
+family for some hundred years. He was thirty-six when he died, and had
+only been married a few months.</p>
+
+<p>Of the mother, unfortunately, we know almost as little. We hear that she
+was recommended by a parishioner to the Rev. Barnabas Smith, an old
+bachelor in search of a wife, as "the widow Newton&mdash;an extraordinary
+good woman:" and so I expect she was, a thoroughly sensible, practical,
+homely, industrious, middle-class, Mill-on-the-Floss sort of woman.
+However, on her second marriage she went to live at North Witham, and
+her mother, old Mrs. Ayscough, came to superintend the farm at
+Woolsthorpe, and take care of young Isaac.</p>
+
+<p><span class='pagenum'><a name="Page_162" id="Page_162">[Pg 162]</a></span></p><p>By her second marriage his mother acquired another piece of land, which
+she settled on her first son; so Isaac found himself heir to two little
+properties, bringing in a rental of about &pound;80 a year.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_56" id="Fig_56"></a>
+<img src="images/fig56.jpg" width="350" height="301" alt="Fig. 56." title="" />
+<span class="caption"><span class="smcap">Fig. 56.</span>&mdash;Manor-house of Woolsthorpe.</span>
+</div>
+
+<p>He had been sent to a couple of village schools to acquire the ordinary
+accomplishments taught at those places, and for three years to the
+grammar school at Grantham, then conducted by an old gentleman named Mr.
+Stokes. He had not been very industrious at school, nor did he feel
+keenly the fascinations of the Latin Grammar, for he tells us that he
+was the last boy in the lowest class but one. He used to pay much more
+attention to the construction of kites and windmills and waterwheels,
+all of which he made to work very well. He also used to tie paper
+lanterns to the tail of his kite, so as to make the country folk fancy
+they saw a comet, and in general to disport himself as a boy should.</p>
+
+<p>It so happened, however, that he succeeded in thrashing, in fair fight,
+a bigger boy who was higher in the school,<span class='pagenum'><a name="Page_163" id="Page_163">[Pg 163]</a></span> and who had given him a
+kick. His success awakened a spirit of emulation in other things than
+boxing, and young Newton speedily rose to be top of the school.</p>
+
+<p>Under these circumstances, at the age of fifteen, his mother, who had
+now returned to Woolsthorpe, which had been rebuilt, thought it was time
+to train him for the management of his land, and to make a farmer and
+grazier of him. The boy was doubtless glad to get away from school, but
+he did not take kindly to the farm&mdash;especially not to the marketing at
+Grantham. He and an old servant were sent to Grantham every week to buy
+and sell produce, but young Isaac used to leave his old mentor to do all
+the business, and himself retire to an attic in the house he had lodged
+in when at school, and there bury himself in books.</p>
+
+<p>After a time he didn't even go through the farce of visiting Grantham at
+all; but stopped on the road and sat under a hedge, reading or making
+some model, until his companion returned.</p>
+
+<p>We hear of him now in the great storm of 1658, the storm on the day
+Cromwell died, measuring the force of the wind by seeing how far he
+could jump with it and against it. He also made a water-clock and set it
+up in the house at Grantham, where it kept fairly good time so long as
+he was in the neighbourhood to look after it occasionally.</p>
+
+<p>At his own home he made a couple of sundials on the side of the wall (he
+began by marking the position of the sun by the shadow of a peg driven
+into the wall, but this gradually developed into a regular dial) one of
+which remained of use for some time; and was still to be seen in the
+same place during the first half of the present century, only with the
+gnomon gone. In 1844 the stone on which it was carved was carefully
+extracted and presented to the Royal Society, who preserve it in their
+library. The letters WTON roughly carved on it are barely visible.</p>
+
+<p>All these pursuits must have been rather trying to his poor mother, and
+she probably complained to her brother,<span class='pagenum'><a name="Page_164" id="Page_164">[Pg 164]</a></span> the rector of Burton Coggles:
+at any rate this gentleman found master Newton one morning under a hedge
+when he ought to have been farming. But as he found him working away at
+mathematics, like a wise man he persuaded his sister to send the boy
+back to school for a short time, and then to Cambridge. On the day of
+his finally leaving school old Mr. Stokes assembled the boys, made them
+a speech in praise of Newton's character and ability, and then dismissed
+him to Cambridge.</p>
+
+<p>At Trinity College a new world opened out before the country-bred lad.
+He knew his classics passably, but of mathematics and science he was
+ignorant, except through the smatterings he had picked up for himself.
+He devoured a book on logic, and another on Kepler's Optics, so fast
+that his attendance at lectures on these subjects became unnecessary. He
+also got hold of a Euclid and of Descartes's Geometry. The Euclid seemed
+childishly easy, and was thrown aside, but the Descartes baffled him for
+a time. However, he set to it again and again and before long mastered
+it. He threw himself heart and soul into mathematics, and very soon made
+some remarkable discoveries. First he discovered the binomial theorem:
+familiar now to all who have done any algebra, unintelligible to others,
+and therefore I say nothing about it. By the age of twenty-one or two he
+had begun his great mathematical discovery of infinite series and
+fluxions&mdash;now known by the name of the Differential Calculus. He wrote
+these things out and must have been quite absorbed in them, but it never
+seems to have occurred to him to publish them or tell any one about
+them.</p>
+
+<p>In 1664 he noticed some halos round the moon, and, as his manner was, he
+measured their angles&mdash;the small ones 3 and 5 degrees each, the larger
+one 22&deg;&middot;35. Later he gave their theory.</p>
+
+<div class="blockquot"><p>Small coloured halos round the moon are often seen, and are said to
+be a sign of rain. They are produced by the action of minute<span class='pagenum'><a name="Page_165" id="Page_165">[Pg 165]</a></span>
+globules of water or cloud particles upon light, and are brightest
+when the particles are nearly equal in size. They are not like the
+rainbow, every part of which is due to light that has entered a
+raindrop, and been refracted and reflected with prismatic
+separation of colours; a halo is caused by particles so small as to
+be almost comparable with the size of waves of light, in a way
+which is explained in optics under the head "diffraction." It may
+be easily imitated by dusting an ordinary piece of window-glass
+over with lycopodium, placing a candle near it, and then looking at
+the candle-flame through the dusty glass from a fair distance. Or
+you may look at the image of a candle in a dusted looking-glass.
+Lycopodium dust is specially suitable, for its granules are
+remarkably equal in size. The large halo, more rarely seen, of
+angular radius 22&deg;&middot;35, is due to another cause again, and is a
+prismatic effect, although it exhibits hardly any colour. The angle
+22&frac12;&deg; is characteristic of refraction in crystals with angles of
+60&deg; and refractive index about the same as water; in other words
+this halo is caused by ice crystals in the higher regions of the
+atmosphere. </p></div>
+
+<p>He also the same year observed a comet, and sat up so late watching it
+that he made himself ill. By the end of the year he was elected to a
+scholarship and took his B.A. degree. The order of merit for that year
+never existed or has not been kept. It would have been interesting, not
+as a testimony to Newton, but to the sense or non-sense of the
+examiners. The oldest Professorship of Mathematics at the University of
+Cambridge, the Lucasian, had not then been long founded, and its first
+occupant was Dr. Isaac Barrow, an eminent mathematician, and a kind old
+man. With him Newton made good friends, and was helpful in preparing a
+treatise on optics for the press. His help is acknowledged by Dr. Barrow
+in the preface, which states that he had corrected several errors and
+made some capital additions of his own. Thus we see that, although the
+chief part of his time was devoted to mathematics, his attention was
+already directed to both optics and astronomy. (Kepler, Descartes,
+Galileo, all combined some optics with astronomy. Tycho and the old ones
+combined alchemy; Newton dabbled in this also.)</p>
+
+<p><span class='pagenum'><a name="Page_166" id="Page_166">[Pg 166]</a></span></p><p>Newton reached the age of twenty-three in 1665, the year of the Great
+Plague. The plague broke out in Cambridge as well as in London, and the
+whole college was sent down. Newton went back to Woolsthorpe, his mind
+teeming with ideas, and spent the rest of this year and part of the next
+in quiet pondering. Somehow or other he had got hold of the notion of
+centrifugal force. It was six years before Huyghens discovered and
+published the laws of centrifugal force, but in some quiet way of his
+own Newton knew about it and applied the idea to the motion of the
+planets.</p>
+
+<p>We can almost follow the course of his thoughts as he brooded and
+meditated on the great problem which had taxed so many previous
+thinkers,&mdash;What makes the planets move round the sun? Kepler had
+discovered how they moved, but why did they so move, what urged them?</p>
+
+<p>Even the "how" took a long time&mdash;all the time of the Greeks, through
+Ptolemy, the Arabs, Copernicus, Tycho: circular motion, epicycles, and
+excentrics had been the prevailing theory. Kepler, with his marvellous
+industry, had wrested from Tycho's observations the secret of their
+orbits. They moved in ellipses with the sun in one focus. Their rate of
+description of area, not their speed, was uniform and proportional to
+time.</p>
+
+<p>Yes, and a third law, a mysterious law of unintelligible import, had
+also yielded itself to his penetrating industry&mdash;a law the discovery of
+which had given him the keenest delight, and excited an outburst of
+rapture&mdash;viz. that there was a relation between the distances and the
+periodic times of the several planets. The cubes of the distances were
+proportional to the squares of the times for the whole system. This law,
+first found true for the six primary planets, he had also extended,
+after Galileo's discovery, to the four secondary planets, or satellites
+of Jupiter (<a href="#Page_81">p. 81</a>).</p>
+
+<p>But all this was working in the dark&mdash;it was only the first step&mdash;this
+empirical discovery of facts; the facts were so, but how came they so?
+What made the planets<span class='pagenum'><a name="Page_167" id="Page_167">[Pg 167]</a></span> move in this particular way? Descartes's vortices
+was an attempt, a poor and imperfect attempt, at an explanation. It had
+been hailed and adopted throughout Europe for want of a better, but it
+did not satisfy Newton. No, it proceeded on a wrong tack, and Kepler had
+proceeded on a wrong tack in imagining spokes or rays sticking out from
+the sun and driving the planets round like a piece of mechanism or mill
+work. For, note that all these theories are based on a wrong idea&mdash;the
+idea, viz., that some force is necessary to maintain a body in motion.
+But this was contrary to the laws of motion as discovered by Galileo.
+You know that during his last years of blind helplessness at Arcetri,
+Galileo had pondered and written much on the laws of motion, the
+foundation of mechanics. In his early youth, at Pisa, he had been
+similarly occupied; he had discovered the pendulum, he had refuted the
+Aristotelians by dropping weights from the leaning tower (which we must
+rejoice that no earthquake has yet injured), and he had returned to
+mechanics at intervals all his life; and now, when his eyes were useless
+for astronomy, when the outer world has become to him only a prison to
+be broken by death, he returns once more to the laws of motion, and
+produces the most solid and substantial work of his life.</p>
+
+<p>For this is Galileo's main glory&mdash;not his brilliant exposition of the
+Copernican system, not his flashes of wit at the expense of a moribund
+philosophy, not his experiments on floating bodies, not even his
+telescope and astronomical discoveries&mdash;though these are the most taking
+and dazzling at first sight. No; his main glory and title to immortality
+consists in this, that he first laid the foundation of mechanics on a
+firm and secure basis of experiment, reasoning, and observation. He
+first discovered the true Laws of Motion.</p>
+
+<p>I said little of this achievement in my lecture on him; for the work was
+written towards the end of his life, and I had no time then. But I knew
+I should have to return to it before we came to Newton, and here we are.</p>
+
+<p><span class='pagenum'><a name="Page_168" id="Page_168">[Pg 168]</a></span></p><p>You may wonder how the work got published when so many of his
+manuscripts were destroyed. Horrible to say, Galileo's own son destroyed
+a great bundle of his father's manuscripts, thinking, no doubt, thereby
+to save his own soul. This book on mechanics was not burnt, however. The
+fact is it was rescued by one or other of his pupils, Toricelli or
+Viviani, who were allowed to visit him in his last two or three years;
+it was kept by them for some time, and then published surreptitiously in
+Holland. Not that there is anything in it bearing in any visible way on
+any theological controversy; but it is unlikely that the Inquisition
+would have suffered it to pass notwithstanding.</p>
+
+<p>I have appended to the summary preceding this lecture (<a href="#Page_160">p. 160</a>) the three
+axioms or laws of motion discovered by Galileo. They are stated by
+Newton with unexampled clearness and accuracy, and are hence known as
+Newton's laws, but they are based on Galileo's work. The first is the
+simplest; though ignorance of it gave the ancients a deal of trouble. It
+is simply a statement that force is needed to change the motion of a
+body; <i>i.e.</i> that if no force act on a body it will continue to move
+uniformly both in speed and direction&mdash;in other words, steadily, in a
+straight line. The old idea had been that some force was needed to
+maintain motion. On the contrary, the first law asserts, some force is
+needed to destroy it. Leave a body alone, free from all friction or
+other retarding forces, and it will go on for ever. The planetary motion
+through empty space therefore wants no keeping up; it is not the motion
+that demands a force to maintain it, it is the curvature of the path
+that needs a force to produce it continually. The motion of a planet is
+approximately uniform so far as speed is concerned, but it is not
+constant in direction; it is nearly a circle. The real force needed is
+not a propelling but a deflecting force.</p>
+
+<p>The second law asserts that when a force acts, the motion changes,
+either in speed or in direction, or both, at a pace proportional to the
+magnitude of the force, and in the same<span class='pagenum'><a name="Page_169" id="Page_169">[Pg 169]</a></span> direction as that in which the
+force acts. Now since it is almost solely in direction that planetary
+motion alters, a deflecting force only is needed; a force at right
+angles to the direction of motion, a force normal to the path.
+Considering the motion as circular, a force along the radius, a radial
+or centripetal force, must be acting continually. Whirl a weight round
+and round by a bit of elastic, the elastic is stretched; whirl it
+faster, it is stretched more. The moving mass pulls at the elastic&mdash;that
+is its centrifugal force; the hand at the centre pulls also&mdash;that is
+centripetal force.</p>
+
+<p>The third law asserts that these two forces are equal, and together
+constitute the tension in the elastic. It is impossible to have one
+force alone, there must be a pair. You can't push hard against a body
+that offers no resistance. Whatever force you exert upon a body, with
+that same force the body must react upon you. Action and reaction are
+always equal and opposite.</p>
+
+<p>Sometimes an absurd difficulty is felt with respect to this, even by
+engineers. They say, "If the cart pulls against the horse with precisely
+the same force as the horse pulls the cart, why should the cart move?"
+Why on earth not? The cart moves because the horse pulls it, and because
+nothing else is pulling it back. "Yes," they say, "the cart is pulling
+back." But what is it pulling back? Not itself, surely? "No, the horse."
+Yes, certainly the cart is pulling at the horse; if the cart offered no
+resistance what would be the good of the horse? That is what he is for,
+to overcome the pull-back of the cart; but nothing is pulling the cart
+back (except, of course, a little friction), and the horse is pulling it
+forward, hence it goes forward. There is no puzzle at all when once you
+realise that there are two bodies and two forces acting, and that one
+force acts on each body.<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a></p>
+
+<p>If, indeed, two balanced forces acted on one body that would be in
+equilibrium, but the two equal forces contemplated<span class='pagenum'><a name="Page_170" id="Page_170">[Pg 170]</a></span> in the third law act
+on two different bodies, and neither is in equilibrium.</p>
+
+<p>So much for the third law, which is extremely simple, though it has
+extraordinarily far-reaching consequences, and when combined with a
+denial of "action at a distance," is precisely the principle of the
+Conservation of Energy. Attempts at perpetual motion may all be regarded
+as attempts to get round this "third law."</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_57" id="Fig_57"></a>
+<img src="images/fig57.jpg" width="400" height="202" alt="Fig. 57." title="" />
+<span class="caption"><span class="smcap">Fig. 57.</span></span>
+</div>
+
+<div class="blockquot"><p>On the subject of the <i>second</i> law a great deal more has to be said
+before it can be in any proper sense even partially appreciated,
+but a complete discussion of it would involve a treatise on
+mechanics. It is <i>the</i> law of mechanics. One aspect of it we must
+attend to now in order to deal with the motion of the planets, and
+that is the fact that the change of motion of a body depends solely
+and simply on the force acting, and not at all upon what the body
+happens to be doing at the time it acts. It may be stationary, or
+it may be moving in any direction; that makes no difference.</p>
+
+<p>Thus, referring back to the summary preceding <a href="#SUMMARY_OF_FACTS_FOR_LECTURES_IV_AND_V">Lecture IV</a>, it is
+there stated that a dropped body falls 16 feet in the first second,
+that in two seconds it falls 64 feet, and so on, in proportion to
+the square of the time. So also will it be the case with a thrown
+body, but the drop must be reckoned from its line of motion&mdash;the
+straight line which, but for gravity, it would describe.</p>
+
+<p>Thus a stone thrown from <i>O</i> with the velocity <i>OA</i> would in one
+second find itself at <i>A</i>, in two seconds at <i>B</i>, in three seconds
+at <i>C</i>, and so on, in accordance with the first law of motion, if
+no force acted. But if gravity acts it will have fallen 16 feet by
+the time it<span class='pagenum'><a name="Page_171" id="Page_171">[Pg 171]</a></span> would have got to <i>A</i>, and so will find itself at <i>P</i>.
+In two seconds it will be at <i>Q</i>, having fallen a vertical height
+of 64 feet; in three seconds it will be at <i>R</i>, 144 feet below <i>C</i>;
+and so on. Its actual path will be a curve, which in this case is a
+parabola. (<a href="#Fig_57">Fig. 57.</a>)</p>
+
+<p>If a cannon is pointed horizontally over a level plain, the cannon
+ball will be just as much affected by gravity as if it were
+dropped, and so will strike the plain at the same instant as
+another which was simply dropped where it started. One ball may
+have gone a mile and the other only dropped a hundred feet or so,
+but the time needed by both for the vertical drop will be the same.
+The horizontal motion of one is an extra, and is due to the powder.</p>
+
+<p>As a matter of fact the path of a projectile in vacuo is only
+approximately a parabola. It is instructive to remember that it is
+really an ellipse with one focus very distant, but not at infinity.
+One of its foci is the centre of the earth. A projectile is really
+a minute satellite of the earth's, and in vacuo it accurately obeys
+all Kepler's laws. It happens not to be able to complete its orbit,
+because it was started inconveniently close to the earth, whose
+bulk gets in its way; but in that respect the earth is to be
+reckoned as a gratuitous obstruction, like a target, but a target
+that differs from most targets in being hard to miss. </p></div>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_58" id="Fig_58"></a>
+<img src="images/fig58.jpg" width="350" height="344" alt="Fig. 58." title="" />
+<span class="caption"><span class="smcap">Fig. 58.</span></span>
+</div>
+
+<div class="blockquot"><p>Now consider circular motion in the same way, say a ball whirled
+round by a string. (<a href="#Fig_58">Fig. 58.</a>)</p>
+
+<p>Attending to the body at <i>O</i>, it is for an instant moving towards
+<i>A</i>, and if no force acted it would get to <i>A</i> in a time which for
+brevity we may call a second. But a force, the pull of the string,
+is continually drawing it towards <i>S</i>, and so it really finds
+itself at <i>P</i>,<span class='pagenum'><a name="Page_172" id="Page_172">[Pg 172]</a></span> having described the circular arc <i>OP</i>, which may
+be considered to be compounded of, and analyzable into the
+rectilinear motion <i>OA</i> and the drop <i>AP</i>. At <i>P</i> it is for an
+instant moving towards <i>B</i>, and the same process therefore carries
+it to <i>Q</i>; in the third second it gets to <i>R</i>; and so on: always
+falling, so to speak, from its natural rectilinear path, towards
+the centre, but never getting any nearer to the centre.</p>
+
+<p>The force with which it has thus to be constantly pulled in towards
+the centre, or, which is the same thing, the force with which it is
+tugging at whatever constraint it is that holds it in, is
+<i>mv<sup>2</sup>/r</i>; where <i>m</i> is the mass of the particle, <i>v</i> its
+velocity, and <i>r</i> the radius of its circle of movement. This is the
+formula first given by Huyghens for centrifugal force.</p>
+
+<p>We shall find it convenient to express it in terms of the time of
+one revolution, say <i>T</i>. It is easily done, since plainly T =
+circumference/speed = <i>2&#960;r/v</i>; so the above expression for
+centrifugal force becomes <i>4&#960;<sup>2</sup>mr/T<sup>2</sup></i>.</p>
+
+<p>As to the fall of the body towards the centre every microscopic
+unit of time, it is easily reckoned. For by Euclid III. 36, and
+<a href="#Fig_58">Fig. 58</a>, <i>AP.AA' = AO<sup>2</sup></i>. Take <i>A</i> very near <i>O</i>, then <i>OA = vt</i>,
+and <i>AA' = 2r</i>; so <i>AP = v<sup>2</sup>t<sup>2</sup>/2r = 2&#960;<sup>2</sup>r
+t<sup>2</sup>/T<sup>2</sup></i>; or the fall per second is <i>2&#960;<sup>2</sup>r/T<sup>2</sup></i>,
+<i>r</i> being its distance from the centre, and <i>T</i> its time of going
+once round.</p>
+
+<p>In the case of the moon for instance, <i>r</i> is 60 earth radii; more
+exactly 60&middot;2; and <i>T</i> is a lunar month, or more precisely 27 days,
+7 hours, 43 minutes, and 11&frac12; seconds. Hence the moon's
+deflection from the tangential or rectilinear path every minute
+comes out as very closely 16 feet (the true size of the earth being
+used). </p></div>
+
+<p>Returning now to the case of a small body revolving round a big one, and
+assuming a force directly proportional to the mass of both bodies, and
+inversely proportional to the square of the distance between them:
+<i>i.e.</i> assuming the known force of gravity, it is</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="Two revolving bodies equation">
+<tr class='tr5'>
+<td align='center'><i>V Mm</i></td>
+</tr>
+<tr>
+ <td align='center'><i>r<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p class="noin">where <i>V</i> is a constant, called the gravitation constant, to be
+determined by experiment.</p>
+
+<p><span class='pagenum'><a name="Page_173" id="Page_173">[Pg 173]</a></span></p><p>If this is the centripetal force pulling a planet or satellite in, it
+must be equal to the centrifugal force of this latter, viz. (see above).</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="Centrifugal force equation">
+<tr class='tr5'>
+<td align='center'><i>4&#960;<sup>2</sup>mr</i></td>
+</tr>
+<tr>
+ <td align='center'><i>T<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p>Equate the two together, and at once we get</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Centrifugal and Revolving body combined equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>r<sup>3</sup></i></td>
+ <td align='center' rowspan='2'>&nbsp;&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>V</i></td>
+ <td class='center' rowspan='2'>&nbsp;M&nbsp;;</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>T<sup>2</sup></i></td>
+ <td align='center'><i>4&#960;<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p class="noin">or, in words, the cube of the distance divided by the square of the
+periodic time for every planet or satellite of the system under
+consideration, will be constant and proportional to the mass of the
+central body.</p>
+
+<p>This is Kepler's third law, with a notable addition. It is stated above
+for circular motion only, so as to avoid geometrical difficulties, but
+even so it is very instructive. The reason of the proportion between
+<i>r<sup>3</sup></i> and <i>T<sup>2</sup></i> is at once manifest; and as soon as the constant <i>V</i>
+became known, <i>the mass of the central body</i>, the sun in the case of a
+planet, the earth in the case of the moon, Jupiter in the case of his
+satellites, was at once determined.</p>
+
+<p>Newton's reasoning at this time might, however, be better displayed
+perhaps by altering the order of the steps a little, as thus:&mdash;</p>
+
+<p>The centrifugal force of a body is proportional to <i>r<sup>3</sup>/T<sup>2</sup></i>, but by
+Kepler's third law <i>r<sup>3</sup>/T<sup>2</sup></i> is constant for all the planets,
+reckoning <i>r</i> from the sun. Hence the centripetal force needed to hold
+in all the planets will be a single force emanating from the sun and
+varying inversely with the square of the distance from that body.</p>
+
+<p>Such a force is at once necessary and sufficient. Such a force would
+explain the motion of the planets.</p>
+
+<p>But then all this proceeds on a wrong assumption&mdash;that<span class='pagenum'><a name="Page_174" id="Page_174">[Pg 174]</a></span> the planetary
+motion is circular. Will it hold for elliptic orbits? Will an inverse
+square law of force keep a body moving in an elliptic orbit about the
+sun in one focus? This is a far more difficult question. Newton solved
+it, but I do not believe that even he could have solved it, except that
+he had at his disposal two mathematical engines of great power&mdash;the
+Cartesian method of treating geometry, and his own method of Fluxions.
+One can explain the elliptic motion now mathematically, but hardly
+otherwise; and I must be content to state that the double fact is
+true&mdash;viz., that an inverse square law will move the body in an ellipse
+or other conic section with the sun in one focus, and that if a body so
+moves it <i>must</i> be acted on by an inverse square law.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_59" id="Fig_59"></a>
+<img src="images/fig59.jpg" width="350" height="275" alt="Fig. 59." title="" />
+<span class="caption"><span class="smcap">Fig. 59.</span></span>
+</div>
+
+<p>This then is the meaning of the first and third laws of Kepler. What
+about the second? What is the meaning of the equable description of
+areas? Well, that rigorously proves that a planet is acted on by a force
+directed to the centre about which the rate of description of areas is
+equable. It proves, in fact, that the sun is the attracting body, and
+that no other force acts.</p>
+
+<div class="blockquot"><p>For first of all if the first law of motion is obeyed, <i>i.e.</i> if no
+force acts, and if the path be equally subdivided to represent
+equal times, and straight lines be drawn from the divisions to any
+point whatever, all these areas thus enclosed will be equal,
+because they are triangles on equal base and of the same height
+(Euclid, I). See <a href="#Fig_59">Fig. 59</a>; <i>S</i> being any point whatever, and <i>A</i>,
+<i>B</i>, <i>C</i>, successive positions of a body.</p>
+
+<p><span class='pagenum'><a name="Page_175" id="Page_175">[Pg 175]</a></span></p><p>Now at each of the successive instants let the body receive a
+sudden blow in the direction of that same point <i>S</i>, sufficient to
+carry it from <i>A</i> to <i>D</i> in the same time as it would have got to
+<i>B</i> if left alone. The result will be that there will be a
+compromise, and it will really arrive at <i>P</i>, travelling along the
+diagonal of the parallelogram <i>AP</i>. The area its radius vector
+sweeps out is therefore <i>SAP</i>, instead of what it would have been,
+<i>SAB</i>. But then these two areas are equal, because they are
+triangles on the same base <i>AS</i>, and between the same parallels
+<i>BP</i>, <i>AS</i>; for by the parallelogram law <i>BP</i> is parallel to <i>AD</i>.
+Hence the area that would have been described is described, and as
+all the areas were equal in the case of no force, they remain equal
+when the body receives a blow at the end of every equal interval of
+time, <i>provided</i> that every blow is actually directed to <i>S</i>, the
+point to which radii vectores are drawn. </p></div>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_60" id="Fig_60"></a>
+<img src="images/fig60.jpg" width="350" height="287" alt="Fig. 60." title="" />
+<span class="caption"><span class="smcap">Fig. 60.</span></span>
+</div>
+
+<div class="figcenter" style="width: 350px;"><br /><a name="Fig_61" id="Fig_61"></a>
+<img src="images/fig61.jpg" width="350" height="368" alt="Fig. 61." title="" />
+<span class="caption"><span class="smcap">Fig. 61.</span></span>
+</div>
+
+<div class="blockquot"><p>It is instructive to see that it does not hold if the blow is any
+otherwise directed; for instance, as in <a href="#Fig_61">Fig. 61</a>, when the blow is
+along <i>AE</i>, the body finds itself at <i>P</i> at the end of the second<span class='pagenum'><a name="Page_176" id="Page_176">[Pg 176]</a></span>
+interval, but the area <i>SAP</i> is by no means equal to <i>SAB</i>, and
+therefore not equal to <i>SOA</i>, the area swept out in the first
+interval.</p>
+
+<p>In order to modify <a href="#Fig_60">Fig. 60</a> so as to represent continuous motion and
+steady forces, we have to take the sides of the polygon <i>OAPQ</i>,
+&amp;c., very numerous and very small; in the limit, infinitely
+numerous and infinitely small. The path then becomes a curve, and
+the series of blows becomes a steady force directed towards <i>S</i>.
+About whatever point therefore the rate of description of areas is
+uniform, that point and no other must be the centre of all the
+force there is. If there be no force, as in <a href="#Fig_59">Fig. 59</a>, well and good,
+but if there be any force however small not directed towards <i>S</i>,
+then the rate of description of areas about <i>S</i> cannot be uniform.
+Kepler, however, says that the rate of description of areas of each
+planet about the sun is, by Tycho's observations, uniform; hence
+the sun is the centre of all the force that acts on them, and there
+is no other force, not even friction. That is the moral of Kepler's
+second law.</p>
+
+<p>We may also see from it that gravity does not travel like light, so
+as to take time on its journey from sun to planet; for, if it did,
+there would be a sort of aberration, and the force on its arrival
+could no longer be accurately directed to the centre of the sun.
+(See <i>Nature</i>, vol. xlvi., p. 497.) It is a matter for accuracy of
+observation, therefore, to decide whether the minutest trace of
+such deviation can be detected, <i>i.e.</i> within what limits of
+accuracy Kepler's second law is now known to be obeyed.</p>
+
+<p>I will content myself by saying that the limits are extremely
+narrow. [Reference may be made also to <a href="#Page_208">p. 208.</a>] </p></div>
+
+<p>Thus then it became clear to Newton that the whole solar system depended
+on a central force emanating from the sun, and varying inversely with
+the square of the distance from him: for by that hypothesis all the laws
+of Kepler concerning these motions were completely accounted for; and,
+in fact, the laws necessitated the hypothesis and established it as a
+theory.</p>
+
+<p>Similarly the satellites of Jupiter were controlled by a force emanating
+from Jupiter and varying according to the same law. And again our moon
+must be controlled by a force from the earth, decreasing with the
+distance according to the same law.</p>
+
+<p>Grant this hypothetical attracting force pulling the<span class='pagenum'><a name="Page_177" id="Page_177">[Pg 177]</a></span> planets towards
+the sun, pulling the moon towards the earth, and the whole mechanism of
+the solar system is beautifully explained.</p>
+
+<p>If only one could be sure there was such a force! It was one thing to
+calculate out what the effects of such a force would be: it was another
+to be able to put one's finger upon it and say, this is the force that
+actually exists and is known to exist. We must picture him meditating in
+his garden on this want&mdash;an attractive force towards the earth.</p>
+
+<p>If only such an attractive force pulling down bodies to the earth
+existed. An apple falls from a tree. Why, it does exist! There is
+gravitation, common gravity that makes bodies fall and gives them their
+weight.</p>
+
+<p>Wanted, a force tending towards the centre of the earth. It is to hand!</p>
+
+<p>It is common old gravity that had been known so long, that was perfectly
+familiar to Galileo, and probably to Archimedes. Gravity that regulates
+the motion of projectiles. Why should it only pull stones and apples?
+Why should it not reach as high as the moon? Why should it not be the
+gravitation of the sun that is the central force acting on all the
+planets?</p>
+
+<p>Surely the secret of the universe is discovered! But, wait a bit; is it
+discovered? Is this force of gravity sufficient for the purpose? It must
+vary inversely with the square of the distance from the centre of the
+earth. How far is the moon away? Sixty earth's radii. Hence the force of
+gravity at the moon's distance can only be <span class="above">1</span>&#8260;<span class="below">3600</span> of what it is on the
+earth's surface. So, instead of pulling it 16 ft. per second, it should
+pull it <span class="above">16</span>&#8260;<span class="below">3600</span> ft. per second, or 16 ft. a minute.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a> How can one
+decide whether such a force is able to pull the moon the actual amount
+required? To Newton this would seem only like a sum in arithmetic. Out
+with a pencil and paper and reckon how much the moon falls toward the
+earth in every second of<span class='pagenum'><a name="Page_178" id="Page_178">[Pg 178]</a></span> its motion. Is it <span class="above">16</span>&#8260;<span class="below">3600</span>? That is what it
+ought to be: but is it? The size of the earth comes into the
+calculation. Sixty miles make a degree, 360 degrees a circumference.
+This gives as the earth's diameter 6,873 miles; work it out.</p>
+
+<p>The answer is not 16 feet a minute, it is 13&middot;9 feet.</p>
+
+<p>Surely a mistake of calculation?</p>
+
+<p>No, it is no mistake: there is something wrong in the theory, gravity is
+too strong.</p>
+
+<p>Instead of falling toward the earth 5&#8531; hundredths of an inch every
+second, as it would under gravity, the moon only falls 4&#8532; hundredths
+of an inch per second.</p>
+
+<p>With such a discovery in his grasp at the age of twenty-three he is
+disappointed&mdash;the figures do not agree, and he cannot make them agree.
+Either gravity is not the force in action, or else something interferes
+with it. Possibly, gravity does part of the work, and the vortices of
+Descartes interfere with it.</p>
+
+<p>He must abandon the fascinating idea for the time. In his own words, "he
+laid aside at that time any further thought of the matter."</p>
+
+<p>So far as is known, he never mentioned his disappointment to a soul. He
+might, perhaps, if he had been at Cambridge, but he was a shy and
+solitary youth, and just as likely he might not. Up in Lincolnshire, in
+the seventeenth century, who was there for him to consult?</p>
+
+<p>True, he might have rushed into premature publication, after our
+nineteenth century fashion, but that was not his method. Publication
+never seemed to have occurred to him.</p>
+
+<p>His reticence now is noteworthy, but later on it is perfectly
+astonishing. He is so absorbed in making discoveries that he actually
+has to be reminded to tell any one about them, and some one else always
+has to see to the printing and publishing for him.</p>
+
+<p>I have entered thus fully into what I conjecture to be the stages of
+this early discovery of the law of gravitation<span class='pagenum'><a name="Page_179" id="Page_179">[Pg 179]</a></span>, as applicable to the
+heavenly bodies, because it is frequently and commonly misunderstood. It
+is sometimes thought that he discovered the force of gravity; I hope I
+have made it clear that he did no such thing. Every educated man long
+before his time, if asked why bodies fell, would reply just as glibly as
+they do now, "Because the earth attracts them," or "because of the force
+of gravity."</p>
+
+<p>His discovery was that the motions of the solar system were due to the
+action of a central force, directed to the body at the centre of the
+system, and varying inversely with the square of the distance from it.
+This discovery was based upon Kepler's laws, and was clear and certain.
+It might have been published had he so chosen.</p>
+
+<p>But he did not like hypothetical and unknown forces; he tried to see
+whether the known force of gravity would serve. This discovery at that
+time he failed to make, owing to a wrong numerical datum. The size of
+the earth he only knew from the common doctrine of sailors that 60 miles
+make a degree; and that threw him out. Instead of falling 16 feet a
+minute, as it ought under gravity, it only fell 13&middot;9 feet, so he
+abandoned the idea. We do not find that he returned to it for sixteen
+years.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_180" id="Page_180">[Pg 180]</a></span></p>
+<h3><a name="LECTURE_VIII" id="LECTURE_VIII"></a>LECTURE VIII</h3>
+
+<h5>NEWTON AND THE LAW OF GRAVITATION</h5>
+
+
+<p><span class="smcap">We</span> left Newton at the age of twenty-three on the verge of discovering
+the mechanism of the solar system, deterred therefrom only by an error
+in the then imagined size of the earth. He had proved from Kepler's laws
+that a centripetal force directed to the sun, and varying as the inverse
+square of the distance from that body, would account for the observed
+planetary motions, and that a similar force directed to the earth would
+account for the lunar motion; and it had struck him that this force
+might be the very same as the familiar force of gravitation which gave
+to bodies their weight: but in attempting a numerical verification of
+this idea in the case of the moon he was led by the then received notion
+that sixty miles made a degree on the earth's surface into an erroneous
+estimate of the size of the moon's orbit. Being thus baffled in
+obtaining such verification, he laid the matter aside for a time.</p>
+
+<p>The anecdote of the apple we learn from Voltaire, who had it from
+Newton's favourite niece, who with her husband lived and kept house for
+him all his later life. It is very like one of those anecdotes which are
+easily invented and believed in, and very often turn out on scrutiny to
+have no foundation. Fortunately this anecdote is well authenticated, and
+moreover is intrinsically probable; I say fortunately, because it is
+always painful to have to give up these child-learnt anecdotes, like
+Alfred and the cakes<span class='pagenum'><a name="Page_181" id="Page_181">[Pg 181]</a></span> and so on. This anecdote of the apple we need not
+resign. The tree was blown down in 1820 and part of its wood is
+preserved.</p>
+
+<p>I have mentioned Voltaire in connection with Newton's philosophy. This
+acute critic at a later stage did a good deal to popularise it
+throughout Europe and to overturn that of his own countryman Descartes.
+Cambridge rapidly became Newtonian, but Oxford remained Cartesian for
+fifty years or more. It is curious what little hold science and
+mathematics have ever secured in the older and more ecclesiastical
+University. The pride of possessing Newton has however no doubt been the
+main stimulus to the special pursuits of Cambridge.</p>
+
+<p>He now began to turn his attention to optics, and, as was usual with
+him, his whole mind became absorbed in this subject as if nothing else
+had ever occupied him. His cash-book for this time has been discovered,
+and the entries show that he is buying prisms and lenses and polishing
+powder at the beginning of 1667. He was anxious to improve telescopes by
+making more perfect lenses than had ever been used before. Accordingly
+he calculated out their proper curves, just as Descartes had also done,
+and then proceeded to grind them as near as he could to those figures.
+But the images did not please him; they were always blurred and rather
+indistinct.</p>
+
+<p>At length, it struck him that perhaps it was not the lenses but the
+light which was at fault. Perhaps light was so composed that it <i>could</i>
+not be focused accurately to a sharp and definite point. Perhaps the law
+of refraction was not quite accurate, but only an approximation. So he
+bought a prism to try the law. He let in sunlight through a small round
+hole in a window shutter, inserted the prism in the light, and received
+the deflected beam on a white screen; turning the prism about till it
+was deviated as little as possible. The patch on the screen was not a
+round disk, as it would have been without the prism, but was an
+elongated<span class='pagenum'><a name="Page_182" id="Page_182">[Pg 182]</a></span> oval and was coloured at its extremities. Evidently
+refraction was not a simple geometrical deflection of a ray, there was a
+spreading out as well.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_63" id="Fig_63"></a>
+<img src="images/fig63.jpg" width="350" height="228" alt="Fig. 63." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 63.</span>&mdash;A prism not only <i>deviates</i> a beam of sunlight,
+but also spreads it out or <i>disperses</i> it.</span>
+</div>
+
+<p>Why did the image thus spread out? If it were due to irregularities in
+the glass a second prism should rather increase them, but a second prism
+when held in appropriate position was able to neutralise the dispersion
+and to reproduce the simple round white spot without deviation.
+Evidently the spreading out of the beam was connected in some definite
+way with its refraction. Could it be that the light particles after
+passing through the prism travelled in variously curved lines, as
+spinning racquet balls do? To examine this he measured the length of the
+oval patch when the screen was at different distances from the prism,
+and found that the two things were directly proportional to each other.
+Doubling the distance of the screen doubled the length of the patch.
+Hence the rays travelled in straight lines from the prism, and the
+spreading out was due to something that occurred within its substance.
+Could it be that white light was compound, was a mixture of several
+constituents, and that its different constituents were differently bent?
+No sooner thought than tried. Pierce the screen to let one of<span class='pagenum'><a name="Page_183" id="Page_183">[Pg 183]</a></span> the
+constituents through and interpose a second prism in its path. If the
+spreading out depended on the prism only it should spread out just as
+much as before, but if it depended on the complex character of white
+light, this isolated simple constituent should be able to spread out no
+more. It did not spread out any more: a prism had no more dispersive
+power over it; it was deflected by the appropriate amount, but it was
+not analysed into constituents. It differed from sunlight in being
+simple. With many ingenious and beautifully simple experiments, which
+are quoted in full in several books on optics, he clinched the argument
+and established his discovery. White light was not simple but compound.
+It could be sorted out by a prism into an infinite number of constituent
+parts which were differently refracted, and the most striking of which
+Newton named violet, indigo, blue, green, yellow, orange, and red.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_64" id="Fig_64"></a>
+<img src="images/fig64.jpg" width="400" height="140" alt="Fig. 64." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 64.</span>&mdash;A single constituent of white light, obtained
+by the use of perforated screens is capable of no more dispersion.</span>
+</div>
+
+<p>At once the true nature of colour became manifest. Colour resided not in
+the coloured object as had till now been thought, but in the light which
+illuminated it. Red glass for instance adds nothing to sunlight. The
+light does not get dyed red by passing through the glass; all that the
+red glass does is to stop and absorb a large part of the sunlight; it is
+opaque to the larger portion, but it is transparent to that particular
+portion which affects our eyes with the sensation of red. The prism acts
+like a sieve sorting out<span class='pagenum'><a name="Page_184" id="Page_184">[Pg 184]</a></span> the different kinds of light. Coloured media
+act like filters, stopping certain kinds but allowing the rest to go
+through. Leonardo's and all the ancient doctrines of colour had been
+singularly wrong; colour is not in the object but in the light.</p>
+
+<p>Goethe, in his <i>Farbenlehre</i>, endeavoured to controvert Newton, and to
+reinstate something more like the old views; but his failure was
+complete.</p>
+
+<p>Refraction analysed out the various constituents of white light and
+displayed them in the form of a series of overlapping images of the
+aperture, each of a different colour; this series of images we call a
+spectrum, and the operation we now call spectrum analysis. The reason of
+the defect of lenses was now plain: it was not so much a defect of the
+lens as a defect of light. A lens acts by refraction and brings rays to
+a focus. If light be simple it acts well, but if ordinary white light
+fall upon a lens, its different constituents have different foci; every
+bright object is fringed with colour, and nothing like a clear image can
+be obtained.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_65" id="Fig_65"></a>
+<img src="images/fig65.jpg" width="400" height="139" alt="Fig. 65." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 65.</span>&mdash;Showing the boundary rays of a parallel beam
+passing through a lens.</span>
+</div>
+
+<p>A parallel beam passing through a lens becomes conical; but instead of a
+single cone it is a sheaf or nest of cones, all having the edge of the
+lens as base, but each having a different vertex. The violet cone is
+innermost, near the lens, the red cone outermost, while the others lie
+between. Beyond the crossing point or focus the order of cones is
+reversed, as the above figure shows. Only the two marginal rays of the
+beam are depicted.</p>
+
+<p>If a screen be held anywhere nearer the lens than the<span class='pagenum'><a name="Page_185" id="Page_185">[Pg 185]</a></span> place marked 1
+there will be a whitish centre to the patch of light and a red and
+orange fringe or border. Held anywhere beyond the region 2, the border
+of the patch will be blue and violet. Held about 3 the colour will be
+less marked than elsewhere, but nowhere can it be got rid of. Each point
+of an object will be represented in the image not by a point but by a
+coloured patch: a fact which amply explains the observed blurring and
+indistinctness.</p>
+
+<p>Newton measured and calculated the distance between the violet and red
+foci&mdash;VR in the diagram&mdash;and showed that it was <span class="above">1</span>&#8260;<span class="below">50</span>th the diameter of
+the lens. To overcome this difficulty (called chromatic aberration)
+telescope glasses were made small and of very long focus: some of them
+so long that they had no tube, all of them egregiously cumbrous. Yet it
+was with such instruments that all the early discoveries were made. With
+such an instrument, for instance, Huyghens discovered the real shape of
+Saturn's ring.</p>
+
+<p>The defects of refractors seemed irremediable, being founded in the
+nature of light itself. So he gave up his "glass works"; and proceeded
+to think of reflexion from metal specula. A concave mirror forms an
+image just as a lens does, but since it does so without refraction or
+transmission through any substance, there is no accompanying dispersion
+or chromatic aberration.</p>
+
+<p>The first reflecting telescope he made was 1 in. diameter and 6 in.
+long, and magnified forty times. It acted as well as a three or four
+feet refractor of that day, and showed Jupiter's moons. So he made a
+larger one, now in the library of the Royal Society, London, with an
+inscription:</p>
+
+<p>"The first reflecting telescope, invented by Sir Isaac Newton, and made
+with his own hands."</p>
+
+<p>This has been the parent of most of the gigantic telescopes of the
+present day. Fifty years elapsed before it was much improved on, and
+then, first by Hadley and afterwards by Herschel and others, large and
+good reflectors were constructed.</p>
+
+<p><span class='pagenum'><a name="Page_186" id="Page_186">[Pg 186]</a></span></p><p>The largest telescope ever made, that of Lord Rosse, is a Newtonian
+reflector, fifty feet long, six feet diameter, with a mirror weighing
+four tons. The sextant, as used by navigators, was also invented by
+Newton.</p>
+
+<p>The year after the plague, in 1667, Newton returned to Trinity College,
+and there continued his experiments on optics. It is specially to be
+noted that at this time, at the age of twenty-four, Newton had laid the
+foundations of all his greatest discoveries:&mdash;</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_66" id="Fig_66"></a>
+<img src="images/fig66.jpg" width="400" height="403" alt="Fig. 66." title="" />
+<span class="caption"><span class="smcap">Fig. 66.</span>&mdash;Newton&#39;s telescope.</span>
+</div>
+
+<p>The Theory of Fluxions; or, the Differential Calculus.</p>
+
+<p>The Law of Gravitation; or, the complete theory of astronomy.</p>
+
+<p>The compound nature of white light; or, the beginning of Spectrum
+Analysis.</p>
+
+<p><span class='pagenum'><a name="Page_187" id="Page_187">[Pg 187]</a></span></p>
+<div class="figcenter" style="width: 450px;"><a name="Fig_67" id="Fig_67"></a>
+<img src="images/fig67.jpg" width="450" height="363" alt="Fig. 67." title="" />
+<span class="caption"><span class="smcap">Fig. 67.</span>&mdash;The sextant, as now made.</span>
+</div>
+
+<p>His later life was to be occupied in working these incipient discoveries
+out. But the most remarkable thing is that no one knew about any one of
+them. However, he was known as an accomplished young mathematician, and
+was made a fellow of his college. You remember that he had a friend
+there in the person of Dr. Isaac Barrow, first Lucasian Professor of
+Mathematics in the University. It happened, about 1669, that a
+mathematical discovery of some interest was being much discussed, and
+Dr. Barrow happened to mention it to Newton, who said yes, he had worked
+out that and a few other similar things some time ago. He accordingly
+went and fetched some papers to Dr. Barrow, who forwarded them to other
+distinguished mathematicians, and it thus appeared that Newton had
+discovered theorems much more general than this special case that was
+exciting so much interest. Dr. Barrow, being anxious to devote his time
+more particularly to theology, resigned his chair the same year in
+favour of Newton, who was<span class='pagenum'><a name="Page_188" id="Page_188">[Pg 188]</a></span> accordingly elected to the Lucasian
+Professorship, which he held for thirty years. This chair is now the
+most famous in the University, and it is commonly referred to as the
+chair of Newton.</p>
+
+<p>Still, however, his method of fluxions was unknown, and still he did not
+publish it. He lectured first on optics, giving an account of his
+experiments. His lectures were afterwards published both in Latin and
+English, and are highly valued to this day.</p>
+
+<p>The fame of his mathematical genius came to the ears of the Royal
+Society, and a motion was made to get him elected a fellow of that body.
+The Royal Society, the oldest and most famous of all scientific
+societies with a continuous existence, took its origin in some private
+meetings, got up in London by the Hon. Robert Boyle and a few scientific
+friends, during all the trouble of the Commonwealth.</p>
+
+<p>After the restoration, Charles II. in 1662 incorporated it under Royal
+Charter; among the original members being Boyle, Hooke, Christopher
+Wren, and other less famous names. Boyle was a great experimenter, a
+worthy follower of Dr. Gilbert. Hooke began as his assistant, but being
+of a most extraordinary ingenuity he rapidly rose so as to exceed his
+master in importance. Fate has been a little unkind to Hooke in placing
+him so near to Newton; had he lived in an ordinary age he would
+undoubtedly have shone as a star of the first magnitude. With great
+ingenuity, remarkable scientific insight, and consummate experimental
+skill, he stands in many respects almost on a level with Galileo. But it
+is difficult to see stars even of the first magnitude when the sun is
+up, and thus it happens that the name and fame of this brilliant man are
+almost lost in the blaze of Newton. Of Christopher Wren I need not say
+much. He is well known as an architect, but he was a most accomplished
+all-round man, and had a considerable taste and faculty for science.</p>
+
+<p><span class='pagenum'><a name="Page_189" id="Page_189">[Pg 189]</a></span></p><p>These then were the luminaries of the Royal Society at the time we are
+speaking of, and to them Newton's first scientific publication was
+submitted. He communicated to them an account of his reflecting
+telescope, and presented them with the instrument.</p>
+
+<p>Their reception of it surprised him; they were greatly delighted with
+it, and wrote specially thanking him for the communication, and assuring
+him that all right should be done him in the matter of the invention.
+The Bishop of Salisbury (Bishop Burnet) proposed him for election as a
+fellow, and elected he was.</p>
+
+<p>In reply, he expressed his surprise at the value they set on the
+telescope, and offered, if they cared for it, to send them an account of
+a discovery which he doubts not will prove much more grateful than the
+communication of that instrument, "being in my judgment the oddest, if
+not the most considerable detection that has recently been made into the
+operations of Nature."</p>
+
+<p>So he tells them about his optical researches and his discovery of the
+nature of white light, writing them a series of papers which were long
+afterwards incorporated and published as his <i>Optics</i>. A magnificent
+work, which of itself suffices to place its author in the first rank of
+the world's men of science.</p>
+
+<p>The nature of white light, the true doctrine of colour, and the
+differential calculus! besides a good number of minor results&mdash;binomial
+theorem, reflecting telescope, sextant, and the like; one would think it
+enough for one man's life-work, but the masterpiece remains still to be
+mentioned. It is as when one is considering Shakspeare: <i>King Lear</i>,
+<i>Macbeth</i>, <i>Othello</i>,&mdash;surely a sufficient achievement,&mdash;but the
+masterpiece remains.</p>
+
+<p>Comparisons in different departments are but little help perhaps,
+nevertheless it seems to me that in his own department, and considered
+simply as a man of science, Newton towers head and shoulders over, not
+only his<span class='pagenum'><a name="Page_190" id="Page_190">[Pg 190]</a></span> contemporaries&mdash;that is a small matter&mdash;but over every other
+scientific man who has ever lived, in a way that we can find no parallel
+for in other departments. Other nations admit his scientific
+pre-eminence with as much alacrity as we do.</p>
+
+<p>Well, we have arrived at the year 1672 and his election to the Royal
+Society. During the first year of his membership there was read at one
+of the meetings a paper giving an account of a very careful
+determination of the length of a degree (<i>i.e.</i> of the size of the
+earth), which had been made by Picard near Paris. The length of the
+degree turned out to be not sixty miles, but nearly seventy miles. How
+soon Newton heard of this we do not learn&mdash;probably not for some
+years,&mdash;Cambridge was not so near London then as it is now, but
+ultimately it was brought to his notice. Armed with this new datum, his
+old speculation concerning gravity occurred to him. He had worked out
+the mechanics of the solar system on a certain hypothesis, but it had
+remained a hypothesis somewhat out of harmony with apparent fact. What
+if it should turn out to be true after all!</p>
+
+<p>He took out his old papers and began again the calculation. If gravity
+were the force keeping the moon in its orbit, it would fall toward the
+earth sixteen feet every minute. How far did it fall? The newly known
+size of the earth would modify the figures: with intense excitement he
+runs through the working, his mind leaps before his hand, and as he
+perceives the answer to be coming out right, all the infinite meaning
+and scope of his mighty discovery flashes upon him, and he can no longer
+see the paper. He throws down the pen; and the secret of the universe
+is, to one man, known.</p>
+
+<p>But of course it had to be worked out. The meaning might flash upon him,
+but its full detail required years of elaboration; and deeper and deeper
+consequences revealed themselves to him as he proceeded.</p>
+
+<p><span class='pagenum'><a name="Page_191" id="Page_191">[Pg 191]</a></span></p><p>For two years he devoted himself solely to this one object. During
+those years he lived but to calculate and think, and the most ludicrous
+stories are told concerning his entire absorption and inattention to
+ordinary affairs of life. Thus, for instance, when getting up in a
+morning he would sit on the side of the bed half-dressed, and remain
+like that till dinner time. Often he would stay at home for days
+together, eating what was taken to him, but without apparently noticing
+what he was doing.</p>
+
+<p>One day an intimate friend, Dr. Stukely, called on him and found on the
+table a cover laid for his solitary dinner. After waiting a long time,
+Dr. Stukely removed the cover and ate the chicken underneath it,
+replacing and covering up the bones again. At length Newton appeared,
+and after greeting his friend, sat down to dinner, but on lifting the
+cover he said in surprise, "Dear me, I thought I had not dined, but I
+see I have."</p>
+
+<p>It was by this continuous application that the <i>Principia</i> was
+accomplished. Probably nothing of the first magnitude can be
+accomplished without something of the same absorbed unconsciousness and
+freedom from interruption. But though desirable and essential for the
+<i>work</i>, it was a severe tax upon the powers of the <i>man</i>. There is, in
+fact, no doubt that Newton's brain suffered temporary aberration after
+this effort for a short time. The attack was slight, and it has been
+denied; but there are letters extant which are inexplicable otherwise,
+and moreover after a year or two he writes to his friends apologizing
+for strange and disjointed epistles, which he believed he had written
+without understanding clearly what he wrote. The derangement was,
+however, both slight and temporary: and it is only instructive to us as
+showing at what cost such a work as the <i>Principia</i> must be produced,
+even by so mighty a mind as that of Newton.</p>
+
+<p>The first part of the work having been done, any ordinary mortal would
+have proceeded to publish it; but the fact is<span class='pagenum'><a name="Page_192" id="Page_192">[Pg 192]</a></span> that after he had sent to
+the Royal Society his papers on optics, there had arisen controversies
+and objections; most of them rather paltry, to which he felt compelled
+to find answers. Many men would have enjoyed this part of the work, and
+taken it as evidence of interest and success. But to Newton's shy and
+retiring disposition these discussions were merely painful. He writes,
+indeed, his answers with great patience and ability, and ultimately
+converts the more reasonable of his opponents, but he relieves his mind
+in the following letter to the secretary of the Royal Society: "I see I
+have made myself a slave to philosophy, but if I get free of this
+present business I will resolutely bid adieu to it eternally, except
+what I do for my private satisfaction or leave to come out after me; for
+I see a man must either resolve to put out nothing new, or to become a
+slave to defend it." And again in a letter to Leibnitz: "I have been so
+persecuted with discussions arising out of my theory of light that I
+blamed my own imprudence for parting with so substantial a blessing as
+my quiet to run after a shadow." This shows how much he cared for
+contemporary fame.</p>
+
+<p>So he locked up the first part of the <i>Principia</i> in his desk, doubtless
+intending it to be published after his death. But fortunately this was
+not so to be.</p>
+
+<p>In 1683, among the leading lights of the Royal Society, the same sort of
+notions about gravity and the solar system began independently to be
+bruited. The theory of gravitation seemed to be in the air, and Wren,
+Hooke, and Halley had many a talk about it.</p>
+
+<p>Hooke showed an experiment with a pendulum, which he likened to a planet
+going round the sun. The analogy is more superficial than real. It does
+not obey Kepler's laws; still it was a striking experiment. They had
+guessed at a law of inverse squares, and their difficulty was to prove
+what curve a body subject to it would describe. They knew it ought to be
+an ellipse if it was to serve to explain the planetary motion, and Hooke
+said he could prove that an<span class='pagenum'><a name="Page_193" id="Page_193">[Pg 193]</a></span> ellipse it was; but he was nothing of a
+mathematician, and the others scarcely believed him. Undoubtedly he had
+shrewd inklings of the truth, though his guesses were based on little
+else than a most sagacious intuition. He surmised also that gravity was
+the force concerned, and asserted that the path of an ordinary
+projectile was an ellipse, like the path of a planet&mdash;which is quite
+right. In fact the beginnings of the discovery were beginning to dawn
+upon him in the well-known way in which things do dawn upon ordinary men
+of genius: and had Newton not lived we should doubtless, by the labours
+of a long chain of distinguished men, beginning with Hooke, Wren, and
+Halley, have been now in possession of all the truths revealed by the
+<i>Principia</i>. We should never have had them stated in the same form, nor
+proved with the same marvellous lucidity and simplicity, but the facts
+themselves we should by this time have arrived at. Their developments
+and completions, due to such men as Clairaut, Euler, D'Alembert,
+Lagrange, Laplace, Airy, Leverrier, Adams, we should of course not have
+had to the same extent; because the lives and energies of these great
+men would have been partially consumed in obtaining the main facts
+themselves.</p>
+
+<p>The youngest of the three questioners at the time we are speaking of was
+Edmund Halley, an able and remarkable man. He had been at Cambridge,
+doubtless had heard Newton lecture, and had acquired a great veneration
+for him.</p>
+
+<p>In January, 1684, we find Wren offering Hooke and Halley a prize, in the
+shape of a book worth forty shillings, if they would either of them
+bring him within two months a demonstration that the path of a planet
+subject to an inverse square law would be an ellipse. Not in two months,
+nor yet in seven, was there any proof forthcoming. So at last, in
+August, Halley went over to Cambridge to speak to Newton about the
+difficult problem and secure his aid. Arriving at his rooms he went
+straight to the point.<span class='pagenum'><a name="Page_194" id="Page_194">[Pg 194]</a></span> He said, "What path will a body describe if it
+be attracted by a centre with a force varying as the inverse square of
+the distance." To which Newton at once replied, "An ellipse." "How on
+earth do you know?" said Halley in amazement. "Why, I have calculated
+it," and began hunting about for the paper. He actually couldn't find it
+just then, but sent it him shortly by post, and with it much more&mdash;in
+fact, what appeared to be a complete treatise on motion in general.</p>
+
+<p>With his valuable burden Halley hastened to the Royal Society and told
+them what he had discovered. The Society at his representation wrote to
+Mr. Newton asking leave that it might be printed. To this he consented;
+but the Royal Society wisely appointed Mr. Halley to see after him and
+jog his memory, in case he forgot about it. However, he set to work to
+polish it up and finish it, and added to it a great number of later
+developments and embellishments, especially the part concerning the
+lunar theory, which gave him a deal of trouble&mdash;and no wonder; for in
+the way he has put it there never was a man yet living who could have
+done the same thing. Mathematicians regard the achievement now as men
+might stare at the work of some demigod of a bygone age, wondering what
+manner of man this was, able to wield such ponderous implements with
+such apparent ease.</p>
+
+<p>To Halley the world owes a great debt of gratitude&mdash;first, for
+discovering the <i>Principia</i>; second, for seeing it through the press;
+and third, for defraying the cost of its publication out of his own
+scanty purse. For though he ultimately suffered no pecuniary loss,
+rather the contrary, yet there was considerable risk in bringing out a
+book which not a dozen men living could at the time comprehend. It is no
+small part of the merit of Halley that he recognized the transcendent
+value of the yet unfinished work, that he brought it to light, and
+assisted in its becoming understood to the best of his ability.</p>
+
+<p><span class='pagenum'><a name="Page_195" id="Page_195">[Pg 195]</a></span></p><p>Though Halley afterwards became Astronomer-Royal, lived to the ripe old
+age of eighty-six, and made many striking observations, yet he would be
+the first to admit that nothing he ever did was at all comparable in
+importance with his discovery of the <i>Principia</i>; and he always used to
+regard his part in it with peculiar pride and pleasure.</p>
+
+<p>And how was the <i>Principia</i> received? Considering the abstruse nature of
+its subject, it was received with great interest and enthusiasm. In less
+than twenty years the edition was sold out, and copies fetched large
+sums. We hear of poor students copying out the whole in manuscript in
+order to possess a copy&mdash;not by any means a bad thing to do, however
+many copies one may possess. The only useful way really to read a book
+like that is to pore over every sentence: it is no book to be skimmed.</p>
+
+<p>While the <i>Principia</i> was preparing for the press a curious incident of
+contact between English history and the University occurred. It seems
+that James II., in his policy of Catholicising the country, ordered both
+Universities to elect certain priests to degrees without the ordinary
+oaths. Oxford had given way, and the Dean of Christ Church was a
+creature of James's choosing. Cambridge rebelled, and sent eight of its
+members, among them Mr. Newton, to plead their cause before the Court of
+High Commission. Judge Jeffreys presided over the Court, and threatened
+and bullied with his usual insolence. The Vice-Chancellor of Cambridge
+was deprived of office, the other deputies were silenced and ordered
+away. From the precincts of this court of justice Newton returned to
+Trinity College to complete the <i>Principia</i>.</p>
+
+<p>By this time Newton was only forty-five years old, but his main work was
+done. His method of fluxions was still unpublished; his optics was
+published only imperfectly; a second edition of the <i>Principia</i>, with
+additions and improvements, had yet to appear; but fame had now come
+upon him, and with fame worries of all kinds.</p>
+
+<p><span class='pagenum'><a name="Page_196" id="Page_196">[Pg 196]</a></span></p><p>By some fatality, principally no doubt because of the interest they
+excited, every discovery he published was the signal for an outburst of
+criticism and sometimes of attack. I shall not go into these matters:
+they are now trivial enough, but it is necessary to mention them,
+because to Newton they evidently loomed large and terrible, and
+occasioned him acute torment.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_68" id="Fig_68"></a>
+<img src="images/fig68.jpg" width="400" height="499" alt="Fig. 68." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 68.</span>&mdash;Newton when young.<br />
+(<i>From an engraving by B. Reading after Sir Peter Lely.</i>)</span>
+</div>
+
+<p>No sooner was the <i>Principia</i> put than Hooke put in his claims for
+priority. And indeed his claims were not<span class='pagenum'><a name="Page_197" id="Page_197">[Pg 197]</a></span> altogether negligible; for
+vague ideas of the same sort had been floating in his comprehensive
+mind, and he doubtless felt indistinctly conscious of a great deal more
+than he could really state or prove.</p>
+
+<p>By indiscreet friends these two great men were set somewhat at
+loggerheads, and worse might have happened had they not managed to come
+to close quarters, and correspond privately in a quite friendly manner,
+instead of acting through the mischievous medium of third parties. In
+the next edition Newton liberally recognizes the claims of both Hooke
+and Wren. However, he takes warning betimes of what he has to expect,
+and writes to Halley that he will only publish the first two books,
+those containing general theorems on motion. The third book&mdash;concerning
+the system of the world, <i>i.e.</i> the application to the solar system&mdash;he
+says "I now design to suppress. Philosophy is such an impertinently
+litigious lady that a man had as good be engaged in law-suits as have to
+do with her. I found it so formerly, and now I am no sooner come near
+her again but she gives me warning. The two books without the third will
+not so well bear the title 'Mathematical Principles of Natural
+Philosophy,' and therefore I had altered it to this, 'On the Free Motion
+of Two Bodies'; but on second thoughts I retain the former title: 'twill
+help the sale of the book&mdash;which I ought not to diminish now 'tis
+yours."</p>
+
+<p>However, fortunately, Halley was able to prevail upon him to publish the
+third book also. It is, indeed, the most interesting and popular of the
+three, as it contains all the direct applications to astronomy of the
+truths established in the other two.</p>
+
+<p>Some years later, when his method of fluxions was published, another and
+a worse controversy arose&mdash;this time with Leibnitz, who had also
+independently invented the differential calculus. It was not so well
+recognized then how frequently it happens that two men independently
+and<span class='pagenum'><a name="Page_198" id="Page_198">[Pg 198]</a></span> unknowingly work at the very same thing at the same time. The
+history of science is now full of such instances; but then the friends
+of each accused the other of plagiarism.</p>
+
+<p>I will not go into the controversy: it is painful and useless. It only
+served to embitter the later years of two great men, and it continued
+long after Newton's death&mdash;long after both their deaths. It can hardly
+be called ancient history even now.</p>
+
+<p>But fame brought other and less unpleasant distractions than
+controversies. We are a curious, practical, and rather stupid people,
+and our one idea of honouring a man is to <i>vote</i> for him in some way or
+other; so they sent Newton to Parliament. He went, I believe, as a Whig,
+but it is not recorded that he spoke. It is, in fact, recorded that he
+was once expected to speak when on a Royal Commission about some
+question of chronometers, but that he would not. However, I dare say he
+made a good average member.</p>
+
+<p>Then a little later it was realized that Newton was poor, that he still
+had to teach for his livelihood, and that though the Crown had continued
+his fellowship to him as Lucasian Professor without the necessity of
+taking orders, yet it was rather disgraceful that he should not be
+better off. So an appeal was made to the Government on his behalf, and
+Lord Halifax, who exerted himself strongly in the matter, succeeding to
+office on the accession of William III., was able to make him ultimately
+Master of the Mint, with a salary of some &pound;1,200 a year. I believe he
+made rather a good Master, and turned out excellent coins: certainly he
+devoted his attention to his work there in a most exemplary manner.</p>
+
+<p>But what a pitiful business it all is! Here is a man sent by Heaven to
+do certain things which no man else could do, and so long as he is
+comparatively unknown he does them; but so soon as he is found out, he
+is clapped into a routine office with a big salary: and there is,
+comparatively speaking, an end of him. It is not to be supposed that he<span class='pagenum'><a name="Page_199" id="Page_199">[Pg 199]</a></span>
+had lost his power, for he frequently solved problems very quickly which
+had been given out by great Continental mathematicians as a challenge to
+the world.</p>
+
+<p>We may ask why Newton allowed himself to be thus bandied about instead
+of settling himself down to the work in which he was so pre-eminently
+great. Well, I expect your truly great man never realizes how great he
+is, and seldom knows where his real strength lies. Certainly Newton did
+not know it. He several times talks of giving up philosophy altogether;
+and though he never really does it, and perhaps the feeling is one only
+born of some temporary overwork, yet he does not sacrifice everything
+else to it as he surely must had he been conscious of his own greatness.
+No; self-consciousness was the last thing that affected him. It is for a
+great man's contemporaries to discover him, to make much of him, and to
+put him in surroundings where he may flourish luxuriantly in his own
+heaven-intended way.</p>
+
+<p>However, it is difficult for us to judge of these things. Perhaps if he
+had been maintained at the national expense to do that for which he was
+preternaturally fitted, he might have worn himself out prematurely;
+whereas by giving him routine work the scientific world got the benefit
+of his matured wisdom and experience. It was no small matter to the
+young Royal Society to be able to have him as their President for
+twenty-four years. His portrait has hung over the President's chair ever
+since, and there I suppose it will continue to hang until the Royal
+Society becomes extinct.</p>
+
+<p>The events of his later life I shall pass over lightly. He lived a calm,
+benevolent life, universally respected and beloved. His silver-white
+hair when he removed his peruke was a venerable spectacle. A lock of it
+is still preserved, with many other relics, in the library of Trinity
+College. He died quietly, after a painful illness, at the ripe age of
+eighty-five. His body lay in state in the Jerusalem Chamber, and he was
+buried in Westminster Abbey, six peers bearing the pall. These things
+are to be mentioned<span class='pagenum'><a name="Page_200" id="Page_200">[Pg 200]</a></span> to the credit of the time and the country; for
+after we have seen the calamitous spectacle of the way Tycho and Kepler
+and Galileo were treated by their ungrateful and unworthy countries, it
+is pleasant to reflect that England, with all its mistakes, yet
+recognized <i>her</i> great man when she received him, and honoured him with
+the best she knew how to give.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_69" id="Fig_69"></a>
+<img src="images/fig69.jpg" width="400" height="469" alt="Fig. 69." title="" />
+<span class="caption"><span class="smcap">Fig. 69.</span>&mdash;Sir Isaac Newton.</span>
+</div>
+
+<p>Concerning his character, one need only say that it was what one would
+expect and wish. It was characterized by a modest, calm, dignified
+simplicity. He lived frugally with his niece and her husband, Mr.
+Conduit, who succeeded him as Master of the Mint. He never married, nor
+apparently<span class='pagenum'><a name="Page_201" id="Page_201">[Pg 201]</a></span> did he ever think of so doing. The idea, perhaps, did not
+naturally occur to him, any more than the idea of publishing his work
+did.</p>
+
+<p>He was always a deeply religious man and a sincere Christian, though
+somewhat of the Arian or Unitarian persuasion&mdash;so, at least, it is
+asserted by orthodox divines who understand these matters. He studied
+theology more or less all his life, and towards the end was greatly
+interested in questions of Biblical criticism and chronology. By some
+ancient eclipse or other he altered the recognized system of dates a few
+hundred years; and his book on the prophecies of Daniel and the
+Revelation of St. John, wherein he identifies the beast with the Church
+of Rome in quite the orthodox way, is still by some admired.</p>
+
+<p>But in all these matters it is probable that he was a merely ordinary
+man, with natural acumen and ability doubtless, but nothing in the least
+superhuman. In science, the impression he makes upon me is only
+expressible by the words inspired, superhuman.</p>
+
+<p>And yet if one realizes his method of work, and the calm, uninterrupted
+flow of all his earlier life, perhaps his achievements become more
+intelligible. When asked how he made his discoveries, he replied: "By
+always thinking unto them. I keep the subject constantly before me, and
+wait till the first dawnings open slowly by little and little into a
+full and clear light." That is the way&mdash;quiet, steady, continuous
+thinking, uninterrupted and unharassed brooding. Much may be done under
+those conditions. Much ought to be sacrificed to obtain those
+conditions. All the best thinking work of the world has been thus
+done.<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a> Buffon said: "Genius is patience." So says Newton: "If I have
+done the public any service this way, it is due<span class='pagenum'><a name="Page_202" id="Page_202">[Pg 202]</a></span> to nothing but industry
+and patient thought." Genius patience? No, it is not quite that, or,
+rather, it is much more than that; but genius without patience is like
+fire without fuel&mdash;it will soon burn itself out.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_203" id="Page_203">[Pg 203]</a></span></p>
+<h4><a name="NOTES_FOR_LECTURE_IX" id="NOTES_FOR_LECTURE_IX"></a>NOTES FOR LECTURE IX</h4>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Newton and The Principia">
+<tr>
+ <td align='left'>The <i>Principia</i> published</td>
+ <td align='right'>1687.</td>
+</tr>
+<tr>
+ <td align='left'>Newton died</td>
+ <td align='right'>1727.</td>
+</tr>
+</table></div>
+
+
+<p><span class="smcap">The Law of Gravitation.</span>&mdash;Every particle of matter attracts every other
+particle of matter with a force proportional to the mass of each and to
+the inverse square of the distance between them.</p>
+
+
+<p class="center"><br /><span class="smcap">Some of Newton's Deductions.</span></p>
+
+<p>1. Kepler's second law (equable description of areas) proves that each
+planet is acted on by a force directed towards the sun as a centre of
+force.</p>
+
+<p>2. Kepler's first law proves that this central force diminishes in the
+same proportion as the square of the distance increases.</p>
+
+<p>3. Kepler's third law proves that all the planets are acted on by the
+same kind of force; of an intensity depending on the mass of the
+sun.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a></p>
+
+<p>4. So by knowing the length of year and distance of any planet from the
+sun, the sun's mass can be calculated, in terms of that of the earth.</p>
+
+<p>5. For the satellites, the force acting depends on the mass of <i>their</i>
+central body, a planet. Hence the mass of any planet possessing a
+satellite becomes known.</p>
+
+<p>6. The force constraining the moon in her orbit is the same gravity as
+gives terrestrial bodies their weight and regulates the motion of
+projectiles. [Because, while a stone drops 16 feet in a second, the
+moon, which is 60 times as far from the centre of the earth, drops 16
+feet in a minute.]</p>
+
+<hr style='width: 5%;' />
+
+<p>7. The moon is attracted not only by the earth, but by the sun also;
+hence its orbit is perturbed, and Newton calculated out the chief of
+these perturbations, viz.:&mdash;</p>
+
+<div class="blockquot"><p class="hang">(The equation of the centre, discovered by Hipparchus.)</p>
+
+<p class="hang">(<i>a</i>) The evection, discovered by Hipparchus and Ptolemy.</p>
+
+<p><span class='pagenum'><a name="Page_204" id="Page_204">[Pg 204]</a></span></p><p class="hang">(<i>b</i>) The variation, discovered by Tycho Brah&eacute;.</p>
+
+<p class="hang">(<i>c</i>) The annual equation, discovered by Tycho Brah&eacute;.</p>
+
+<p class="hang">(<i>d</i>) The retrogression of the nodes, then being observed at
+Greenwich by Flamsteed.</p>
+
+<p class="hang">(<i>e</i>) The variation of inclination, then being observed at
+Greenwich by Flamsteed.</p>
+
+<p class="hang">(<i>f</i>) The progression of the apses (with an error of one-half).</p>
+
+<p class="hang">(<i>g</i>) The inequality of apogee, previously unknown.</p>
+
+<p class="hang">(<i>h</i>) The inequality of nodes, previously unknown. </p></div>
+
+<p>8. Each planet is attracted not only by the sun but by the other
+planets, hence their orbits are slightly affected by each other. Newton
+began the theory of planetary perturbations.</p>
+
+<p>9. He recognized the comets as members of the solar system, obedient to
+the same law of gravity and moving in very elongated ellipses; so their
+return could be predicted (<i>e.g.</i> Halley's comet).</p>
+
+<p>10. Applying the idea of centrifugal force to the earth considered as a
+rotating body, he perceived that it could not be a true sphere, and
+calculated its oblateness, obtaining 28 miles greater equatorial than
+polar diameter.</p>
+
+<p>11. Conversely, from the observed shape of Jupiter, or any planet, the
+length of its day could be estimated.</p>
+
+<p>12. The so-calculated shape of the earth, in combination with
+centrifugal force, causes the weight of bodies to vary with latitude;
+and Newton calculated the amount of this variation. 194 lbs. at pole
+balance 195 lbs. at equator.</p>
+
+<p>13. A homogeneous sphere attracts as if its mass were concentrated at
+its centre. For any other figure, such as an oblate spheroid, this is
+not exactly true. A hollow concentric spherical shell exerts no force on
+small bodies inside it.</p>
+
+<p>14. The earth's equatorial protuberance, being acted on by the
+attraction of the sun and moon, must disturb its axis of rotation in a
+calculated manner; and thus is produced the precession of the equinoxes.
+[The attraction of the planets on the same protuberance causes a smaller
+and rather different kind of precession.]</p>
+
+<p>15. The waters of the ocean are attracted towards the sun and moon on
+one side, and whirled a little further away than the solid earth on the
+other side: hence Newton explained all the main phenomena of the tides.</p>
+
+<p>16. The sun's mass being known, he calculated the height of the solar
+tide.</p>
+
+<p>17. From the observed heights of spring and neap tides he determined the
+lunar tide, and thence made an estimate of the mass of the moon.</p>
+
+<p><span class='pagenum'><a name="Page_205" id="Page_205">[Pg 205]</a></span></p>
+
+<p class="center"><span class="smcap">Reference Table of Numerical Data.</span></p>
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Reference Table of Numerical Data.">
+<tr class='tr4'>
+ <td class='tdcbrbl'>&nbsp;</td>
+ <td class='tdcbr'><small>Masses in Solar<br />System.</small></td>
+ <td class='tdcbr'><small>Height dropped by a<br />stone in first second.</small></td>
+ <td class='tdcbr'><small>Length of Day or<br />time of rotation.</small></td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>Mercury</td>
+ <td class='tdrbrpr1'>&middot;065</td>
+ <td class='tdrbrpr1'>7&middot;0 feet</td>
+ <td class='tdrbrpr1'>24 hours</td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>Venus</td>
+ <td class='tdrbrpr1'>&middot;885</td>
+ <td class='tdrbrpr1'>15&middot;8&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>23&frac12;&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>Earth</td>
+ <td class='tdrbrpr1'>1&middot;000</td>
+ <td class='tdrbrpr1'>16&middot;1&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>24&nbsp;&nbsp;&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>Mars</td>
+ <td class='tdrbrpr1'>&middot;108</td>
+ <td class='tdrbrpr1'>6&middot;2&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>24&frac12;&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>Jupiter</td>
+ <td class='tdrbrpr1'>300&middot;8&nbsp;&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>45&middot;0&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>10&nbsp;&nbsp;&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>Saturn</td>
+ <td class='tdrbrpr1'>89&middot;7&nbsp;&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>18&middot;4&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>10&frac12;&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+<tr>
+ <td class='tdlbrblpr2'>The Sun</td>
+ <td class='tdrbrpr1'>316000&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>436&middot;0&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>608&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+<tr class='tr5'>
+ <td class='tdlbrblpr2'>The Moon</td>
+ <td class='tdrbrpr1'>about &middot;012&nbsp;</td>
+ <td class='tdrbrpr1'>3&middot;7&nbsp;&nbsp;"&nbsp;&nbsp;&nbsp;</td>
+ <td class='tdrbrpr1'>702&nbsp;&nbsp;"&nbsp;&nbsp;</td>
+</tr>
+</table></div>
+
+<p class="center"><small>The mass of the earth, taken above as unity, is 6,000 trillion tons.</small></p>
+
+<p><br /><i>Observatories.</i>&mdash;Uraniburg flourished from 1576 to 1597; the
+Observatory of Paris was founded in 1667; Greenwich Observatory in 1675.</p>
+
+<p><i>Astronomers-Royal.</i>&mdash;Flamsteed, Halley, Bradley, Bliss, Maskelyne,
+Pond, Airy, Christie.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_206" id="Page_206">[Pg 206]</a></span></p>
+<h3><a name="LECTURE_IX" id="LECTURE_IX"></a>LECTURE IX</h3>
+
+<h5><span class="smcap">Newton's "Principia"</span></h5>
+
+
+<p><span class="smcap">The</span> law of gravitation, above enunciated, in conjunction with the laws
+of motion rehearsed at the end of the preliminary notes of <a href="#SUMMARY_OF_FACTS_FOR_LECTURES_VII_AND_VIII">Lecture VII.</a>,
+now supersedes the laws of Kepler and includes them as special cases.
+The more comprehensive law enables us to criticize Kepler's laws from a
+higher standpoint, to see how far they are exact and how far they are
+only approximations. They are, in fact, not precisely accurate, but the
+reason for every discrepancy now becomes abundantly clear, and can be
+worked out by the theory of gravitation.</p>
+
+<p>We may treat Kepler's laws either as immediate consequences of the law
+of gravitation, or as the known facts upon which that law was founded.
+Historically, the latter is the more natural plan, and it is thus that
+they are treated in the first three statements of the above notes; but
+each proposition may be worked inversely, and we might state them
+thus:&mdash;</p>
+
+<p>1. The fact that the force acting on each planet is directed to the sun,
+necessitates the equable description of areas.</p>
+
+<p>2. The fact that the force varies as the inverse square of the distance,
+necessitates motion in an ellipse, or some other conic section, with the
+sun in one focus.</p>
+
+<p>3. The fact that one attracting body acts on all the planets with an
+inverse square law, causes the cubes of their<span class='pagenum'><a name="Page_207" id="Page_207">[Pg 207]</a></span> mean distances to be
+proportional to the squares of their periodic times.</p>
+
+<p>Not only these but a multitude of other deductions follow rigorously
+from the simple datum that every particle of matter attracts every other
+particle with a force directly proportional to the mass of each and to
+the inverse square of their mutual distance. Those dealt with in the
+<i>Principia</i> are summarized above, and it will be convenient to run over
+them in order, with the object of giving some idea of the general
+meaning of each, without attempting anything too intricate to be readily
+intelligible.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_70" id="Fig_70"></a>
+<img src="images/fig70.jpg" width="400" height="471" alt="Fig. 70." title="" />
+<span class="caption"><span class="smcap">Fig. 70.</span></span>
+</div>
+
+<p>No. 1. Kepler's second law (equable description of areas) proves that
+each planet is acted on by a force directed towards the sun as a centre
+of force.</p>
+
+<p>The equable description of areas about a centre of force has already
+been fully, though briefly, established. (<a href="#Page_175">p. 175.</a>) It is undoubtedly of
+fundamental importance, and is the<span class='pagenum'><a name="Page_208" id="Page_208">[Pg 208]</a></span> earliest instance of the serious
+discussion of central forces, <i>i.e.</i> of forces directed always to a
+fixed centre.</p>
+
+<p>We may put it afresh thus:&mdash;OA has been the motion of a particle in a
+unit of time; at A it receives a knock towards C, whereby in the next
+unit it travels along AD instead of AB. Now the area of the triangle
+CAD, swept out by the radius vector in unit time, is &frac12;<i>bh</i>; <i>h</i> being
+the perpendicular height of the triangle from the base AC. (<a href="#Fig_70">Fig. 70.</a>)
+Now the blow at A, being along the base, has no effect upon <i>h</i>; and
+consequently the area remains just what it would have been without the
+blow. A blow directed to any point other than C would at once alter the
+area of the triangle.</p>
+
+<p>One interesting deduction may at once be drawn. If gravity were a
+radiant force emitted from the sun with a velocity like that of light,
+the moving planet would encounter it at a certain apparent angle
+(aberration), and the force experienced would come from a point a little
+in advance of the sun. The rate of description of areas would thus tend
+to increase; whereas in reality it is constant. Hence the force of
+gravity, if it travel at all, does so with a speed far greater than that
+of light. It appears to be practically instantaneous. (Cf. "Modern Views
+of Electricity," &sect; 126, end of chap. xii.) Again, anything like a
+retarding effect of the medium through which the planets move would
+constitute a tangential force, entirely un-directed towards the sun.
+Hence no such frictional or retarding force can appreciably exist. It
+is, however, conceivable that both these effects might occur and just
+neutralize each other. The neutralization is unlikely to be exact for
+all the planets; and the fact is, that no trace of either effect has as
+yet been discovered. (<a href="#Page_176">See also p. 176.</a>)</p>
+
+<p>The planets are, however, subject to forces not directed towards the
+sun, viz. their attractions for each other; and these perturbing forces
+do produce a slight discrepancy from Kepler's second law, but a
+discrepancy which is completely subject to calculation.</p>
+
+<p><span class='pagenum'><a name="Page_209" id="Page_209">[Pg 209]</a></span></p><p>No. 2. Kepler's first law proves that this central force diminishes in
+the same proportion as the square of the distance increases.</p>
+
+<p>To prove the connection between the inverse-square law of distance, and
+the travelling in a conic section with the centre of force in one focus
+(the other focus being empty), is not so simple. It obviously involves
+some geometry, and must therefore be left to properly armed students.
+But it may be useful to state that the inverse-square law of distance,
+although the simplest possible law for force emanating from a point or
+sphere, is not to be regarded as self-evident or as needing no
+demonstration. The force of a magnetic pole on a magnetized steel scrap,
+for instance, varies as the inverse cube of the distance; and the curve
+described by such a particle would be quite different from a conic
+section&mdash;it would be a definite class of spiral (called Cotes's spiral).
+Again, on an iron filing the force of a single pole might vary more
+nearly as the inverse fifth power; and so on. Even when the thing
+concerned is radiant in straight lines, like light, the law of inverse
+squares is not universally true. Its truth assumes, first, that the
+source is a point or sphere; next, that there is no reflection or
+refraction of any kind; and lastly, that the medium is perfectly
+transparent. The law of inverse squares by no means holds from a prairie
+fire for instance, or from a lighthouse, or from a street lamp in a fog.</p>
+
+<p>Mutual perturbations, especially the pull of Jupiter, prevent the path
+of a planet from being really and truly an ellipse, or indeed from being
+any simple re-entrant curve. Moreover, when a planet possesses a
+satellite, it is not the centre of the planet which ever attempts to
+describe the Keplerian ellipse, but it is the common centre of gravity
+of the two bodies. Thus, in the case of the earth and moon, the point
+which really does describe a close attempt at an ellipse is a point
+displaced about 3000 miles from the centre<span class='pagenum'><a name="Page_210" id="Page_210">[Pg 210]</a></span> of the earth towards the
+moon, and is therefore only 1000 miles beneath the surface.</p>
+
+<p>No. 3. Kepler's third law proves that all the planets are acted on by
+the same kind of force; of an intensity depending on the mass of the
+sun.</p>
+
+<p>The third law of Kepler, although it requires geometry to state and
+establish it for elliptic motion (for which it holds just as well as it
+does for circular motion), is very easy to establish for circular
+motion, by any one who knows about centrifugal force. If <i>m</i> is the mass
+of a planet, <i>v</i> its velocity, <i>r</i> the radius of its orbit, and <i>T</i> the
+time of describing it; 2&#960;<i>r</i> = <i>vT</i>, and the centripetal force
+needed to hold it in its orbit is</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Centripetal force to hold orbit equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>mv<sup>2</sup></i></td>
+ <td align='center' rowspan='2'>&nbsp;&nbsp;or&nbsp;&nbsp;</td>
+ <td class='tdcbb'><i>4&#960;<sup>2</sup>mr</i></td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>r</i></td>
+ <td align='center'><i>T<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p>Now the force of gravitative attraction between the planet and the sun
+is</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Planet and Sun gravitational attraction equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>VmS</i></td>
+ <td align='center' rowspan='2'>&nbsp;,</td>
+ <td class='tdcbb'></td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>r<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p class="noin">where <i>v</i> is a fixed quantity called the gravitation-constant, to be
+determined if possible by experiment once for all. Now, expressing the
+fact that the force of gravitation <i>is</i> the force holding the planet in,
+we write,</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Gravitational Force equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>4&#960;<sup>2</sup>mr</i></td>
+ <td align='center' rowspan='2'>&nbsp;&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>VmS</i></td>
+ <td class='center' rowspan='2'>&nbsp;,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>T<sup>2</sup></i></td>
+ <td align='center'><i>r<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p class="noin">whence, by the simplest algebra,</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Gravitational Force simplified algebraic equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>r<sup>3</sup>mr</i></td>
+ <td align='center' rowspan='2'>&nbsp;&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>VS</i></td>
+ <td class='center' rowspan='2'>&nbsp;.</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>T<sup>2</sup></i></td>
+ <td align='center'><i>4&#960;<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p>The mass of the planet has been cancelled out; the mass of the sun
+remains, multiplied by the gravitation-constant, and is seen to be
+proportional to the cube of the distance divided by the square of the
+periodic time: a ratio, which is therefore<span class='pagenum'><a name="Page_211" id="Page_211">[Pg 211]</a></span> the same for all planets
+controlled by the sun. Hence, knowing <i>r</i> and <i>T</i> for any single planet,
+the value of <i>VS</i> is known.</p>
+
+<p>No. 4. So by knowing the length of year and distance of any planet from
+the sun, the sun's mass can be calculated, in terms of that of the
+earth.</p>
+
+<p>No. 5. For the satellites, the force acting depends on the mass of
+<i>their</i> central body, a planet. Hence the mass of any planet possessing
+a satellite becomes known.</p>
+
+<p>The same argument holds for any other system controlled by a central
+body&mdash;for instance, for the satellites of Jupiter; only instead of <i>S</i>
+it will be natural to write <i>J</i>, as meaning the mass of Jupiter. Hence,
+knowing <i>r</i> and <i>T</i> for any one satellite of Jupiter, the value of <i>VJ</i>
+is known.</p>
+
+<p>Apply the argument also to the case of moon and earth. Knowing the
+distance and time of revolution of our moon, the value of <i>VE</i> is at
+once determined; <i>E</i> being the mass of the earth. Hence, <i>S</i> and <i>J</i>,
+and in fact the mass of any central body possessing a visible satellite,
+are now known in terms of <i>E</i>, the mass of the earth (or, what is
+practically the same thing, in terms of <i>V</i>, the gravitation-constant).
+Observe that so far none of these quantities are known absolutely. Their
+relative values are known, and are tabulated at the end of the Notes
+above, but the finding of their absolute values is another matter, which
+we must defer.</p>
+
+<p>But, it may be asked, if Kepler's third law only gives us the mass of a
+<i>central</i> body, how is the mass of a <i>satellite</i> to be known? Well, it
+is not easy; the mass of no satellite is known with much accuracy. Their
+mutual perturbations give us some data in the case of the satellites of
+Jupiter; but to our own moon this method is of course inapplicable. Our
+moon perturbs at first sight nothing, and accordingly its mass is not
+even yet known with exactness. The mass of comets, again, is quite
+unknown. All that we can be sure of is that they are smaller than a
+certain limit, else they would perturb the planets they pass near.
+Nothing of this sort has ever<span class='pagenum'><a name="Page_212" id="Page_212">[Pg 212]</a></span> been detected. They are themselves
+perturbed plentifully, but they perturb nothing; hence we learn that
+their mass is small. The mass of a comet may, indeed, be a few million
+or even billion tons; but that is quite small in astronomy.</p>
+
+<p>But now it may be asked, surely the moon perturbs the earth, swinging it
+round their common centre of gravity, and really describing its own
+orbit about this point instead of about the earth's centre? Yes, that is
+so; and a more precise consideration of Kepler's third law enables us to
+make a fair approximation to the position of this common centre of
+gravity, and thus practically to "weigh the moon," i.e. to compare its
+mass with that of the earth; for their masses will be inversely as their
+respective distances from the common centre of gravity or balancing
+point&mdash;on the simple steel-yard principle.</p>
+
+<p>Hitherto we have not troubled ourselves about the precise point about
+which the revolution occurs, but Kepler's third law is not precisely
+accurate unless it is attended to. The bigger the revolving body the
+greater is the discrepancy: and we see in the table preceding Lecture
+<span class="smcap">III</span>., <a href="#Page_57">on page 57</a>, that Jupiter exhibits an error which, though very
+slight, is greater than that of any of the other planets, when the sun
+is considered the fixed centre.</p>
+
+<div class="blockquot"><p>Let the common centre of gravity of earth and moon be displaced a
+distance <i>x</i> from the centre of the earth, then the moon's distance
+from the real centre of revolution is not <i>r</i>, but <i>r-x</i>; and the
+equation of centrifugal force to gravitative-attraction is strictly</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Gravitational Force with CG equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>4&#960;<sup>2</sup></i></td>
+ <td align='center' rowspan='2'>(<i>r &ndash; x</i>)&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>VE</i></td>
+ <td class='center' rowspan='2'>&nbsp;,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>T<sup>2</sup></i></td>
+ <td align='center'><i>r<sup>2</sup></i></td>
+</tr>
+</table></div>
+
+<p class="noin">instead of what is in the text above; and this gives a slightly
+modified "third law." From this equation, if we have any distinct
+method of determining <i>VE</i> (and the next section gives such a
+method), we can calculate <i>x</i> and thus roughly weigh the moon,
+since</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Rough Weight of the Moon equation">
+<tr class='tr2'>
+ <td class='tdcbb'><i>r &ndash; x</i></td>
+ <td align='center' rowspan='2'>&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>E</i></td>
+ <td class='center' rowspan='2'>&nbsp;,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>r</i></td>
+ <td align='center'>E + M</td>
+</tr>
+</table></div>
+
+<p class="noin">but to get anything like a reasonable result the data must be very
+precise. </p></div>
+
+<p><span class='pagenum'><a name="Page_213" id="Page_213">[Pg 213]</a></span></p><p>No. 6. The force constraining the moon in her orbit is the same gravity
+as gives terrestrial bodies their weight and regulates the motion of
+projectiles.</p>
+
+<p>Here we come to the Newtonian verification already several times
+mentioned; but because of its importance I will repeat it in other
+words. The hypothesis to be verified is that the force acting on the
+moon is the same kind of force as acts on bodies we can handle and
+weigh, and which gives them their weight. Now the weight of a mass <i>m</i>
+is commonly written <i>mg</i>, where <i>g</i> is the intensity of terrestrial
+gravity, a thing easily measured; being, indeed, numerically equal to
+twice the distance a stone drops in the first second of free fall. [See
+table <a href="#Page_205">p. 205</a>.] Hence, expressing that the weight of a body is due to
+gravity, and remembering that the centre of the earth's attraction is
+distant from us by one earth's radius (R), we can write</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Substituting gravity for weight equation">
+<tr class='tr2'>
+ <td align='center' rowspan='2'><i>mg</i>&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>Vm</i>E</td>
+ <td class='center' rowspan='2'>&nbsp;,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'>R<sup>2</sup></td>
+</tr>
+</table></div>
+
+<p class="noin">or</p>
+
+<p><i>V</i>E = <i>g</i>R<sup>2</sup> = 95,522 cubic miles-per-second per second.</p>
+
+<p>But we already know <i>v</i>E, in terms of the moon's motion, as 4&#960;<sup>2</sup><i>r</i><sup>3</sup>/<i>T</i><sup>2</sup>
+approximately, [more accurately, see preceding note, this quantity is
+<i>V</i>(E + M)]; hence we can easily see if the two determinations of this
+quantity agree.<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a></p>
+
+<p><span class='pagenum'><a name="Page_214" id="Page_214">[Pg 214]</a></span></p><p>All these deductions are fundamental, and may be considered as the
+foundation of the <i>Principia</i>. It was these that flashed upon Newton
+during that moment of excitement when he learned the real size of the
+earth, and discovered his speculations to be true.</p>
+
+<p>The next are elaborations and amplifications of the theory, such as in
+ordinary times are left for subsequent generations of theorists to
+discover and work out.</p>
+
+<p>Newton did not work out these remoter consequences of his theory
+completely by any means: the astronomical and mathematical world has
+been working them out ever since; but he carried the theory a great way,
+and here it is that his marvellous power is most conspicuous.</p>
+
+<p>It is his treatment of No. 7, the perturbations of the moon, that
+perhaps most especially has struck all future mathematicians with
+amazement. No. 7, No. 14, No. 15, these are the most inspired of the
+whole.</p>
+
+<p>No. 7. The moon is attracted not only by the earth, but by the sun also;
+hence its orbit is perturbed, and Newton calculated out the chief of
+these perturbations.</p>
+
+<p>Now running through the perturbations (<a href="#Page_203">p. 203</a>) in order:&mdash;The first is
+in parenthesis, because it is mere excentricity. It is not a true
+perturbation at all, and more properly belongs to Kepler.</p>
+
+<p>(<i>a</i>) The first true perturbation is what Ptolemy called "the evection,"
+the principal part of which is a periodic change in the ellipticity or
+excentricity of the moon's orbit, owing to the pull of the sun. It is a
+complicated matter, and Newton only partially solved it. I shall not
+attempt to give an account of it.</p>
+
+<p>(<i>b</i>) The next, "the variation," is a much simpler affair. It is caused
+by the fact that as the moon revolves round the earth it is half the
+time nearer to the sun than the earth is, and so gets pulled more than
+the average, while for the other fortnight it is further from the sun
+than the earth is, and so gets pulled less. For the week during<span class='pagenum'><a name="Page_215" id="Page_215">[Pg 215]</a></span> which
+it is changing from a decreasing half to a new moon it is moving in the
+direction of the extra pull, and hence becomes new sooner than would
+have been expected. All next week it is moving against the same extra
+pull, and so arrives at quadrature (half moon) somewhat late. For the
+next fortnight it is in the region of too little pull, the earth gets
+pulled more than it does; the effect of this is to hurry it up for the
+third week, so that the full moon occurs a little early, and to retard
+it for the fourth week, so that the decreasing half moon like the
+increasing half occurs behind time again. Thus each syzygy (as new and
+full are technically called) is too early; each quadrature is too late;
+the maximum hurrying and slackening force being felt at the octants, or
+intermediate 45&deg; points.</p>
+
+<p>(<i>c</i>) The "annual equation" is a fluctuation introduced into the other
+perturbations by reason of the varying distance of the disturbing body,
+the sun, at different seasons of the year. Its magnitude plainly depends
+simply on the excentricity of the earth's orbit.</p>
+
+<p>Both these perturbations, (<i>b</i>) and (<i>c</i>), Newton worked out completely.</p>
+
+<p>(<i>d</i>) and (<i>e</i>) Next come the retrogression of the nodes and the
+variation of the inclination, which at the time were being observed at
+Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged
+for data that he might complete their theory while he had his mind upon
+it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see
+list at end of notes above], and then Newton would have no difficulty in
+gaining such information as the national Observatory could give.</p>
+
+<p>The "inclination" meant is the angle between the plane of the moon's
+orbit and that of the earth. The plane of the earth's orbit round the
+sun is called the ecliptic; the plane of the moon's orbit round the
+earth is inclined to it at a certain angle, which is slowly changing,
+though in a periodic manner. Imagine a curtain ring bisected by a<span class='pagenum'><a name="Page_216" id="Page_216">[Pg 216]</a></span> sheet
+of paper, and tilted to a certain angle; it may be likened to the moon's
+orbit, cutting the plane of the ecliptic. The two points at which the
+plane is cut by the ring are called "nodes"; and these nodes are not
+stationary, but are slowly regressing, <i>i.e.</i> travelling in a direction
+opposite to that of the moon itself. Also the angle of tilt is varying
+slowly, oscillating up and down in the course of centuries.</p>
+
+<p>(<i>f</i>) The two points in the moon's elliptic orbit where it comes nearest
+to or farthest from the earth, <i>i.e.</i> the points at the extremity of the
+long axis of the ellipse, are called separately perigee and apogee, or
+together "the apses." Now the pull of the sun causes the whole orbit to
+slowly revolve in its own plane, and consequently these apses
+"progress," so that the true path is not quite a closed curve, but a
+sort of spiral with elliptic loops.</p>
+
+<p>But here comes in a striking circumstance. Newton states with reference
+to this perturbation that theory only accounts for 1&frac12;&deg; per annum,
+whereas observation gives 3&deg;, or just twice as much.</p>
+
+<p>This is published in the <i>Principia</i> as a fact, without comment. It was
+for long regarded as a very curious thing, and many great mathematicians
+afterwards tried to find an error in the working. D'Alembert, Clairaut,
+and others attacked the problem, but were led to just the same result.
+It constituted the great outstanding difficulty in the way of accepting
+the theory of gravitation. It was suggested that perhaps the inverse
+square law was only a first approximation; that perhaps a more complete
+expression, such as</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Inverse Square Law Equation">
+<tr class='tr2'>
+ <td class='tdcbb'>A</td>
+ <td align='center' rowspan='2'>&nbsp;&nbsp;+&nbsp;&nbsp;</td>
+ <td class='tdcbb'>B</td>
+ <td class='center' rowspan='2'>&nbsp;,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'><i>r<sup>2</sup></i></td>
+ <td align='center'><i>r<sup>4</sup></i></td>
+</tr>
+</table></div>
+
+<p class="noin">must be given for it; and so on.</p>
+
+<p>Ultimately, Clairaut took into account a whole series of neglected
+terms, and it came out correct; thus verifying the theory.</p>
+
+<p><span class='pagenum'><a name="Page_217" id="Page_217">[Pg 217]</a></span></p><p>But the strangest part of this tale is to come. For only a few years
+ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was
+editing various old papers of Newton's, now in the possession of the
+Duke of Portland, and he found manuscripts bearing on this very point,
+and discovered that Newton had reworked out the calculations himself,
+had found the cause of the error, had taken into account the terms
+hitherto neglected, and so, fifty years before Clairaut, had completely,
+though not publicly, solved this long outstanding problem of the
+progression of the apses.</p>
+
+<p>(<i>g</i>) and (<i>h</i>) Two other inequalities he calculated out and predicted,
+viz. variation in the motions of the apses and the nodes. Neither of
+these had then been observed, but they were afterwards detected and
+verified.</p>
+
+<p>A good many other minor irregularities are now known&mdash;some thirty, I
+believe; and altogether the lunar theory, or problem of the moon's exact
+motion, is one of the most complicated and difficult in astronomy; the
+perturbations being so numerous and large, because of the enormous mass
+of the perturbing body.</p>
+
+<p>The disturbances experienced by the planets are much smaller, because
+they are controlled by the sun and perturbed by each other. The moon is
+controlled only by the earth, and perturbed by the sun. Planetary
+perturbations can be treated as a series of disturbances with some
+satisfaction: not so those of the moon. And yet it is the only way at
+present known of dealing with the lunar theory.</p>
+
+<p>To deal with it satisfactorily would demand the solution of such a
+problem as this:&mdash;Given three rigid spherical masses thrown into empty
+space with any initial motions whatever, and abandoned to gravity: to
+determine their subsequent motions. With two masses the problem is
+simple enough, being pretty well summed up in Kepler's laws; but with
+three masses, strange to say, it is so complicated as to be beyond the
+reach of even modern<span class='pagenum'><a name="Page_218" id="Page_218">[Pg 218]</a></span> mathematics. It is a famous problem, known as that
+of "the three bodies," but it has not yet been solved. Even when it is
+solved it will be only a close approximation to the case of earth, moon,
+and sun, for these bodies are not spherical, and are not rigid. One may
+imagine how absurdly and hopelessly complicated a complete treatment of
+the motions of the entire solar system would be.</p>
+
+<p>No. 8. Each planet is attracted not only by the sun but by the other
+planets, hence their orbits are slightly affected by each other.</p>
+
+<p>The subject of planetary perturbation was only just begun by Newton.
+Gradually (by Laplace and others) the theory became highly developed;
+and, as everybody knows, in 1846 Neptune was discovered by means of it.</p>
+
+<p>No. 9. He recognized the comets as members of the solar system, obedient
+to the same law of gravity and moving in very elongated ellipses; so
+their return could be predicted.</p>
+
+<p>It was a long time before Newton recognized the comets as real members
+of the solar system, and subject to gravity like the rest. He at first
+thought they moved in straight lines. It was only in the second edition
+of the <i>Principia</i> that the theory of comets was introduced.</p>
+
+<p>Halley observed a fine comet in 1682, and calculated its orbit on
+Newtonian principles. He also calculated when it ought to have been seen
+in past times; and he found the year 1607, when one was seen by Kepler;
+also the year 1531, when one was seen by Appian; again, he reckoned
+1456, 1380, 1305. All these appearances were the same comet, in all
+probability, returning every seventy-five or seventy-six years. The
+period was easily allowed to be not exact, because of perturbing
+planets. He then predicted its return for 1758, or perhaps 1759, a date
+he could not himself hope to see. He lived to a great age, but he died
+sixteen years before this date.</p>
+
+<p>As the time drew nigh, three-quarters of a century afterwards,
+astronomers were greatly interested in this first<span class='pagenum'><a name="Page_219" id="Page_219">[Pg 219]</a></span> cometary prediction,
+and kept an eager look-out for "Halley's comet." Clairaut, a most
+eminent mathematician and student of Newton, proceeded to calculate out
+more exactly the perturbing influence of Jupiter, near which it had
+passed. After immense labour (for the difficulty of the calculation was
+extreme, and the mass of mere figures something portentous), he
+predicted its return on the 13th of April, 1759, but he considered that
+he might have made a possible error of a month. It returned on the 13th
+of March, 1759, and established beyond all doubt the rule of the
+Newtonian theory over comets.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_71" id="Fig_71"></a>
+<img src="images/fig71.jpg" width="400" height="222" alt="Fig. 71." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 71.</span>&mdash;Well-known model exhibiting the oblate
+spheroidal form as a consequence of spinning about a central axis. The
+brass strip <i>a</i> looks like a transparent globe when whirled, and bulges
+out equatorially.</span>
+</div>
+
+<p>No. 10. Applying the idea of centrifugal force to the earth considered
+as a rotating body, he perceived that it could not be a true sphere, and
+calculated its oblateness, obtaining 28 miles greater equatorial than
+polar diameter.</p>
+
+<p>Here we return to one of the more simple deductions. A spinning body of
+any kind tends to swell at its circumference (or equator), and shrink
+along its axis (or poles). If the body is of yielding material, its
+shape must alter under the influence of centrifugal force; and if a
+globe of yielding substance subject to known forces rotates at a
+definite pace, its shape can be calculated. Thus a<span class='pagenum'><a name="Page_220" id="Page_220">[Pg 220]</a></span> plastic sphere the
+size of the earth, held together by its own gravity, and rotating once a
+day, can be shown to have its equatorial diameter twenty-eight miles
+greater than its polar diameter: the two diameters being 8,000 and 8,028
+respectively. Now we have no guarantee that the earth is of yielding
+material: for all Newton could tell it might be extremely rigid. As a
+matter of fact it is now very nearly rigid. But he argued thus. The
+water on it is certainly yielding, and although the solid earth might
+decline to bulge at the equator in deference to the diurnal rotation,
+that would not prevent the ocean from flowing from the poles to the
+equator and piling itself up as an equatorial ocean fourteen miles deep,
+leaving dry land everywhere near either pole. Nothing of this sort is
+observed: the distribution of land and water is not thus regulated.
+Hence, whatever the earth may be now, it must once have been plastic
+enough to accommodate itself perfectly to the centrifugal forces, and to
+take the shape appropriate to a perfectly plastic body. In all
+probability it was once molten, and for long afterwards pasty.</p>
+
+<p>Thus, then, the shape of the earth can be calculated from the length of
+its day and the intensity of its gravity. The calculation is not
+difficult: it consists in imagining a couple of holes bored to the
+centre of the earth, one from a pole and one from the equator; filling
+these both with water, and calculating how much higher the water will
+stand in one leg of the gigantic <big>V</big> tube so formed than in the other. The
+answer comes out about fourteen miles.</p>
+
+<p>The shape of the earth can now be observed geodetically, and it accords
+with calculation, but the observations are extremely delicate; in
+Newton's time the <i>size</i> was only barely known, the <i>shape</i> was not
+observed till long after; but on the principles of mechanics, combined
+with a little common-sense reasoning, it could be calculated with
+certainty and accuracy.</p>
+
+<p><span class='pagenum'><a name="Page_221" id="Page_221">[Pg 221]</a></span></p><p>No. 11. From the observed shape of Jupiter or any planet the length of
+its day could be estimated.</p>
+
+<p>Jupiter is much more oblate than the earth. Its two diameters are to one
+another as 17 is to 16; the ellipticity of its disk is manifest to
+simple inspection. Hence we perceive that its whirling action must be
+more violent&mdash;it must rotate quicker. As a matter of fact its day is ten</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_72" id="Fig_72"></a>
+<img src="images/fig72.jpg" width="400" height="283" alt="Fig. 72." title="" />
+<span class="caption"><span class="smcap">Fig. 72.</span>&mdash;Jupiter.</span>
+</div>
+
+<p>hours long&mdash;five hours daylight and five hours night. The times of
+rotation of other bodies in the solar system are recorded in a table
+above.</p>
+
+<p>No. 12. The so-calculated shape of the earth, in combination with
+centrifugal force, causes the weight of bodies to vary with latitude;
+and Newton calculated the amount of this variation. 194 lbs. at pole
+balance 195 lbs. at equator.</p>
+
+<p>But following from the calculated shape of the earth follow several
+interesting consequences. First of all, the intensity of gravity will
+not be the same everywhere; for at the equator a stone is further from
+the average<span class='pagenum'><a name="Page_222" id="Page_222">[Pg 222]</a></span> bulk of the earth (say the centre) than it is at the poles,
+and owing to this fact a mass of 590 pounds at the pole; would suffice
+to balance 591 pounds at the equator, if the two could be placed in the
+pans of a gigantic balance whose beam straddled along an earth's
+quadrant. This is a <i>true</i> variation of gravity due to the shape of the
+earth. But besides this there is a still larger <i>apparent</i> variation due
+to centrifugal force, which affects all bodies at the equator but not
+those at the poles. From this cause, even if the earth were a true
+sphere, yet if it were spinning at its actual pace, 288 pounds at the
+pole could balance 289 pounds at the equator; because at the equator the
+true weight of the mass would not be fully appreciated, centrifugal
+force would virtually diminish it by <span class="above">1</span>&#8260;<span class="below">289</span>th of its amount.</p>
+
+<p>In actual fact both causes co-exist, and accordingly the total variation
+of gravity observed is compounded of the real and the apparent effects;
+the result is that 194 pounds at a pole weighs as much as 195 pounds at
+the equator.</p>
+
+<p>No. 13. A homogeneous sphere attracts as if its mass were concentrated
+at its centre. For any other figure, such as an oblate spheroid, this is
+not exactly true. A hollow concentric spherical shell exerts no force on
+small bodies inside it.</p>
+
+<p>A sphere composed of uniform material, or of materials arranged in
+concentric strata, can be shown to attract external bodies as if its
+mass were concentrated at its centre. A hollow sphere, similarly
+composed, does the same, but on internal bodies it exerts no force at
+all.</p>
+
+<p>Hence, at all distances above the surface of the earth, gravity
+decreases in inverse proportion as the square of the distance from the
+centre of the earth increases; but, if you descend a mine, gravity
+decreases in this case also as you leave the surface, though not at the
+same rate as when you went up. For as you penetrate the crust you get
+inside a concentric shell, which is thus powerless to act upon you, and
+the earth you are now outside is a smaller one. At<span class='pagenum'><a name="Page_223" id="Page_223">[Pg 223]</a></span> what rate the force
+decreases depends on the distribution of density; if the density were
+uniform all through, the law of variation would be the direct distance,
+otherwise it would be more complicated. Anyhow, the intensity of gravity
+is a maximum at the surface of the earth, and decreases as you travel
+from the surface either up or down.</p>
+
+<p>No. 14. The earth's equatorial protuberance, being acted on by the
+attraction of the sun and moon, must disturb its axis of rotation in a
+calculated manner; and thus is produced the precession of the equinoxes.</p>
+
+<p>Here we come to a truly awful piece of reasoning. A sphere attracts as
+if its mass were concentrated at its centre (No. 12), but a spheroid
+does not. The earth is a spheroid, and hence it pulls and is pulled by
+the moon with a slightly uncentric attraction. In other words, the line
+of pull does not pass through its precise centre. Now when we have a
+spinning body, say a top, overloaded on one side so that gravity acts on
+it unsymmetrically, what happens? The axis of rotation begins to rotate
+cone-wise, at a pace which depends on the rate of spin, and on the shape
+and mass of the top, as well as on the amount and leverage of the
+overloading.</p>
+
+<p>Newton calculated out the rapidity of this conical motion of the axis of
+the earth, produced by the slightly unsymmetrical pull of the moon, and
+found that it would complete a revolution in 26,000 years&mdash;precisely
+what was wanted to explain the precession of the equinoxes. In fact he
+had discovered the physical cause of that precession.</p>
+
+<p>Observe that there were three stages in this discovery of precession:&mdash;</p>
+
+<p>First, the observation by Hipparchus, that the nodes, or intersections
+of the earth's orbit (the sun's apparent orbit) with the plane of the
+equator, were not stationary, but slowly moved.</p>
+
+<p>Second, the description of this motion by Copernicus, by<span class='pagenum'><a name="Page_224" id="Page_224">[Pg 224]</a></span> the statement
+that it was due to a conical motion of the earth's axis of rotation
+about its centre as a fixed point.</p>
+
+<p>Third, the explanation of this motion by Newton as due to the pull of
+the moon on the equatorial protuberance of the earth.</p>
+
+<p>The explanation <i>could</i> not have been previously suspected, for the
+shape of the earth, on which the whole theory depends, was entirely
+unknown till Newton calculated it.</p>
+
+<p>Another and smaller motion of a somewhat similar kind has been worked
+out since: it is due to the unsymmetrical attraction of the other
+planets for this same equatorial protuberance. It shows itself as a
+periodic change in the obliquity of the ecliptic, or so-called recession
+of the apses, rather than as a motion of the nodes.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a></p>
+
+<p>No. 15. The waters of the ocean are attracted towards the sun and moon
+on one side, and whirled a little farther away than the solid earth on
+the other side: hence Newton explained all the main phenomena of the
+tides.</p>
+
+<p>And now comes another tremendous generalization. The tides had long been
+an utter mystery. Kepler likens the earth to an animal, and the tides to
+his breathings and inbreathings, and says they follow the moon.</p>
+
+<p>Galileo chaffs him for this, and says that it is mere superstition to
+connect the moon with the tides.</p>
+
+<p>Descartes said the moon pressed down upon the waters by the centrifugal
+force of its vortex, and so produced a low tide under it.</p>
+
+<p>Everything was fog and darkness on the subject. The legend goes that an
+astronomer threw himself into the sea in despair of ever being able to
+explain the flux and reflux of its waters.</p>
+
+<p><span class='pagenum'><a name="Page_225" id="Page_225">[Pg 225]</a></span></p><p>Newton now with consummate skill applied his theory to the effect of
+the moon upon the ocean, and all the main details of tidal action
+gradually revealed themselves to him.</p>
+
+<p>He treated the water, rotating with the earth once a day, somewhat as if
+it were a satellite acted on by perturbing forces. The moon as it
+revolves round the earth is perturbed by the sun. The ocean as it
+revolves round the earth (being held on by gravitation just as the moon
+is) is perturbed by both sun and moon.</p>
+
+<p>The perturbing effect of a body varies directly as its mass, and
+inversely as the cube of its distance. (The simple law of inverse square
+does not apply, because a perturbation is a differential effect: the
+satellite or ocean when nearer to the perturbing body than the rest of
+the earth, is attracted more, and when further off it is attracted less
+than is the main body of the earth; and it is these differences alone
+which constitute the perturbation.) The moon is the more powerful of the
+two perturbing bodies, hence the main tides are due to the moon; and its
+chief action is to cause a pair of low waves or oceanic humps, of
+gigantic area, to travel round the earth once in a lunar day, <i>i.e.</i> in
+about 24 hours and 50 minutes. The sun makes a similar but still lower
+pair of low elevations to travel round once in a solar day of 24 hours.
+And the combination of the two pairs of humps, thus periodically
+overtaking each other, accounts for the well-known spring and neap
+tides,&mdash;spring tides when their maxima agree, neap tides when the
+maximum of one coincides with the minimum of the other: each of which
+events happens regularly once a fortnight.</p>
+
+<p>These are the main effects, but besides these there are the effects of
+varying distances and obliquity to be taken into account; and so we have
+a whole series of minor disturbances, very like those discussed in No.
+7, under the lunar theory, but more complex still, because there are two
+perturbing bodies instead of only one.</p>
+
+<p><span class='pagenum'><a name="Page_226" id="Page_226">[Pg 226]</a></span></p><p>The subject of the tides is, therefore, very recondite; and though one
+may give some elementary account of its main features, it will be best
+to defer this to a separate lecture (<a href="#NOTES_FOR_LECTURE_XVII">Lecture XVII</a>).</p>
+
+<p>I had better, however, here say that Newton did not limit himself to the
+consideration of the primary oceanic humps: he pursued the subject into
+geographical detail. He pointed out that, although the rise and fall of
+the tide at mid-ocean islands would be but small, yet on stretches of
+coast the wave would fling itself, and by its momentum would propel the
+waters, to a much greater height&mdash;for instance, 20 or 30 feet;
+especially in some funnel-shaped openings like the Bristol Channel and
+the Bay of Fundy, where the concentrated impetus of the water is
+enormous.</p>
+
+<p>He also showed how the tidal waves reached different stations in
+successive regular order each day; and how some places might be fed with
+tide by two distinct channels; and that if the time of these channels
+happened to differ by six hours, a high tide might be arriving by one
+channel and a low tide by the other, so that the place would only feel
+the difference, and so have a very small observed rise and fall;
+instancing a port in China (in the Gulf of Tonquin) where that
+approximately occurs.</p>
+
+<p>In fact, although his theory was not, as we now know, complete or final,
+yet it satisfactorily explained a mass of intricate detail as well as
+the main features of the tides.</p>
+
+<p>No. 16. The sun's mass being known, he calculated the height of the
+solar tide.</p>
+
+<p>No. 17. From the observed heights of spring and neap tides he determined
+the lunar tide, and thence made an estimate of the mass of the moon.</p>
+
+<p>Knowing the sun's mass and distance, it was not difficult for Newton to
+calculate the height of the protuberance caused by it in a pasty ocean
+covering the<span class='pagenum'><a name="Page_227" id="Page_227">[Pg 227]</a></span> whole earth. I say pasty, because, if there was any
+tendency for impulses to accumulate, as timely pushes given to a
+pendulum accumulate, the amount of disturbance might become excessive,
+and its calculation would involve a multitude of data. The Newtonian
+tide ignored this, thus practically treating the motion as either
+dead-beat, or else the impulses as very inadequately timed. With this
+reservation the mid-ocean tide due to the action of the sun alone comes
+out about one foot, or let us say one foot for simplicity. Now the
+actual tide observed in mid-Atlantic is at the springs about four feet,
+at the neaps about two. The spring tide is lunar plus solar; the neap
+tide is lunar minus solar. Hence it appears that the tide caused by the
+moon alone must be about three feet, when unaffected by momentum. From
+this datum Newton made the first attempt to approximately estimate the
+mass of the moon. I said that the masses of satellites must be
+estimated, if at all, by the perturbation they are able to cause. The
+lunar tide is a perturbation in the diurnal motion of the sea, and its
+amount is therefore a legitimate mode of calculating the moon's mass.
+The available data were not at all good, however; nor are they even now
+very perfect; and so the estimate was a good way out. It is now
+considered that the mass of the moon is about one-eightieth that of the
+earth.</p>
+
+<hr style='width: 15%;' />
+
+<p>Such are some of the gems extracted from their setting in the
+<i>Principia</i>, and presented as clearly as I am able before you.</p>
+
+<p>Do you realize the tremendous stride in knowledge&mdash;not a stride, as
+Whewell says, nor yet a leap, but a flight&mdash;which has occurred between
+the dim gropings of Kepler, the elementary truths of Galileo, the
+fascinating but wild speculations of Descartes, and this magnificent and
+comprehensive system of ordered knowledge. To some his<span class='pagenum'><a name="Page_228" id="Page_228">[Pg 228]</a></span> genius seemed
+almost divine. "Does Mr. Newton eat, drink, sleep, like other men?" said
+the Marquis de l'H&ocirc;pital, a French mathematician of no mean eminence; "I
+picture him to myself as a celestial genius, entirely removed from the
+restrictions of ordinary matter." To many it seemed as if there was
+nothing more to be discovered, as if the universe were now explored, and
+only a few fragments of truth remained for the gleaner. This is the
+attitude of mind expressed in Pope's famous epigram:&mdash;</p>
+
+<p class="poem">
+"Nature and Nature's laws lay hid in Night,<br />
+God said, Let Newton be, and all was light."<br />
+</p>
+
+<p>This feeling of hopelessness and impotence was very natural after the
+advent of so overpowering a genius, and it prevailed in England for
+fully a century. It was very natural, but it was very mischievous; for,
+as a consequence, nothing of great moment was done by England in
+science, and no Englishman of the first magnitude appeared, till some
+who are either living now or who have lived within the present century.</p>
+
+<p>It appeared to his contemporaries as if he had almost exhausted the
+possibility of discovery; but did it so appear to Newton? Did it seem to
+him as if he had seen far and deep into the truths of this great and
+infinite universe? It did not. When quite an old man, full of honour and
+renown, venerated, almost worshipped, by his contemporaries, these were
+his words:&mdash;</p>
+
+<p>"I know not what the world will think of my labours, but to myself it
+seems that I have been but as a child playing on the sea-shore; now
+finding some pebble rather more polished, and now some shell rather more
+agreeably variegated than another, while the immense ocean of truth
+extended itself unexplored before me."</p>
+
+<p>And so it must ever seem to the wisest and greatest of men when brought
+into contact with the great things of<span class='pagenum'><a name="Page_229" id="Page_229">[Pg 229]</a></span> God&mdash;that which they know is as
+nothing, and less than nothing, to the infinitude of which they are
+ignorant.</p>
+
+<p>Newton's words sound like a simple and pleasing echo of the words of
+that great unknown poet, the writer of the book of Job:&mdash;</p>
+
+<p class="poem">
+"Lo, these are parts of His ways,<br />
+But how little a portion is heard of Him;<br />
+The thunder of His power, who can understand?"<br />
+</p>
+
+<p class="center"><small>END OF PART I.</small><span class='pagenum'><a name="Page_230" id="Page_230">[Pg 230]</a></span></p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_231" id="Page_231">[Pg 231]</a></span></p>
+<h3><a name="PART_II" id="PART_II"></a>PART II<br />
+
+<i>A COUPLE OF CENTURIES' PROGRESS.</i></h3>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_232" id="Page_232">[Pg 232]</a></span></p>
+<h4><a name="NOTES_TO_LECTURE_X" id="NOTES_TO_LECTURE_X"></a>NOTES TO LECTURE X</h4>
+
+<p class="center"><i>Science during the century after Newton</i><br /></p>
+
+<p class="center">The <i>Principia</i> published, 1687</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Science after Newton">
+<tr>
+ <td align='left'>Roemer</td>
+ <td align='left'>1644&ndash;1710</td>
+</tr>
+<tr>
+ <td align='left'>James Bradley</td>
+ <td align='left'>1692&ndash;1762</td>
+</tr>
+<tr>
+ <td align='left'>Clairaut</td>
+ <td align='left'>1713&ndash;1765</td>
+</tr>
+<tr>
+ <td align='left'>Euler</td>
+ <td align='left'>1707&ndash;1783</td>
+</tr>
+<tr>
+ <td align='left'>D'Alembert</td>
+ <td align='left'>1717&ndash;1783</td>
+</tr>
+<tr>
+ <td align='left'>Lagrange</td>
+ <td align='left'>1736&ndash;1813</td>
+</tr>
+<tr>
+ <td align='left'>Laplace</td>
+ <td align='left'>1749&ndash;1827</td>
+</tr>
+<tr>
+ <td align='left'>William Herschel</td>
+ <td align='left'>1738&ndash;1822</td>
+</tr>
+</table></div>
+
+
+<p><i>Olaus Roemer</i> was born in Jutland, and studied at Copenhagen. Assisted
+Picard in 1671 to determine the exact position of Tycho's observatory on
+Huen. Accompanied Picard to Paris, and in 1675 read before the Academy
+his paper "On Successive Propagation of Light as revealed by a certain
+inequality in the motion of Jupiter's First Satellite." In 1681 he
+returned to Copenhagen as Professor of Mathematics and Astronomy, and
+died in 1710. He invented the transit instrument, mural circle,
+equatorial mounting for telescopes, and most of the other principal
+instruments now in use in observatories. He made as many observations as
+Tycho Brah&eacute;, but the records of all but the work of three days were
+destroyed by a great fire in 1728.</p>
+
+<p><i>Bradley</i>, Professor of Astronomy at Oxford, discovered the aberration
+of light in 1729, while examining stars for parallax, and the nutation
+of the earth's axis in 1748. Was appointed Astronomer-Royal in 1742.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_233" id="Page_233">[Pg 233]</a></span></p>
+<h3><a name="LECTURE_X" id="LECTURE_X"></a>LECTURE X</h3>
+
+<h5>ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT</h5>
+
+
+<p><span class="smcap">At</span> Newton's death England stood pre-eminent among the nations of Europe
+in the sphere of science. But the pre-eminence did not last long. Two
+great discoveries were made very soon after his decease, both by
+Professor Bradley, of Oxford, and then there came a gap. A moderately
+great man often leaves behind him a school of disciples able to work
+according to their master's methods, and with a healthy spirit of
+rivalry which stimulates and encourages them. Newton left, indeed, a
+school of disciples, but his methods of work were largely unknown to
+them, and such as were known were too ponderous to be used by ordinary
+men. Only one fresh result, and that a small one, has ever been attained
+by other men working according to the methods of the <i>Principia</i>. The
+methods were studied and commented on in England to the exclusion of all
+others for nigh a century, and as a consequence no really important work
+was done.</p>
+
+<p>On the Continent, however, no such system of slavish imitation
+prevailed. Those methods of Newton's which had been simultaneously
+discovered by Leibnitz were more thoroughly grasped, modified, extended,
+and improved. There arose a great school of French and German
+mathematicians, and the laurels of scientific discovery passed to France
+and Germany&mdash;more especially,<span class='pagenum'><a name="Page_234" id="Page_234">[Pg 234]</a></span> perhaps, at this time to France. England
+has never wholly recovered them. During the present century this country
+has been favoured with some giants who, as they become distant enough
+for their true magnitude to be perceived, may possibly stand out as
+great as any who have ever lived; but for the mass and bulk of
+scientific work at the present day we have to look to Germany, with its
+enlightened Government and extensive intellectual development. England,
+however, is waking up, and what its Government does not do, private
+enterprise is beginning to accomplish. The establishment of centres of
+scientific and literary activity in the great towns of England, though
+at present they are partially encumbered with the supply of education of
+an exceedingly rudimentary type, is a movement that in the course of
+another century or so will be seen to be one of the most important and
+fruitful steps ever taken by this country. On the Continent such centres
+have long existed; almost every large town is the seat of a University,
+and they are now liberally endowed. The University of Bologna (where,
+you may remember, Copernicus learnt mathematics) has recently celebrated
+its 800th anniversary.</p>
+
+<p>The scientific history of the century after Newton, summarized in the
+above table of dates, embraces the labours of the great mathematicians
+Clairaut, Euler, D'Alembert, and especially of Lagrange and Laplace.</p>
+
+<p>But the main work of all these men was hardly pioneering work. It was
+rather the surveying, and mapping out, and bringing into cultivation, of
+lands already discovered. Probably Herschel may be justly regarded as
+the next true pioneer. We shall not, however, properly appreciate the
+stages through which astronomy has passed, nor shall we be prepared
+adequately to welcome the discoveries of modern times unless we pay some
+attention to the intervening age. Moreover, during this era several
+facts of great moment gradually came into recognition; and the<span class='pagenum'><a name="Page_235" id="Page_235">[Pg 235]</a></span>
+importance of the discovery we have now to speak of can hardly be
+over-estimated.</p>
+
+<p>Our whole direct knowledge of the planetary and stellar universe, from
+the early observations of the ancients down to the magnificent
+discoveries of a Herschel, depends entirely upon our happening to
+possess a sense of sight. To no other of our senses do any other worlds
+than our own in the slightest degree appeal. We touch them or hear them
+never. Consequently, if the human race had happened to be blind, no
+other world but the one it groped its way upon could ever have been
+known or imagined by it. The outside universe would have existed, but
+man would have been entirely and hopelessly ignorant of it. The bare
+idea of an outside universe beyond the world would have been
+inconceivable, and might have been scouted as absurd. We do possess the
+sense of sight; but is it to be supposed that we possess every sense
+that can be possessed by finite beings? There is not the least ground
+for such an assumption. It is easy to imagine a deaf race or a blind
+race: it is not so easy to imagine a race more highly endowed with
+senses than our own; and yet the sense of smell in animals may give us
+some aid in thinking of powers of perception which transcend our own in
+particular directions. If there were a race with higher or other senses
+than our own, or if the human race should ever in the process of
+development acquire such extra sense-organs, a whole universe of
+existent fact might become for the first time perceived by us, and we
+should look back upon our past state as upon a blind chrysalid form of
+existence in which we had been unconscious of all this new wealth of
+perception.</p>
+
+<p>It cannot be too clearly and strongly insisted on and brought home to
+every mind, that the mode in which the universe strikes us, our view of
+the universe, our whole idea of matter, and force, and other worlds, and
+even of consciousness, depends upon the particular set of sense-organs
+with which we, as men, happen to be endowed. The<span class='pagenum'><a name="Page_236" id="Page_236">[Pg 236]</a></span> senses of force, of
+motion, of sound, of light, of touch, of heat, of taste, and of
+smell&mdash;these we have, and these are the things we primarily know. All
+else is inference founded upon these sensations. So the world appears to
+us. But given other sense-organs, and it might appear quite otherwise.
+What it is actually and truly like, therefore, is quite and for ever
+beyond us&mdash;so long as we are finite beings.</p>
+
+<p>Without eyes, astronomy would be non-existent. Light it is which conveys
+all the information we possess, or, as it would seem, ever can possess,
+concerning the outer and greater universe in which this small world
+forms a speck. Light is the channel, the messenger of information; our
+eyes, aided by telescopes, spectroscopes, and many other "scopes" that
+may yet be invented, are the means by which we read the information that
+light brings.</p>
+
+<p>Light travels from the stars to our eyes: does it come instantaneously?
+or does it loiter by the way? for if it lingers it is not bringing us
+information properly up to date&mdash;it is only telling us what the state of
+affairs was when it started on its long journey.</p>
+
+<p>Now, it is evidently a matter of interest to us whether we see the sun
+as he is now, or only as he was some three hundred years ago. If the
+information came by express train it would be three hundred years behind
+date, and the sun might have gone out in the reign of Queen Anne without
+our being as yet any the wiser. The question, therefore, "At what rate
+does our messenger travel?" is evidently one of great interest for
+astronomers, and many have been the attempts made to solve it. Very
+likely the ancient Greeks pondered over this question, but the earliest
+writer known to me who seriously discussed the question is Galileo. He
+suggests a rough experimental means of attacking it. First of all, it
+plainly comes quicker than sound. This can be perceived by merely
+watching distant hammering, or by noticing that the flash of a pistol is
+seen<span class='pagenum'><a name="Page_237" id="Page_237">[Pg 237]</a></span> before its report is heard, or by listening to the noise of a
+flash of lightning. Sound takes five seconds to travel a mile&mdash;it has
+about the same speed as a rifle bullet; but light is much quicker than
+that.</p>
+
+<p>The rude experiment suggested by Galileo was to send two men with
+lanterns and screens to two distant watch-towers or neighbouring
+mountain tops, and to arrange that each was to watch alternate displays
+and obscurations of the light made by the other, and to imitate them as
+promptly as possible. Either man, therefore, on obscuring or showing his
+own light would see the distant glimmer do the same, and would be able
+to judge if there was any appreciable interval between his own action
+and the response of the distant light. The experiment was actually tried
+by the Florentine Academicians,<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a> with the result that, as practice
+improved, the interval became shorter and shorter, so that there was no
+reason to suppose that there was any real interval at all. Light, in
+fact, seemed to travel instantaneously.</p>
+
+<p>Well might they have arrived at this result. Even if they had made far
+more perfect arrangements&mdash;for instance, by arranging a looking-glass at
+one of the stations in which a distant observer might see the reflection
+of his own lantern, and watch the obscurations and flashings made by
+himself, without having to depend on the response of human
+mechanism&mdash;even then no interval whatever could have been detected.</p>
+
+<p>If, by some impossibly perfect optical arrangement, a lighthouse here
+were made visible to us after reflection in a mirror erected at New
+York, so that the light would have to travel across the Atlantic and
+back before it could be seen, even then the appearance of the light on
+removing a shutter, or the eclipse on interposing it, would seem to
+happen<span class='pagenum'><a name="Page_238" id="Page_238">[Pg 238]</a></span> quite instantaneously. There would certainly be an interval: the
+interval would be the fiftieth part of a second (the time a stone takes
+to drop <span class="above">1</span>&#8260;<span class="below">13</span>th of an inch), but that is too short to be securely
+detected without mechanism. With mechanism the thing might be managed,
+for a series of shutters might be arranged like the teeth of a large
+wheel; so that, when the wheel rotates, eclipses follow one another very
+rapidly; if then an eye looked through the same opening as that by which
+the light goes on its way to the distant mirror, a tooth might have
+moved sufficiently to cover up this space by the time the light
+returned; in which case the whole would appear dark, for the light would
+be stopped by a tooth, either at starting or at returning, continually.
+At higher speeds of rotation some light would reappear, and at lower
+speeds it would also reappear; by noticing, therefore, the precise speed
+at which there was constant eclipse the velocity of light could be
+determined.</p>
+
+<div class="figcenter" style="width: 450px;"><a name="Fig_73" id="Fig_73"></a>
+<img src="images/fig73.jpg" width="400" height="334" alt="Fig. 73." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 73.</span>&mdash;Diagram of eye looking at a light reflected in
+a distant mirror through the teeth of a revolving wheel.</span>
+</div>
+
+<p>This experiment has now been made in a highly refined form by Fizeau,
+and repeated by M. Cornu with prodigious care and accuracy. But with
+these recent matters we have no concern at present. It may be
+instructive to say, however,<span class='pagenum'><a name="Page_239" id="Page_239">[Pg 239]</a></span> that if the light had to travel two miles
+altogether, the wheel would have to possess 450 teeth and to spin 100
+times a second (at the risk of flying to pieces) in order that the ray
+starting through any one of the gaps might be stopped on returning by
+the adjacent tooth.</p>
+
+<p>Well might the velocity of light be called instantaneous by the early
+observers. An ordinary experiment seemed (and was) hopeless, and light
+was supposed to travel at an infinite speed. But a phenomenon was
+noticed in the heavens by a quick-witted and ingenious Danish
+astronomer, which was not susceptible of any ordinary explanation, and
+which he perceived could at once be explained if light had a certain
+rate of travel&mdash;great, indeed, but something short of infinite. This
+phenomenon was connected with the satellites of Jupiter, and the
+astronomer's name was Roemer. I will speak first of the observation and
+then of the man.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_74" id="Fig_74"></a>
+<img src="images/fig74.jpg" width="400" height="159" alt="Fig. 74." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 74.</span>&mdash;Fizeau&#39;s wheel, shewing the appearance of
+distant image seen through its teeth. 1st, when stationary, next when
+revolving at a moderate speed, last when revolving at the high speed
+just sufficient to cause eclipse.</div>
+</div>
+
+<p>Jupiter's satellites are visible, precisely as our own moon is, by
+reason of the shimmer of sunlight which they reflect. But as they
+revolve round their great planet they plunge into his shadow at one part
+of their course, and so become eclipsed from sunshine and invisible to
+us. The moment of disappearance can be sharply observed.</p>
+
+<p>Take the first satellite as an example. The interval between successive
+eclipses ought to be its period of<span class='pagenum'><a name="Page_240" id="Page_240">[Pg 240]</a></span> revolution round Jupiter. Observe
+this period. It was not uniform. On the average it was 42 hours 47
+minutes, but it seemed to depend on the time of year. When Roemer
+observed in spring it was less, and in autumn it was more than usual.
+This was evidently a puzzling fact: what on earth can our year have to
+do with the motion of a moon of Jupiter's? It was probably, therefore,
+only an apparent change, caused either by our greater or less distance
+from Jupiter, or else by our greater or less speed of travelling to or
+from him. Considering it thus, he was led to see that, when the time of
+revolution seemed longest, we were receding fastest from Jupiter, and
+when shortest, approaching fastest.</p>
+
+<p><i>If</i>, then, light took time on its journey, <i>if</i> it travelled
+progressively, the whole anomaly would be explained.</p>
+
+<p>In a second the earth goes nineteen miles; therefore in 42&frac34; hours
+(the time of revolution of Jupiter's first satellite) it goes 2&middot;9
+million (say three million) miles. The eclipse happens punctually, but
+we do not see it till the light conveying the information has travelled
+the extra three million miles and caught up the earth. Evidently,
+therefore, by observing how much the apparent time of revolution is
+lengthened in one part of the earth's orbit and shortened in another,
+getting all the data accurately, and assuming the truth of our
+hypothetical explanation, we can calculate the velocity of light. This
+is what Roemer did.</p>
+
+<p>Now the maximum amount of retardation is just about fifteen seconds.
+Hence light takes this time to travel three million miles; therefore its
+velocity is three million divided by fifteen, say 200,000, or, as we now
+know more exactly, 186,000 miles every second. Note that the delay does
+not depend on our <i>distance</i>, but on our <i>speed</i>. One can tell this by
+common-sense as soon as we grasp the general idea of the explanation. A
+velocity cannot possibly depend on a distance only.</p>
+
+<p><span class='pagenum'><a name="Page_241" id="Page_241">[Pg 241]</a></span></p>
+<div class="figcenter" style="width: 500px;"><a name="Fig_75" id="Fig_75"></a>
+<img src="images/fig75.jpg" width="400" height="613" alt="Fig. 75." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 75.</span>&mdash;Eclipses of one of Jupiter&#39;s satellites. A
+diagram intended to illustrate the dependence of its apparent time of
+revolution (from eclipse to eclipse) on the motion of the earth; but not
+illustrating the matter at all well. TT&#39; T&#39;&#39; are successive positions of
+the earth, while JJ&#39; J&#39;&#39; are corresponding positions of Jupiter.</div>
+</div>
+
+<p><span class='pagenum'><a name="Page_242" id="Page_242">[Pg 242]</a></span></p><p>Roemer's explanation of the anomaly was not accepted by astronomers. It
+excited some attention, and was discussed, but it was found not
+obviously applicable to any of the satellites except the first, and not
+very simply and satisfactorily even to that. I have, of course, given
+you the theory in its most elementary and simple form. In actual fact a
+host of disturbing and complicated considerations come in&mdash;not so
+violently disturbing for the first satellite as for the others, because
+it moves so quickly, but still complicated enough.</p>
+
+<p>The fact is, the real motion of Jupiter's satellites is a most difficult
+problem. The motion even of our own moon (the lunar theory) is difficult
+enough: perturbed as its motion is by the sun. You know that Newton said
+it cost him more labour than all the rest of the <i>Principia</i>. But the
+motion of Jupiter's satellites is far worse. No one, in fact, has yet
+worked their theory completely out. They are perturbed by the sun, of
+course, but they also perturb each other, and Jupiter is far from
+spherical. The shape of Jupiter, and their mutual attractions, combine
+to make their motions most peculiar and distracting.</p>
+
+<p>Hence an error in the time of revolution of a satellite was not
+<i>certainly</i> due to the cause Roemer suggested, unless one could be sure
+that the inequality was not a real one, unless it could be shown that
+the theory of gravitation was insufficient to account for it. This had
+not then been done; so the half-made discovery was shelved, and properly
+shelved, as a brilliant but unverified speculation. It remained on the
+shelf for half a century, and was no doubt almost forgotten.</p>
+
+<p><span class='pagenum'><a name="Page_243" id="Page_243">[Pg 243]</a></span></p>
+<div class="figcenter" style="width: 500px;"><a name="Fig_76" id="Fig_76"></a>
+<img src="images/fig76.jpg" width="400" height="256" alt="Fig. 76." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 76.</span>&mdash;A Transit-instrument for the British
+astronomical expedition, 1874. Shewing in its essential features the
+simplest form of such an instrument.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_244" id="Page_244">[Pg 244]</a></span></p><p>Now a word or two about the man. He was a Dane, educated at Copenhagen,
+and learned in the mathematics. We first hear of him as appointed to
+assist Picard, the eminent French geodetic surveyor (whose admirable
+work in determining the length of a degree you remember in connection
+with Newton), who had come over to Denmark with the object of fixing the
+exact site of the old and extinct Tychonic observatory in the island of
+Huen. For of course the knowledge of the exact latitude and longitude of
+every place whence numerous observations have been taken must be an
+essential to the full interpretation of those observations. The
+measurements being finished, young Roemer accompanied Picard to Paris,
+and here it was, a few years after, that he read his famous paper
+concerning "An Inequality in the Motion of Jupiter's First Satellite,"
+and its explanation by means of an hypothesis of "the successive
+propagation of light."</p>
+
+<p>The later years of his life he spent in Copenhagen as a professor in the
+University and an enthusiastic observer of the heavens,&mdash;not a
+descriptive observer like Herschel, but a measuring observer like Sir
+George Airy or Tycho Brah&eacute;. He was, in fact, a worthy follower of Tycho,
+and the main work of his life is the development and devising of new and
+more accurate astronomical instruments. Many of the large and accurate
+instruments with which a modern observatory is furnished are the
+invention of this Dane. One of the finest observatories in the world is
+the Russian one at Pulkowa, and a list of the instruments there reads
+like an extended catalogue of Roemer's inventions.</p>
+
+<p>He not only <i>invented</i> the instruments, he had them made, being allowed
+money for the purpose; and he used them vigorously, so that at his death
+he left great piles of manuscript stored in the national observatory.</p>
+
+<p>Unfortunately this observatory was in the heart of the city, and was
+thus exposed to a danger from which such places ought to be as far as
+possible exempt.</p>
+
+<p>Some eighteen years after Roemer's death a great conflagration broke out
+in Copenhagen, and ruined large portions of the city. The successor to
+Roemer, Horrebow by name, fled from his house, with such valuables as he
+possessed, to the observatory, and there went on with his work. But
+before long the wind shifted, and to his horror<span class='pagenum'><a name="Page_245" id="Page_245">[Pg 245]</a></span> he saw the flames
+coming his way. He packed up his own and his predecessor's manuscript
+observations in two cases, and prepared to escape with them, but the
+neighbours had resorted to the observatory as a place of safety, and so
+choked up the staircase with their property that he was barely able to
+escape himself, let alone the luggage, and everything was lost.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_77" id="Fig_77"></a>
+<img src="images/fig77.jpg" width="400" height="484" alt="Fig. 77." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 77.</span>&mdash;Diagram of equatorially mounted telescope; CE
+is the polar axis parallel to the axis of the earth; AB the declination
+axis. The diurnal motion is compensated by motion about the polar axis
+only, the other being clamped.</div>
+</div>
+
+<p>Of all the observations, only three days' work remains, and these were
+carefully discussed by Dr. Galle, of Berlin, in 1845, and their
+nutriment extracted. These ancient observations are of great use for
+purposes of comparison with the present state of the heavens, and throw
+light upon possible changes that are going on. Of course nowadays such a
+series of observations would be printed<span class='pagenum'><a name="Page_246" id="Page_246">[Pg 246]</a></span> and distributed in many
+libraries, and so made practically indestructible.</p>
+
+<p>Sad as the disaster was to the posthumous fame of the great observer, a
+considerable compensation was preparing. The very year that the fire
+occurred in Denmark a quiet philosopher in England was speculating and
+brooding on a remarkable observation that he had made concerning the
+apparent motion of certain stars, and he was led thereby to a discovery
+of the first magnitude concerning the speed of light&mdash;a discovery which
+resuscitated the old theory of Roemer about Jupiter's satellites, and
+made both it and him immortal.</p>
+
+<p>James Bradley lived a quiet, uneventful, studious life, mainly at Oxford
+but afterwards at the National Observatory at Greenwich, of which he was
+third Astronomer-Royal, Flamsteed and Halley having preceded him in that
+office. He had taken orders, and lectured at Oxford as Savilian
+Professor. It is said that he pondered his great discovery while pacing
+the Long Walk at Magdalen College&mdash;and a beautiful place it is to
+meditate in.</p>
+
+<p>Bradley was engaged in making observations to determine if possible the
+parallax of some of the fixed stars. Parallax means the apparent
+relative shift of bodies due to a change in the observer's position. It
+is parallax which we observe when travelling by rail and looking out of
+window at the distant landscape. Things at different distances are left
+behind at different apparent rates, and accordingly they seem to move
+relatively to each other. The most distant objects are least affected;
+and anything enormously distant, like the moon, is not subject to this
+effect, but would retain its position however far we travelled, unless
+we had some extraordinarily precise means of observation.</p>
+
+<p>So with the fixed stars: they were being observed from a moving
+carriage&mdash;viz. the earth&mdash;and one moving at the rate of nineteen miles a
+second. Unless they were infinitely distant, or unless they were all at
+the same distance, they<span class='pagenum'><a name="Page_247" id="Page_247">[Pg 247]</a></span> must show relative apparent motions among
+themselves. Seen from one point of the earth's orbit, and then in six
+months from an opposite point, nearly 184 million miles away, surely
+they must show some difference of aspect.</p>
+
+<p>Remember that the old Copernican difficulty had never been removed. If
+the earth revolved round the sun, how came it that the fixed stars
+showed no parallax? The fact still remained a surprise, and the question
+a challenge. Picard, like other astronomers, supposed that it was only
+because the methods of observation had not been delicate enough; but now
+that, since the invention of the telescope and the founding of National
+Observatories, accuracy hitherto undreamt of was possible, why not
+attack the problem anew? This, then, he did, watching the stars with
+great care to see if in six months they showed any change in absolute
+position with reference to the pole of the heavens; any known secular
+motion of the pole, such as precession, being allowed for. Already he
+thought he detected a slight parallax for several stars near the pole,
+and the subject was exciting much interest.</p>
+
+<p>Bradley determined to attempt the same investigation. He was not
+destined to succeed in it. Not till the present century was success in
+that most difficult observation achieved; and even now it cannot be done
+by the absolute methods then attempted; but, as so often happens,
+Bradley, in attempting one thing, hit upon another, and, as it happened,
+one of still greater brilliance and importance. Let us trace the stages
+of his discovery.</p>
+
+<p>Atmospheric refraction made horizon observations useless for the
+delicacy of his purpose, so he chose stars near the zenith, particularly
+one&mdash;&#947; Draconis. This he observed very carefully at different
+seasons of the year by means of an instrument specially adapted for
+zenith observations, viz. a zenith sector. The observations were made in
+conjunction with a friend of his, an amateur astronomer named<span class='pagenum'><a name="Page_248" id="Page_248">[Pg 248]</a></span> Molyneux,
+and they were made at Kew. Molyneux was shortly made First Lord of the
+Admiralty, or something important of that sort, and gave up frivolous
+pursuits. So Bradley observed alone. They observed the star accurately
+early in the month of December, and then intended to wait six months.
+But from curiosity Bradley observed it again only about a week later. To
+his surprise, he found that it had already changed its position. He
+recorded his observation on the back of an old envelope: it was his wont
+thus to use up odd scraps of paper&mdash;he was not, I regret to say, a tidy
+or methodical person&mdash;and this odd piece of paper turned up long
+afterwards among his manuscripts. It has been photographed and preserved
+as an historical relic.</p>
+
+<p>Again and again he repeated the observation of the star, and continually
+found it moving still a little further and further south, an excessively
+small motion, but still an appreciable one&mdash;not to be set down to errors
+of observation. So it went on till March. It then waited, and after a
+bit longer began to return, until June. By September it was displaced as
+much to the north as it had been to the south, and by December it had
+got back to its original position. It had described, in fact, a small
+oscillation in the course of the year. The motion affected neighbouring
+stars in a similar way, and was called an "aberration," or wandering
+from their true place.</p>
+
+<p>For a long time Bradley pondered over this observation, and over others
+like them which he also made. He found one group of stars describing
+small circles, while others at a distance from them were oscillating in
+straight lines, and all the others were describing ellipses. Unless this
+state of things were cleared up, accurate astronomy was impossible. The
+fixed stars!&mdash;they were not fixed a bit. To refined and accurate
+observation, such as was now possible, they were all careering about in
+little orbits having a reference to the earth's year, besides any proper
+motion which they might<span class='pagenum'><a name="Page_249" id="Page_249">[Pg 249]</a></span> really have of their own, though no such motion
+was at present known. Not till Herschel was that discovered; not till
+this extraordinary aberration was allowed for could it be discovered.
+The effect observed by Bradley and Molyneux must manifestly be only an
+apparent motion: it was absurd to suppose a real stellar motion
+regulating itself according to the position of the earth. Parallax could
+not do it, for that would displace stars relatively among each other&mdash;it
+would not move similarly a set of neighbouring stars.</p>
+
+<p>At length, four years after the observation, the explanation struck him,
+while in a boat upon the Thames. He noticed the apparent direction of
+the wind changed whenever the boat started. The wind veered when the
+boat's motion changed. Of course the cause of this was obvious
+enough&mdash;the speed of the wind and the speed of the boat were compounded,
+and gave an apparent direction of the wind other than the true
+direction. But this immediately suggested a cause for what he had
+observed in the heavens. He had been observing an apparent direction of
+the stars other than the true direction, because he was observing from a
+moving vehicle. The real direction was doubtless fixed: the apparent
+direction veered about with the motion of the earth. It must be that
+light did not travel instantaneously, but gradually, as Roemer had
+surmised fifty years ago; and that the motion of the light was
+compounded with the motion of the earth.</p>
+
+<p>Think of a stream of light or anything else falling on a moving
+carriage. The carriage will run athwart the stream, the occupants of the
+carriage will mistake its true direction. A rifle fired through the
+windows of a railway carriage by a man at rest outside would make its
+perforations not in the true line of fire unless the train is
+stationary. If the train is moving, the line joining the holes will
+point to a place in advance of where the rifle is really located.</p>
+
+<p>So it is with the two glasses of a telescope, the object-glass<span class='pagenum'><a name="Page_250" id="Page_250">[Pg 250]</a></span> and
+eye-piece, which are pierced by the light; an astronomer, applying his
+eye to the tube and looking for the origin of the disturbance, sees it
+apparently, but not in its real position&mdash;its apparent direction is
+displaced in the direction of the telescope's motion; by an amount
+depending on the ratio of the velocity of the earth to the velocity of
+light, and on the angle between those two directions.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_78" id="Fig_78"></a>
+<img src="images/fig78.jpg" width="400" height="469" alt="Fig. 78." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 78.</span>&mdash;Aberration diagram. The light-ray L penetrates
+the object-glass of the moving telescope at O, but does not reach the
+eye-piece until the telescope has travelled to the second position.
+Consequently a moving telescope does not point out the true direction of
+the light, but aims at a point a little in advance.</div>
+</div>
+
+<p>But how minute is the displacement! The greatest effect is obtained when
+the two motions are at right angles to each other, <i>i.e.</i> when the star
+seen is at right angles to the direction of the earth's motion, but even
+then it is only 20", or <span class="above">1</span>&#8260;<span class="below">180</span>th part of a degree; one-ninetieth of the
+moon's apparent diameter. It could not be detected without a cross-wire
+in the telescope, and would only appear as a<span class='pagenum'><a name="Page_251" id="Page_251">[Pg 251]</a></span> slight displacement from
+the centre of the field, supposing the telescope accurately pointed to
+the true direction.</p>
+
+<p>But if this explanation be true, it at once gives a method of
+determining the velocity of light. The maximum angle of deviation,
+represented as a ratio of arc &divide; radius, amounts to</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Light velocity gradient equation">
+<tr class='tr2'>
+ <td class='tdcbb'>1</td>
+ <td align='center' rowspan='2'>&nbsp;&ndash;&nbsp;&nbsp;&middot;0001&nbsp;=&nbsp;</td>
+ <td class='tdcbb'>1</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'>180 &times; 57&#8531;</td>
+ <td align='center'>10,000</td>
+</tr>
+</table></div>
+
+<p class="noin">(a gradient of 1 foot in two miles). In other words, the velocity of
+light must be 10,000 times as great as the velocity of the earth in its
+orbit. This amounts to a speed of 190,000 miles a second&mdash;not so very
+different from what Roemer had reckoned it in order to explain the
+anomalies of Jupiter's first satellite.</p>
+
+<p>Stars in the direction in which the earth was moving would not be thus
+affected; there would be nothing in mere approach or recession to alter
+direction or to make itself in any way visible. Stars at right angles to
+the earth's line of motion would be most affected, and these would be
+all displaced by the full amount of 20 seconds of arc. Stars in
+intermediate directions would be displaced by intermediate amounts.</p>
+
+<p>But the line of the earth's motion is approximately a circle round the
+sun, hence the direction of its advance is constantly though slowly
+changing, and in one year it goes through all the points of the compass.
+The stars, being displaced always in the line of advance, must similarly
+appear to describe little closed curves, always a quadrant in advance of
+the earth, completing their orbits once a year. Those near the pole of
+the ecliptic will describe circles, being always at right angles to the
+motion. Those in the plane of the ecliptic (near the zodiac) will be
+sometimes at right angles to the motion, but at other times will be
+approached or receded from; hence these will oscillate like pendulums
+once a year; and intermediate stars will have intermediate motions&mdash;that
+is to say, will describe ellipses<span class='pagenum'><a name="Page_252" id="Page_252">[Pg 252]</a></span> of varying excentricity, but all
+completed in a year, and all with the major axis 20". This agreed very
+closely with what was observed.</p>
+
+<p>The main details were thus clearly and simply explained by the
+hypothesis of a finite velocity for light, "the successive propagation
+of light in time." This time there was no room for hesitation, and
+astronomers hailed the discovery with enthusiasm.</p>
+
+<p>Not yet, however, did Bradley rest. The finite velocity of light
+explained the major part of the irregularities he had observed, but not
+the whole. The more carefully he measured the amount of the deviation,
+the less completely accurate became its explanation.</p>
+
+<p>There clearly was a small outstanding error or discrepancy; the stars
+were still subject to an unexplained displacement&mdash;not, indeed, a
+displacement that repeated itself every year, but one that went through
+a cycle of changes in a longer period.</p>
+
+<p>The displacement was only about half that of aberration, and having a
+longer period was rather more difficult to detect securely. But the
+major difficulty was the fact that the two sorts of disturbances were
+co-existent, and the skill of disentangling them, and exhibiting the
+true and complete cause of each inequality, was very brilliant.</p>
+
+<p>For nineteen years did Bradley observe this minor displacement, and in
+that time he saw it go through a complete cycle. Its cause was now clear
+to him; the nineteen-year period suggested the explanation. It is the
+period in which the moon goes through all her changes&mdash;a period known to
+the ancients as the lunar cycle, or Metonic cycle, and used by them to
+predict eclipses. It is still used for the first rough approximation to
+the prediction of eclipses, and to calculate Easter. The "Golden Number"
+of the Prayer-book is the number of the year in this cycle.</p>
+
+<p>The cause of the second inequality, or apparent periodic motion of the
+stars, Bradley made out to be a nodding motion of the earth's axis.</p>
+
+<p><span class='pagenum'><a name="Page_253" id="Page_253">[Pg 253]</a></span></p><p>The axis of the earth describes its precessional orbit or conical
+motion every 26,000 years, as had long been known; but superposed upon
+this great movement have now been detected minute nods, each with a
+period of nineteen years.</p>
+
+<p>The cause of the nodding is completely accounted for by the theory of
+gravitation, just as the precession of the equinoxes was. Both
+disturbances result from the attraction of the moon on the non-spherical
+earth&mdash;on its protuberant equator.</p>
+
+<p>"Nutation" is, in fact, a small perturbation of precession. The motion
+may be observed in a non-sleeping top. The slow conical motion of the
+top's slanting axis represents the course of precession. Sometimes this
+path is loopy, and its little nods correspond to nutation.</p>
+
+<p>The probable existence of some such perturbation had not escaped the
+sagacity of Newton, and he mentions something about it in the
+<i>Principia</i>, but thinks it too small to be detected by observation. He
+was thinking, however, of a solar disturbance rather than a lunar one,
+and this is certainly very small, though it, too, has now been observed.</p>
+
+<p>Newton was dead before Bradley made these great discoveries, else he
+would have been greatly pleased to hear of them.</p>
+
+<p>These discoveries of aberration and nutation, says Delambre, the great
+French historian of science, secure to their author a distinguished
+place after Hipparchus and Kepler among the astronomers of all ages and
+all countries.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_254" id="Page_254">[Pg 254]</a></span></p>
+<h4><a name="NOTES_TO_LECTURE_XI" id="NOTES_TO_LECTURE_XI"></a>NOTES TO LECTURE XI</h4>
+
+
+<p><i>Lagrange</i> and <i>Laplace</i>, both tremendous mathematicians, worked very
+much in alliance, and completed Newton's work. The <i>M&eacute;canique C&eacute;leste</i>
+contains the higher intricacies of astronomy mathematically worked out
+according to the theory of gravitation. They proved the solar system to
+be stable; all its inequalities being periodic, not cumulative. And
+Laplace suggested the "nebular hypothesis" concerning the origin of sun
+and planets: a hypothesis previously suggested, and to some extent,
+elaborated, by Kant.</p>
+
+<p>A list of some of the principal astronomical researches of Lagrange and
+Laplace:&mdash;Libration of the moon. Long inequality of Jupiter and Saturn.
+Perturbations of Jupiter's satellites. Perturbations of comets.
+Acceleration of the moon's mean motion. Improved lunar theory.
+Improvements in the theory of the tides. Periodic changes in the form
+and obliquity of the earth's orbit. Stability of the solar system
+considered as an assemblage of rigid bodies subject to gravity.</p>
+
+<p>The two equations which establish the stability of the solar system
+are:&mdash;</p>
+
+<p class="center">
+<i>Sum (me<sup>2</sup>&#8730;d) = constant,</i><br />
+<br />
+and<br />
+<br />
+<i>Sum (m tan<sup>2</sup>&#952;&#8730;d) = constant;</i><br />
+</p>
+
+<p class="noin">where <i>m</i> is the mass of each planet, <i>d</i> its mean distance from the
+sun, <i>e</i> the excentricity of its orbit, and <i>&#952;</i> the inclination
+of its plane. However the expressions above formulated may change for
+individual planets, the sum of them for all the planets remains
+invariable.</p>
+
+<p>The period of the variations in excentricity of the earth's orbit is
+86,000 years; the period of conical revolution of the earth's axis is
+25,800 years. About 18,000 years ago the excentricity was at a maximum.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_255" id="Page_255">[Pg 255]</a></span></p>
+<h3><a name="LECTURE_XI" id="LECTURE_XI"></a>LECTURE XI</h3>
+
+<h5>LAGRANGE AND LAPLACE&mdash;THE STABILITY OF THE SOLAR SYSTEM, AND THE NEBULAR
+HYPOTHESIS</h5>
+
+
+<p><span class="smcap">Laplace</span> was the son of a small farmer or peasant of Normandy. His
+extraordinary ability was noticed by some wealthy neighbours, and by
+them he was sent to a good school. From that time his career was one
+brilliant success, until in the later years of his life his prominence
+brought him tangibly into contact with the deteriorating influence of
+politics. Perhaps one ought rather to say trying than deteriorating; for
+they seem trying to a strong character, deteriorating to a weak one&mdash;and
+unfortunately, Laplace must be classed in this latter category.</p>
+
+<p>It has always been the custom in France for its high scientific men to
+be conspicuous also in politics. It seems to be now becoming the fashion
+in this country also, I regret to say.</p>
+
+<p>The <i>life</i> of Laplace is not specially interesting, and I shall not go
+into it. His brilliant mathematical genius is unquestionable, and almost
+unrivalled. He is, in fact, generally considered to come in this respect
+next after Newton. His talents were of a more popular order than those
+of Lagrange, and accordingly he acquired fame and rank, and rose to the
+highest dignities. Nevertheless, as a man and a politician he hardly
+commands our respect, and in time-serving adjustability he is comparable
+to the redoubtable<span class='pagenum'><a name="Page_256" id="Page_256">[Pg 256]</a></span> Vicar of Bray. His scientific insight and genius
+were however unquestionably of the very highest order, and his work has
+been invaluable to astronomy.</p>
+
+<p>I will give a short sketch of some of his investigations, so far as they
+can be made intelligible without overmuch labour. He worked very much in
+conjunction with Lagrange, a more solid though a less brilliant man, and
+it is both impossible and unnecessary for us to attempt to apportion
+respective shares of credit between these two scientific giants, the
+greatest scientific men that France ever produced.</p>
+
+<p>First comes a research into the libration of the moon. This was
+discovered by Galileo in his old age at Arcetri, just before his
+blindness. The moon, as every one knows, keeps the same face to the
+earth as it revolves round it. In other words, it does not rotate with
+reference to the earth, though it does rotate with respect to outside
+bodies. Its libration consists in a sort of oscillation, whereby it
+shows us now a little more on one side, now a little more on the other,
+so that altogether we are cognizant of more than one-half of its
+surface&mdash;in fact, altogether of about three-fifths. It is a simple and
+unimportant matter, easily explained.</p>
+
+<div class="blockquot"><p>The motion of the moon may be analyzed into a rotation about its
+own axis combined with a revolution about the earth. The speed of
+the rotation is quite uniform, the speed of the revolution is not
+quite uniform, because the orbit is not circular but elliptical,
+and the moon has to travel faster in perigee than in apogee (in
+accordance with Kepler's second law). The consequence of this is
+that we see a little too far round the body of the moon, first on
+one side, then on the other. Hence it <i>appears</i> to oscillate
+slightly, like a lop-sided fly-wheel whose revolutions have been
+allowed to die away so that they end in oscillations of small
+amplitude.<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> Its axis of rotation, too, is not precisely
+perpendicular to its plane of revolution, and therefore we
+sometimes see a few hundred miles beyond its north<span class='pagenum'><a name="Page_257" id="Page_257">[Pg 257]</a></span> pole, sometimes
+a similar amount beyond its south. Lastly, there is a sort of
+parallax effect, owing to the fact that we see the rising moon from
+one point of view, and the setting moon from a point 8,000 miles
+distant; and this base-line of the earth's diameter gives us again
+some extra glimpses. This diurnal or parallactic libration is
+really more effective than the other two in extending our vision
+into the space-facing hemisphere of the moon.</p>
+
+<p>These simple matters may as well be understood, but there is
+nothing in them to dwell upon. The far side of the moon is probably
+but little worth seeing. Its features are likely to be more blurred
+with accumulations of meteoric dust than are those of our side, but
+otherwise they are likely to be of the same general character. </p></div>
+
+<p>The thing of real interest is the fact that the moon does turn the same
+face towards us; <i>i.e.</i> has ceased to rotate with respect to the earth
+(if ever it did so). The stability of this state of things was shown by
+Lagrange to depend on the shape of the moon. It must be slightly
+egg-shape, or prolate&mdash;extended in the direction of the earth; its
+earth-pointing diameter being a few hundred feet longer than its visible
+diameter; a cause slight enough, but nevertheless sufficient to maintain
+stability, except under the action of a distinct disturbing cause. The
+prolate or lemon-like shape is caused by the gravitative pull of the
+earth, balanced by the centrifugal whirl. The two forces balance each
+other as regards motion, but between them they have strained the moon a
+trifle out of shape. The moon has yielded as if it were perfectly
+plastic; in all probability it once was so.</p>
+
+<p>It may be interesting to note for a moment the correlative effect of
+this aspect of the moon, if we transfer ourselves to its surface in
+imagination, and look at the earth (cf. <a href="#Fig_41">Fig. 41</a>). The earth would be
+like a gigantic moon of four times our moon's diameter, and would go
+through its phases in regular order. But it would not rise or set: it
+would be fixed in the sky, and subject only to a minute oscillation to
+and fro once a month, by reason of the "libration" we have been speaking
+of. Its aspect, as seen by<span class='pagenum'><a name="Page_258" id="Page_258">[Pg 258]</a></span> markings on its surface, would rapidly
+change, going through a cycle in twenty-four hours; but its permanent
+features would be usually masked by lawless accumulations of cloud,
+mainly aggregated in rude belts parallel to the equator. And these
+cloudy patches would be the most luminous, the whitest portions; for of
+course it would be their silver lining that we would then be looking
+on.<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a></p>
+
+<p>Next among the investigations of Lagrange and Laplace we will mention
+the long inequality of Jupiter and Saturn. Halley had found that Jupiter
+was continually lagging behind its true place as given by the theory of
+gravitation; and, on the other hand, that Saturn was being accelerated.
+The lag on the part of Jupiter amounted to about 34&frac12; minutes in a
+century. Overhauling ancient observations, however, Halley found signs
+of the opposite state of things, for when he got far enough back Jupiter
+was accelerated and Saturn was being retarded.</p>
+
+<p>Here was evidently a case of planetary perturbation, and Laplace and
+Lagrange undertook the working of it out. They attacked it as a case of
+the problem of three bodies, viz. the sun, Jupiter, and Saturn; which
+are so enormously the biggest of the known bodies in the system that
+insignificant masses like the Earth, Mars, and the rest, may be wholly
+neglected. They succeeded brilliantly, after a long and complex
+investigation: succeeded, not in solving the problem of the three
+bodies, but, by considering their mutual<span class='pagenum'><a name="Page_259" id="Page_259">[Pg 259]</a></span> action as perturbations
+superposed on each other, in explaining the most conspicuous of the
+observed anomalies of their motion, and in laying the foundation of a
+general planetary theory.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_79" id="Fig_79"></a>
+<img src="images/fig79.jpg" width="400" height="380" alt="Fig. 79." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 79.</span>&mdash;Shewing the three conjunction places in the
+orbits of Jupiter and Saturn. The two planets are represented as leaving
+one of the conjunctions where Jupiter was being pulled back and Saturn
+being pulled forward by their mutual attraction.</div>
+</div>
+
+<div class="blockquot"><p>One of the facts that plays a large part in the result was known to
+the old astrologers, viz. that Jupiter and Saturn come into
+conjunction with a certain triangular symmetry; the whole scheme
+being called a trigon, and being mentioned several times by Kepler.
+It happens that five of Jupiter's years very nearly equal two of
+Saturn's,<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a> so that they get very nearly into conjunction three
+times in every five Jupiter years, but not exactly. The result of
+this close approach is that periodically one pulls the other on and
+is itself pulled back; but since the three points progress, it is
+not always the same planet which gets pulled back. The complete
+theory shows that in the year 1560 there was no marked
+perturbation: before that it was in one direction, while afterwards
+it was in the other direction, and the period of the whole cycle of
+disturbances<span class='pagenum'><a name="Page_260" id="Page_260">[Pg 260]</a></span> is 929 of our years. The solution of this long
+outstanding puzzle by the theory of gravitation was hailed with the
+greatest enthusiasm by astronomers, and it established the fame of
+the two French mathematicians.</p></div>
+
+<p>Next they attacked the complicated problem of the motions of Jupiter's
+satellites. They succeeded in obtaining a theory of their motions which
+represented fact very nearly indeed, and they detected the following
+curious relationship between the satellites:&mdash;The speed of the first
+satellite + twice the speed of the second is equal to the speed of the
+third.</p>
+
+<p>They found this, not empirically, after the manner of Kepler, but as a
+deduction from the law of gravitation; for they go on to show that even
+if the satellites had not started with this relation they would sooner
+or later, by mutual perturbation, get themselves into it. One singular
+consequence of this, and of another quite similar connection between
+their positions, is that all three satellites can never be eclipsed at
+once.</p>
+
+<p>The motion of the fourth satellite is less tractable; it does not so
+readily form an easy system with the others.</p>
+
+<p>After these great successes the two astronomers naturally proceeded to
+study the mutual perturbations of all other bodies in the solar system.
+And one very remarkable discovery they made concerning the earth and
+moon, an account of which will be interesting, though the details and
+processes of calculation are quite beyond us in a course like this.</p>
+
+<p>Astronomical theory had become so nearly perfect by this time, and
+observations so accurate, that it was possible to calculate many
+astronomical events forwards or backwards, over even a thousand years or
+more, with admirable precision.</p>
+
+<p>Now, Halley had studied some records of ancient eclipses, and had
+calculated back by means of the lunar theory to see whether the
+calculation of the time they ought to occur<span class='pagenum'><a name="Page_261" id="Page_261">[Pg 261]</a></span> would agree with the record
+of the time they did occur. To his surprise he found a discrepancy, not
+a large one, but still one quite noticeable. To state it as we know it
+now:&mdash;An eclipse a century ago happened twelve seconds later than it
+ought to have happened by theory; two centuries back the error amounted
+to forty-eight seconds, in three centuries it would be 108 seconds, and
+so on; the lag depending on the square of the time. By research, and
+help from scholars, he succeeded in obtaining the records of some very
+ancient eclipses indeed. One in Egypt towards the end of the tenth
+century <span class="ampm">A.D.</span>; another in 201 <span class="ampm">A.D.</span>; another a little before Christ; and
+one, the oldest of all of which any authentic record has been preserved,
+observed by the Chald&aelig;an astronomers in Babylon in the reign of
+Hezekiah.</p>
+
+<p>Calculating back to this splendid old record of a solar eclipse, over
+the intervening 2,400 years, the calculated and the observed times were
+found to disagree by nearly two hours. Pondering over an explanation of
+the discrepancy, Halley guessed that it must be because the moon's
+motion was not uniform, it must be going quicker and quicker, gaining
+twelve seconds each century on its previous gain&mdash;a discovery announced
+by him as "the acceleration of the moon's mean motion." The month was
+constantly getting shorter.</p>
+
+<p>What was the physical cause of this acceleration according to the theory
+of gravitation? Many attacked the question, but all failed. This was the
+problem Laplace set himself to work out. A singular and beautiful result
+rewarded his efforts.</p>
+
+<p>You know that the earth describes an elliptic orbit round the sun: and
+that an ellipse is a circle with a certain amount of flattening or
+"excentricity."<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a> Well, Laplace found that the excentricity of the
+earth's orbit must be changing,<span class='pagenum'><a name="Page_262" id="Page_262">[Pg 262]</a></span> getting slightly less; and that this
+change of excentricity would have an effect upon the length of the
+month. It would make the moon go quicker.</p>
+
+<p>One can almost see how it comes about. A decrease in excentricity means
+an increase in mean distance of the earth from the sun. This means to
+the moon a less solar perturbation. Now one effect of the solar
+perturbation is to keep the moon's orbit extra large: if the size of its
+orbit diminishes, its velocity must increase, according to Kepler's
+third law.</p>
+
+<p>Laplace calculated the amount of acceleration so resulting, and found it
+ten seconds a century; very nearly what observation required; for,
+though I have quoted observation as demanding twelve seconds per
+century, the facts were not then so distinctly and definitely
+ascertained.</p>
+
+<p>This calculation for a long time seemed thoroughly satisfactory, but it
+is not the last word on the subject. Quite lately an error has been
+found in the working, which diminishes the theoretical
+gravitation-acceleration to six seconds a century instead of ten, thus
+making it insufficient to agree exactly with fact. The theory of
+gravitation leaves an outstanding error. (The point is now almost
+thoroughly understood, and we shall return to it in <a href="#NOTES_FOR_LECTURE_XVIII">Lecture XVIII</a>).</p>
+
+<p>But another question arises out of this discussion. I have spoken of the
+excentricity of the earth's orbit as decreasing. Was it always
+decreasing? and if so, how far back was it so excentric that at
+perihelion the earth passed quite near the sun? If it ever did thus pass
+near the sun, the inference is manifest&mdash;the earth must at one time have
+been thrown off, or been separated off, from the sun.</p>
+
+<p>If a projectile could be fired so fast that it described an orbit round
+the earth&mdash;and the speed of fire to attain this lies between five and
+seven miles a second (not less than the one, nor more than the
+other)&mdash;it would ever<span class='pagenum'><a name="Page_263" id="Page_263">[Pg 263]</a></span> afterwards pass through its point of projection
+as one point of its elliptic orbit; and its periodic return through that
+point would be the sign of its origin. Similarly, if a satellite does
+<i>not</i> come near its central orb, and can be shown never to have been
+near it, the natural inference is that it has <i>not</i> been born from it,
+but has originated in some other way.</p>
+
+<p>The question which presented itself in connexion with the variable
+ellipticity of the earth's orbit was the following:&mdash;Had it always been
+decreasing, so that once it was excentric enough just to graze the sun
+at perihelion as a projected body would do?</p>
+
+<p>Into the problem thus presented Lagrange threw himself, and he succeeded
+in showing that no such explanation of the origin of the earth is
+possible. The excentricity of the orbit, though now decreasing, was not
+always decreasing; ages ago it was increasing: it passes through
+periodic changes. Eighteen thousand years ago its excentricity was a
+maximum; since then it has been diminishing, and will continue to
+diminish for 25,000 years more, when it will be an almost perfect
+circle; it will then begin to increase again, and so on. The obliquity
+of the ecliptic is also changing periodically, but not greatly: the
+change is less than three degrees.</p>
+
+<p>This research has, or ought to have, the most transcendent interest for
+geologists and geographers. You know that geologists find traces of
+extraordinary variations of temperature on the surface of the earth.
+England was at one time tropical, at another time glacial. Far away
+north, in Spitzbergen, evidence of the luxuriant vegetation of past ages
+has been found; and the explanation of these great climatic changes has
+long been a puzzle. Does not the secular variation in excentricity of
+the earth's orbit, combined with the precession of the equinoxes, afford
+a key? And if a key at all, it will be an accurate key, and enable us to
+calculate back with some precision to the date of the<span class='pagenum'><a name="Page_264" id="Page_264">[Pg 264]</a></span> glacial epoch;
+and again to the time when a tropical flora flourished in what is now
+northern Europe, <i>i.e.</i> to the date of the Carboniferous era.</p>
+
+<p>This aspect of the subject has recently been taught with vigour and
+success by Dr. Croll in his book "Climate and Time."</p>
+
+<div class="blockquot"><p>A brief and partial explanation of the matter may be given, because
+it is a point of some interest and is also one of fair simplicity.</p>
+
+<p>Every one knows that the climatic conditions of winter and summer
+are inverted in the two hemispheres, and that at present the sun is
+nearest to us in our (northern) winter. In other words, the earth's
+axis is inclined so as to tilt its north pole away from the sun at
+perihelion, or when the earth is at the part of its elliptic orbit
+nearest the sun's focus; and to tilt it towards the sun at
+aphelion. The result of this present state of things is to diminish
+the intensity of the average northern winter and of the average
+northern summer, and on the other hand to aggravate the extremes of
+temperature in the southern hemisphere; all other things being
+equal. Of course other things are not equal, and the distribution
+of land and sea is a still more powerful climatic agent than is the
+three million miles or so extra nearness of the sun. But it is
+supposed that the Antarctic ice-cap is larger than the northern,
+and increased summer radiation with increased winter cold would
+account for this.</p>
+
+<p>But the present state of things did not always obtain. The conical
+movement of the earth's axis (now known by a curious perversion of
+phrase as "precession") will in the course of 13,000 years or so
+cause the tilt to be precisely opposite, and then we shall have the
+more extreme winters and summers instead of the southern
+hemisphere.</p>
+
+<p>If the change were to occur now, it might not be overpowering,
+because now the excentricity is moderate. But if it happened some
+time back, when the excentricity was much greater, a decidedly
+different arrangement of climate may have resulted. There is no
+need to say <i>if</i> it happened some time back: it did happen, and
+accordingly an agent for affecting the distribution of mean
+temperature on the earth is to hand; though whether it is
+sufficient to achieve all that has been observed by geologists is a
+matter of opinion.</p>
+
+<p>Once more, the whole diversity of the seasons depends on the tilt
+of the earth's axis, the 23&deg; by which it is inclined to a
+perpendicular to the orbital plane; and this obliquity or tilt is
+subject to slow fluctuations. Hence there will come eras when all
+causes combine<span class='pagenum'><a name="Page_265" id="Page_265">[Pg 265]</a></span> to produce a maximum extremity of seasons in the
+northern hemisphere, and other eras when it is the southern
+hemisphere which is subject to extremes.</p></div>
+
+<p>But a grander problem still awaited solution&mdash;nothing less than the fate
+of the whole solar system. Here are a number of bodies of various sizes
+circulating at various rates round one central body, all attracted by
+it, and all attracting each other, the whole abandoned to the free play
+of the force of gravitation: what will be the end of it all? Will they
+ultimately approach and fall into the sun, or will they recede further
+and further from him, into the cold of space? There is a third possible
+alternative: may they not alternately approach and recede from him, so
+as on the whole to maintain a fair approximation to their present
+distances, without great and violent extremes of temperature either way?</p>
+
+<p>If any one planet of the system were to fall into the sun, more
+especially if it were a big one like Jupiter or Saturn, the heat
+produced would be so terrific that life on this earth would be
+destroyed, even at its present distance; so that we are personally
+interested in the behaviour of the other planets as well as in the
+behaviour of our own.</p>
+
+<p>The result of the portentously difficult and profoundly interesting
+investigation, here sketched in barest outline, is that the solar system
+is stable: that is to say, that if disturbed a little it will oscillate
+and return to its old state; whereas if it were unstable the slightest
+disturbance would tend to accumulate, and would sooner or later bring
+about a catastrophe. A hanging pendulum is stable, and oscillates about
+a mean position; its motion is periodic. A top-heavy load balanced on a
+point is unstable. All the changes of the solar system are periodic,
+<i>i.e.</i> they repeat themselves at regular intervals, and they never
+exceed a certain moderate amount.</p>
+
+<p>The period is something enormous. They will not have gone through all
+their changes until a period of 2,000,000<span class='pagenum'><a name="Page_266" id="Page_266">[Pg 266]</a></span> years has elapsed. This is
+the period of the planetary oscillation: "a great pendulum of eternity
+which beats ages as our pendulums beat seconds." Enormous it seems; and
+yet we have reason to believe that the earth has existed through many
+such periods.</p>
+
+<div class="blockquot"><p>The two laws of stability discovered and stated by Lagrange and
+Laplace I can state, though they may be difficult to understand:&mdash;</p>
+
+<p>Represent the masses of the several planets by <i>m<sub>1</sub></i>, <i>m<sub>2</sub></i>, &amp;c.; their
+mean distances from the sun (or radii vectores) by <i>r<sub>1</sub></i>, <i>r<sub>2</sub></i>, &amp;c.;
+the excentricities of their orbits by <i>e<sub>1</sub></i>, <i>e<sub>2</sub></i>, &amp;c.; and the
+obliquity of the planes of these orbits, reckoned from a single
+plane of reference or "invariable plane," by <i>&#952;<sub>1</sub></i>, <i>&#952;<sub>2</sub></i>,
+&amp;c.; then all these quantities (except m) are liable to
+fluctuate; but, however much they change, an increase for one
+planet will be accompanied by a decrease for some others; so that,
+taking all the planets into account, the sum of a set of terms like
+these, <i>m<sub>1</sub>e<sub>2</sub><sup>2</sup>&#8730;r<sub>1</sub></i> + <i>m<sub>2</sub>e<sub>2</sub><sup>2</sup>&#8730;r<sub>2</sub></i>
++ &amp;c., will remain always the same. This is summed up briefly in
+the following statement:</p>
+
+<p class="center">
+<i>&#931;(me<sup>2</sup>&#8730;r)</i> = constant.<br />
+</p>
+
+<p>That is one law, and the other is like it, but with inclination of
+orbit instead of excentricity, viz.:</p>
+
+<p class="center">
+<i>&#931;(m&#952;<sup>2</sup>&#8730;r)</i> = constant.<br />
+</p>
+
+<p>The value of each of these two constants can at any time be
+calculated. At present their values are small. Hence they always
+were and always will be small; being, in fact, invariable. Hence
+neither <i>e</i> nor <i>r</i> nor &#952; can ever become infinite, nor can
+their average value for the system ever become zero. </p></div>
+
+<p>The planets may share the given amount of total excentricity and
+obliquity in various proportions between themselves; but even if it were
+all piled on to one planet it would not be very excessive, unless the
+planet were so small a one as Mercury; and it would be most improbable
+that one planet should ever have all the excentricity of the solar
+system heaped upon itself. The earth, therefore, never has been, nor
+ever will be, enormously nearer the sun than it is at present: nor can
+it ever get very much<span class='pagenum'><a name="Page_267" id="Page_267">[Pg 267]</a></span> further off. Its changes are small and are
+periodic&mdash;an increase is followed by a decrease, like the swing of a
+pendulum.</p>
+
+<p>The above two laws have been called the Magna Charta of the solar
+system, and were long supposed to guarantee its absolute permanence. So
+far as the theory of gravitation carries us, they do guarantee its
+permanence; but something more remains to be said on the subject in a
+future lecture (<a href="#NOTES_FOR_LECTURE_XVIII">XVIII</a>).</p>
+
+<p>And now, finally, we come to a sublime speculation, thrown out by
+Laplace, not as the result of profound calculation, like the results
+hitherto mentioned, not following certainly from the theory of
+gravitation, or from any other known theory, and therefore not to be
+accepted as more than a brilliant hypothesis, to be confirmed or
+rejected as our knowledge extends. This speculation is the "Nebular
+hypothesis." Since the time of Laplace the nebular hypothesis has had
+ups and downs of credence, sometimes being largely believed in,
+sometimes being almost ignored. At the present time it holds the field
+with perhaps greater probability of ultimate triumph than has ever
+before seemed to belong to it&mdash;far greater than belonged to it when
+first propounded.</p>
+
+<p>It had been previously stated clearly and well by the philosopher Kant,
+who was intensely interested in "the starry heavens" as well as in the
+"mind of man," and who shewed in connexion with astronomy also a most
+surprising genius. The hypothesis ought by rights perhaps to be known
+rather by his name than by that of Laplace.</p>
+
+<p>The data on which it was founded are these:&mdash;Every motion in the solar
+system known at that time took place in one direction, and in one
+direction only. Thus the planets revolve round the sun, all going the
+same way round; moons revolve round the planets, still maintaining the
+same direction of rotation, and all the bodies that were known to rotate
+on their own axis did so with still the<span class='pagenum'><a name="Page_268" id="Page_268">[Pg 268]</a></span> same kind of spin. Moreover,
+all these motions take place in or near a single plane. The ancients
+knew that sun moon and planets all keep near to the ecliptic, within a
+belt known as the zodiac: none strays away into other parts of the sky.
+Satellites also, and rings, are arranged in or near the same plane; and
+the plane of diurnal spin, or equator of the different bodies, is but
+slightly tilted.</p>
+
+<p>Now all this could not be the result of chance. What could have caused
+it? Is there any connection or common ancestry possible, to account for
+this strange family likeness? There is no connection now, but there may
+have been once. Must have been, we may almost say. It is as though they
+had once been parts of one great mass rotating as a whole; for if such a
+rotating mass broke up, its parts would retain its direction of
+rotation. But such a mass, filling all space as far as or beyond Saturn,
+although containing the materials of the whole solar system in itself,
+must have been of very rare consistency. Occupying so much bulk it could
+not have been solid, nor yet liquid, but it might have been gaseous.</p>
+
+<p>Are there any such gigantic rotating masses of gas in the heaven now?
+Certainly there are; there are the nebul&aelig;. Some of the nebul&aelig; are now
+known to be gaseous, and some of them at least are in a state of
+rotation. Laplace could not have known this for certain, but he
+suspected it. The first distinctly spiral nebula was discovered by the
+telescope of Lord Rosse; and quite recently a splendid photograph of the
+great Andromeda nebula, by our townsman, Mr. Isaac Roberts, reveals what
+was quite unsuspected&mdash;and makes it clear that this prodigious mass also
+is in a state of extensive and majestic whirl.</p>
+
+<p>Very well, then, put this problem:&mdash;A vast mass of rotating gas is left
+to itself to cool for ages and to condense as it cools: how will it
+behave? A difficult mathematical problem, worthy of being attacked
+to-day; not yet at all adequately treated. There are those who believe
+that by<span class='pagenum'><a name="Page_269" id="Page_269">[Pg 269]</a></span> the complete treatment of such a problem all the history of the
+solar system could be evolved.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_80" id="Fig_80"></a>
+<img src="images/fig80.jpg" width="400" height="355" alt="Fig. 80." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 80.</span>&mdash;Lord Rosse&#39;s drawing of the spiral nebula in
+Canes Venatici, with the stub marks of the draughtsman unduly emphasised
+into features by the engraver.</div>
+</div>
+
+<p>Laplace pictured to himself this mass shrinking and thereby whirling
+more and more rapidly. A spinning body shrinking in size and retaining
+its original amount of rotation, as it will unless a brake is applied,
+must spin more and more rapidly as it shrinks. It has what
+mathematicians call a constant moment of momentum; and what it loses in
+leverage, as it shrinks, it gains in speed. The mass is held together by
+gravitation, every particle attracting every other particle; but since
+all the particles are describing curved paths, they will tend to fly off
+tangentially, and only a small excess of the gravitation force over the
+centrifugal is left to pull the particles in, and slowly to concentrate
+the nebula. The mutual gravitation of the parts is opposed by the
+centrifugal force of the whirl. At length a point is reached where<span class='pagenum'><a name="Page_270" id="Page_270">[Pg 270]</a></span> the
+two forces balance. A portion outside a certain line will be in
+equilibrium; it will be left behind, and the rest must contract without
+it. A ring is formed, and away goes the inner nucleus contracting
+further and further towards a centre. After a time another ring will be
+left behind in the same way, and so on. What happens to these rings?
+They rotate with the motion they possess when thrown or shrunk off; but
+will they remain rings? If perfectly regular they may; if there be any
+irregularity they are liable to break up. They will break into one or
+two or more large masses, which are ultimately very likely to collide
+and become one. The revolving body so formed is still a rotating gaseous
+mass; and it will go on shrinking and cooling and throwing off rings,
+like the larger nucleus by which it has been abandoned. As any nucleus
+gets smaller, its rate of rotation increases, and so the rings last
+thrown off will be spinning faster than those thrown off earliest. The
+final nucleus or residual central body will be rotating fastest of all.</p>
+
+<p>The nucleus of the whole original mass we now see shrunk up into what we
+call the sun, which is spinning on its axis once every twenty-five days.
+The rings successively thrown off by it are now the planets&mdash;some large,
+some small&mdash;those last thrown off rotating round him comparatively
+quickly, those outside much more slowly. The rings thrown off by the
+planetary gaseous masses as they contracted have now become satellites;
+except one ring which has remained without breaking up, and is to be
+seen rotating round Saturn still.</p>
+
+<p>One other similar ring, an abortive attempt at a planet, is also left
+round the sun (the zone of asteroids).</p>
+
+<p>Such, crudely and baldly, is the famous nebular hypothesis of Laplace.
+It was first stated, as has been said above, by the philosopher Kant,
+but it was elaborated into much fuller detail by the greatest of French
+mathematicians and astronomers.</p>
+
+<p>The contracting masses will condense and generate great<span class='pagenum'><a name="Page_271" id="Page_271">[Pg 271]</a></span> quantities of
+heat by their own shrinkage; they will at a certain stage condense to
+liquid, and after a time will begin to cool and congeal with a
+superficial crust, which will get thicker and thicker; but for ages they
+will remain hot, even after they have become thoroughly solid. The small
+ones will cool fastest; the big ones will retain their heat for an
+immense time. Bullets cool quickly, cannon-balls take hours or days to
+cool, planets take millions of years. Our moon may be nearly cold, but
+the earth is still warm&mdash;indeed, very hot inside. Jupiter is believed by
+some observers still to glow with a dull red heat; and the high
+temperature of the much larger and still liquid mass of the sun is
+apparent to everybody. Not till it begins to scum over will it be
+perceptibly cooler.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_81" id="Fig_81"></a>
+<img src="images/fig81.jpg" width="400" height="333" alt="Fig. 81." title="" />
+<span class="caption"><span class="smcap">Fig. 81.</span>&mdash;Saturn.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_272" id="Page_272">[Pg 272]</a></span></p><p>Many things are now known concerning heat which were not known to
+Laplace (in the above paragraph they are only hinted at), and these
+confirm and strengthen the general features of his hypothesis in a
+striking way; so do the most recent telescopic discoveries. But fresh
+possibilities have now occurred to us, tidal phenomena are seen to have
+an influence then wholly unsuspected, and it will be in a modified and
+amplified form that the philosopher of next century will still hold to
+the main features of this famous old Nebular Hypothesis respecting the
+origin of the sun and planets&mdash;the Evolution of the solar system.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_273" id="Page_273">[Pg 273]</a></span></p>
+<h4><a name="NOTES_TO_LECTURE_XII" id="NOTES_TO_LECTURE_XII"></a>NOTES TO LECTURE XII</h4>
+
+
+<p>The subject of stellar astronomy was first opened up by Sir William
+Herschel, the greatest observing astronomer.</p>
+
+<p><i>Frederick William Herschel</i> was born in Hanover in 1738, and brought up
+as a musician. Came to England in 1756. First saw a telescope in 1773.
+Made a great many himself, and began a survey of the heavens. His sister
+Caroline, born in 1750, came to England in 1772, and became his devoted
+assistant to the end of his life. Uranus discovered in 1781. Music
+finally abandoned next year, and the 40-foot telescope begun. Discovered
+two moons of Saturn and two of Uranus. Reviewed, described, and gauged
+all the visible heavens. Discovered and catalogued 2,500 nebul&aelig; and 806
+double stars. Speculated concerning the Milky Way, the nebulosity of
+stars, the origin and growth of solar systems. Discovered that the stars
+were in motion, not fixed, and that the sun as one of them was
+journeying towards a point in the constellation Hercules. Died in 1822,
+eighty-four years old. Caroline Herschel discovered eight comets, and
+lived on to the age of ninety-eight.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_274" id="Page_274">[Pg 274]</a></span></p>
+<h3><a name="LECTURE_XII" id="LECTURE_XII"></a>LECTURE XII</h3>
+
+<h5>HERSCHEL AND THE MOTION OF THE FIXED STARS</h5>
+
+
+<p><span class="smcap">We</span> may admit, I think, that, with a few notable exceptions, the work of
+the great men we have been recently considering was rather to complete
+and round off the work of Newton, than to strike out new and original
+lines.</p>
+
+<p>This was the whole tendency of eighteenth century astronomy. It appeared
+to be getting into an adult and uninteresting stage, wherein everything
+could be calculated and predicted. Labour and ingenuity, and a severe
+mathematical training, were necessary to work out the remote
+consequences of known laws, but nothing fresh seemed likely to turn up.
+Consequently men's minds began turning in other directions, and we find
+chemistry and optics largely studied by some of the greatest minds,
+instead of astronomy.</p>
+
+<p>But before the century closed there was destined to arise one remarkable
+exception&mdash;a man who was comparatively ignorant of that which had been
+done before&mdash;a man unversed in mathematics and the intricacies of
+science, but who possessed such a real and genuine enthusiasm and love
+of Nature that he overcame the force of adverse circumstances, and
+entering the territory of astronomy by a by-path, struck out a new line
+for himself, and infused into the science a healthy spirit of fresh life
+and activity.</p>
+
+<p>This man was William Herschel.</p>
+
+<p><span class='pagenum'><a name="Page_275" id="Page_275">[Pg 275]</a></span></p><p>"The rise of Herschel," says Miss Clerke, "is the one conspicuous
+anomaly in the otherwise somewhat quiet and prosy eighteenth century. It
+proved decisive of the course of events in the nineteenth. It was
+unexplained by anything that had gone before, yet all that came after
+hinged upon it. It gave a new direction to effort; it lent a fresh
+impulse to thought. It opened a channel for the widespread public
+interest which was gathering towards astronomical subjects to flow in."</p>
+
+<p>Herschel was born at Hanover in 1738, the son of an oboe player in a
+military regiment. The father was a good musician, and a cultivated man.
+The mother was a German <i>Frau</i> of the period, a strong, active,
+business-like woman, of strong character and profound ignorance. Herself
+unable to write, she set her face against learning and all new-fangled
+notions. The education of the sons she could not altogether control,
+though she lamented over it, but the education of her two daughters she
+strictly limited to cooking, sewing, and household management. These,
+however, she taught them well.</p>
+
+<p>It was a large family, and William was the fourth child. We need only
+remember the names of his younger brother Alexander, and of his much
+younger sister Caroline.</p>
+
+<p>They were all very musical&mdash;the youngest boy was once raised upon a
+table to play the violin at a public performance. The girls were
+forbidden to learn music by their mother, but their father sometimes
+taught them a little on the sly. Alexander was besides an ingenious
+mechanician.</p>
+
+<p>At the age of seventeen, William became oboist to the Hanoverian Guards,
+shortly before the regiment was ordered to England. Two years later he
+removed himself from the regiment, with the approval of his parents,
+though probably without the approbation or consent of the commanding
+officer, by whom such removal would be regarded as simple desertion,
+which indeed it was; and George III. long afterwards handed him an
+official pardon for it.</p>
+
+<p><span class='pagenum'><a name="Page_276" id="Page_276">[Pg 276]</a></span></p><p>At the age of nineteen, he was thus launched in England with an outfit
+of some French, Latin, and English, picked up by himself; some skill in
+playing the hautboy, the violin, and the organ, as taught by his father;
+and some good linen and clothing, and an immense stock of energy,
+provided by his mother.</p>
+
+<p>He lived as musical instructor to one or two militia bands in Yorkshire,
+and for three years we hear no more than this of him. But, at the end of
+that time, a noted organist, Dr. Miller, of Durham, who had heard his
+playing, proposed that he should come and live with him and play at
+concerts, which he was very glad to do. He next obtained the post of
+organist at Halifax; and some four or five years later he was invited to
+become organist at the Octagon Chapel in Bath, and soon led the musical
+life of that then very fashionable place.</p>
+
+<p>About this time he went on a short visit to his family at Hanover, by
+all of whom he was very much beloved, especially by his young sister
+Caroline, who always regarded him as specially her own brother. It is
+rather pitiful, however, to find that her domestic occupations still
+unfairly repressed and blighted her life. She says:&mdash;</p>
+
+<div class="blockquot"><p>"Of the joys and pleasures which all felt at this long-wished-for
+meeting with my&mdash;let me say my dearest&mdash;brother, but a small
+portion could fall to my share; for with my constant attendance at
+church and school, besides the time I was employed in doing the
+drudgery of the scullery, it was but seldom I could make one in the
+group when the family were assembled together." </p></div>
+
+<p>While at Bath he wrote many musical pieces&mdash;glees, anthems, chants,
+pieces for the harp, and an orchestral symphony. He taught a large
+number of pupils, and lived a hard and successful life. After fourteen
+hours or so spent in teaching and playing, he would retire at night to
+instruct his mind with a study of mathematics, optics, Italian, or
+Greek, in all of which he managed to make some<span class='pagenum'><a name="Page_277" id="Page_277">[Pg 277]</a></span> progress. He also about
+this time fell in with some book on astronomy.</p>
+
+<p>In 1763 his father was struck with paralysis, and two years later he
+died.</p>
+
+<p>William then proposed that Alexander should come over from Hanover and
+join him at Bath, which was done. Next they wanted to rescue their
+sister Caroline from her humdrum existence, but this was a more
+difficult matter. Caroline's journal gives an account of her life at
+this time that is instructive. Here are a few extracts from it:&mdash;</p>
+
+<div class="blockquot"><p>"My father wished to give me something like a polished education,
+but my mother was particularly determined that it should be a
+rough, but at the same time a useful one; and nothing further she
+thought was necessary but to send me two or three months to a
+sempstress to be taught to make household linen....</p>
+
+<p>"My mother would not consent to my being taught French, ... so all
+my father could do for me was to indulge me (and please himself)
+sometimes with a short lesson on the violin, when my mother was
+either in good humour or out of the way.... She had cause for
+wishing me not to know more than was necessary for being useful in
+the family; for it was her certain belief that my brother William
+would have returned to his country, and my eldest brother not have
+looked so high, if they had had a little less learning."</p></div>
+
+<p>However, seven years after the death of their father, William went over
+to Germany and returned to England in triumph, bringing Caroline with
+him: she being then twenty-two.</p>
+
+<p>So now began a busy life in Bath. For Caroline the work must have been
+tremendous. For, besides having to learn singing, she had to learn
+English. She had, moreover, to keep accounts and do the marketing.</p>
+
+<p>When the season at Bath was over, she hoped to get rather more of her
+brother William's society; but he was deep in optics and astronomy, used
+to sleep with the books under his pillow, read them during meals, and
+scarcely ever thought of anything else.</p>
+
+<p><span class='pagenum'><a name="Page_278" id="Page_278">[Pg 278]</a></span></p><p>He was determined to see for himself all the astronomical wonders; and
+there being a small Gregorian reflector in one of the shops, he hired
+it. But he was not satisfied with this, and contemplated making a
+telescope 20 feet long. He wrote to opticians inquiring the price of a
+mirror suitable, but found there were none so large, and that even the
+smaller ones were beyond his means. Nothing daunted, he determined to
+make some for himself. Alexander entered into his plans: tools, hones,
+polishers, and all sorts of rubbish were imported into the house, to the
+sister's dismay, who says:&mdash;</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_82" id="Fig_82"></a>
+<img src="images/fig82.jpg" width="400" height="175" alt="Fig. 82." title="" />
+<span class="caption"><span class="smcap">Fig. 82.</span>&mdash;Principle of Newtonian reflector.</span>
+</div>
+
+<div class="blockquot"><p>"And then, to my sorrow, I saw almost every room turned into a
+workshop. A cabinet-maker making a tube and stands of all
+descriptions in a handsomely furnished drawing-room; Alex. putting
+up a huge turning-machine (which he had brought in the autumn from
+Bristol, where he used to spend the summer) in a bed-room, for
+turning patterns, grinding glasses, and turning eye-pieces, &amp;c. At
+the same time music durst not lie entirely dormant during the
+summer, and my brother had frequent rehearsals at home."</p></div>
+
+<p>Finally, in 1774, at the age of thirty-six, he had made himself a
+5&frac12;-foot telescope, and began to view the heavens. So attached was he
+to the instrument that he would run from the concert-room between the
+parts, and take a look at the stars.</p>
+
+<p><span class='pagenum'><a name="Page_279" id="Page_279">[Pg 279]</a></span></p><p>He soon began another telescope, and then another. He must have made
+some dozen different telescopes, always trying to get them bigger and
+bigger; at last he got a 7-foot and then a 10-foot instrument, and began
+a systematic survey of the heavens; he also began to communicate his
+results to the Royal Society.</p>
+
+<p>He now took a larger house, with more room for workshops, and a grass
+plot for a 20-foot telescope, and still he went on grinding
+mirrors&mdash;literally hundreds of them.</p>
+
+<p>I read another extract from the diary of his sister, who waited on him
+and obeyed him like a spaniel:&mdash;</p>
+
+<div class="blockquot"><p>"My time was taken up with copying music and practising, besides
+attendance on my brother when polishing, since by way of keeping
+him alive I was constantly obliged to feed him by putting the
+victuals by bits into his mouth. This was once the case when, in
+order to finish a 7-foot mirror, he had not taken his hands from it
+for sixteen hours together. In general he was never unemployed at
+meals, but was always at those times contriving or making drawings
+of whatever came in his mind. Generally I was obliged to read to
+him whilst he was at the turning-lathe, or polishing mirrors&mdash;<i>Don
+Quixote</i>, <i>Arabian Nights' Entertainments</i>, the novels of Sterne,
+Fielding, &amp;c.; serving tea and supper without interrupting the work
+with which he was engaged, ... and sometimes lending a hand. I
+became, in time, as useful a member of the workshop as a boy might
+be to his master in the first year of his apprenticeship.... But as
+I was to take a part the next year in the oratorios, I had, for a
+whole twelvemonth, two lessons per week from Miss Fleming, the
+celebrated dancing-mistress, to drill me for a gentlewoman (God
+knows how she succeeded). So we lived on without interruption. My
+brother Alex. was absent from Bath for some months every summer,
+but when at home he took much pleasure in executing some turning or
+clockmaker's work for his brother."</p></div>
+
+<p>The music, and the astronomy, and the making of telescopes, all went on
+together, each at high pressure, and enough done in each to satisfy any
+ordinary activity. But<span class='pagenum'><a name="Page_280" id="Page_280">[Pg 280]</a></span> the Herschels knew no rest. Grinding mirrors by
+day, concerts and oratorios in the evening, star-gazing at night. It is
+strange his health could stand it.</p>
+
+<p>The star-gazing, moreover, was no <i>dilettante</i> work; it was based on a
+serious system&mdash;a well thought out plan of observation. It was nothing
+less than this&mdash;to pass the whole heavens steadily and in order through
+the telescope, noting and describing and recording every object that
+should be visible, whether previously known or unknown. The operation is
+called sweeping; but it is not a rapid passage from one object to
+another, as the term might suggest; it is a most tedious business, and
+consists in following with the telescope a certain field of view for
+some minutes, so as to be sure that nothing is missed, then shifting it
+to the next overlapping field, and watching again. And whatever object
+appears must be scrutinized anxiously to see what there is peculiar
+about it. If a star, it may be double, or it may be coloured, or it may
+be nebulous; or again it may be variable, and so its brightness must be
+estimated in order to compare with a subsequent observation.</p>
+
+<p>Four distinct times in his life did Herschel thus pass the whole visible
+heavens under review; and each survey occupied him several years. He
+discovered double stars, variable stars, nebul&aelig;, and comets; and Mr.
+William Herschel, of Bath, the amateur astronomer, was gradually
+emerging from his obscurity, and becoming a known man.</p>
+
+<p>Tuesday, the 13th of March, 1781, is a date memorable in the annals of
+astronomy. "On this night," he writes to the Royal Society, "in
+examining the small stars near <i>&#951;</i> Geminorum, I perceived one
+visibly larger than the rest. Struck with its uncommon appearance, I
+compared it to <i>&#951;</i> Geminorum and another star, and finding it so
+much larger than either, I suspected it to be a comet."</p>
+
+<p>The "comet" was immediately observed by professional astronomers, and
+its orbit was computed by some of them.<span class='pagenum'><a name="Page_281" id="Page_281">[Pg 281]</a></span> It was thus found to move in
+nearly a circle instead of an elongated ellipse, and to be nearly twice
+as far from the sun as Saturn. It was no comet, it was a new planet;
+more than 100 times as big as the earth, and nearly twice as far away as
+Saturn. It was presently christened "Uranus."</p>
+
+<p>This was a most striking discovery, and the news sped over Europe. To
+understand the interest it excited we must remember that such a
+discovery was unique. Since the most ancient times of which men had any
+knowledge, the planets Mercury, Venus, Mars, Jupiter, Saturn, had been
+known, and there had been no addition to their number. Galileo and
+others had discovered satellites indeed, but a new primary planet was an
+entire and utterly unsuspected novelty.</p>
+
+<p>One of the most immediate consequences of the event was the discovery of
+Herschel himself. The Royal Society made him a Fellow the same year. The
+University of Oxford dubbed him a doctor; and the King sent for him to
+bring his telescope and show it at Court. So to London and Windsor he
+went, taking with him his best telescope. Maskelyne, the then
+Astronomer-Royal, compared it with the National one at Greenwich, and
+found Herschel's home-made instrument far the better of the two. He had
+a stand made after Herschel's pattern, but was so disgusted with his own
+instrument now that he scarcely thought it worthy of the stand when it
+was made. At Windsor, George III. was very civil, and Mr. Herschel was
+in great request to show the ladies of the Court Saturn and other
+objects of interest. Mr. Herschel exhibited a piece of worldly wisdom
+under these circumstances, that recalls faintly the behaviour of Tycho
+Brah&eacute; under similar circumstances. The evening when the exhibition was
+to take place threatened to become cloudy and wet, so Herschel rigged up
+an artificial Saturn, constructed of card and tissue paper, with a lamp
+behind it, in the distant wall of a garden; and, when the time came, his
+new titled friends were regaled with a view of this<span class='pagenum'><a name="Page_282" id="Page_282">[Pg 282]</a></span> imitation Saturn
+through the telescope&mdash;the real one not being visible. They went away
+much pleased.</p>
+
+<p>He stayed hovering between Windsor and Greenwich, and uncertain what was
+to be the outcome of all this regal patronizing. He writes to his sister
+that he would much rather be back grinding mirrors at Bath. And she
+writes begging him to come, for his musical pupils were getting
+impatient. They had to get the better of their impatience, however, for
+the King ultimately appointed him astronomer or rather telescope-maker
+to himself, and so Caroline and the whole household were sent for, and
+established in a small house at Datchet.</p>
+
+<p>From being a star-gazing musician, Herschel thus became a practical
+astronomer. Henceforth he lived in his observatory; only on wet and
+moonlight nights could he be torn away from it. The day-time he devoted
+to making his long-contemplated 20-foot telescope.</p>
+
+<p>Not yet, however, were all their difficulties removed. The house at
+Datchet was a tumble-down barn of a place, chosen rather as a workshop
+and observatory than as a dwelling-house. And the salary allowed him by
+George III. was scarcely a princely one. It was, as a matter of fact,
+&pound;200 a year. The idea was that he would earn his living by making
+telescopes, and so indeed he did. He made altogether some hundreds.
+Among others, four for the King. But this eternal making of telescopes
+for other people to use or play with was a weariness to the flesh. What
+he wanted was to observe, observe, observe.</p>
+
+<p>Sir William Watson, an old friend of his, and of some influence at
+Court, expressed his mind pretty plainly concerning Herschel's position;
+and as soon as the King got to understand that there was anything the
+matter, he immediately offered &pound;2,000 for a gigantic telescope to be
+made for Herschel's own use. Nothing better did he want in life. The
+whole army of carpenters and craftsmen resident in Datchet were pressed
+into the service. Furnaces for the<span class='pagenum'><a name="Page_283" id="Page_283">[Pg 283]</a></span> speculum metal were built, stands
+erected, and the 40-foot telescope fairly begun. It cost &pound;4,000 before
+it was finished, but the King paid the whole.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_83" id="Fig_83"></a>
+<img src="images/fig83.jpg" width="400" height="428" alt="Fig. 83." title="" />
+<span class="caption"><span class="smcap">Fig. 83.</span>&mdash;Herschel&#39;s 40-foot telescope.</span>
+</div>
+
+<p>With it he discovered two more satellites to Saturn (five hitherto had
+been known), and two moons to his own planet Uranus. These two are now
+known as Oberon and Titania. They were not seen again till some forty
+years after, when his son, Sir John Herschel, reobserved them. And in
+1847, Mr. Lassell, at his house, "Starfield," near Liverpool, discovered
+two more, called Ariel and Umbriel, making the<span class='pagenum'><a name="Page_284" id="Page_284">[Pg 284]</a></span> number four, as now
+known. Mr. Lassell also discovered, with a telescope of his own making,
+an eighth satellite of Saturn&mdash;Hyperion&mdash;and a satellite to Neptune.</p>
+
+<p>A letter from a foreign astronomer about this period describes Herschel
+and his sister's method of work:&mdash;</p>
+
+<div class="blockquot"><p>"I spent the night of the 6th of January at Herschel's, in Datchet,
+near Windsor, and had the good luck to hit on a fine evening. He
+has his 20-foot Newtonian telescope in the open air, and mounted in
+his garden very simply and conveniently. It is moved by an
+assistant, who stands below it.... Near the instrument is a clock
+regulated to sidereal time.... In the room near it sits Herschel's
+sister, and she has Flamsteed's atlas open before her. As he gives
+her the word, she writes down the declination and right ascension,
+and the other circumstances of the observation. In this way
+Herschel examines the whole sky without omitting the least part. He
+commonly observes with a magnifying power of one hundred and fifty,
+and is sure that after four or five years he will have passed in
+review every object above our horizon. He showed me the book in
+which his observations up to this time are written, and I am
+astonished at the great number of them. Each sweep covers 2&deg; 15' in
+declination, and he lets each star pass at least three times
+through the field of his telescope, so that it is impossible that
+anything can escape him. He has already found about 900 double
+stars, and almost as many nebul&aelig;. I went to bed about one o'clock,
+and up to that time he had found that night four or five new
+nebul&aelig;. The thermometer in the garden stood at 13&deg; Fahrenheit; but,
+in spite of this, Herschel observes the whole night through, except
+that he stops every three or four hours and goes into the room for
+a few moments. For some years Herschel has observed the heavens
+every hour when the weather is clear, and this always in the open
+air, because he says that the telescope only performs well when it
+is at the same temperature as the air. He protects himself against
+the weather by putting on more clothing. He has an excellent
+constitution, and thinks about nothing else in the world but the
+celestial bodies. He has promised me in the most cordial way,
+entirely in the service of astronomy, and without thinking of his
+own interest, to see to the telescopes I have ordered for European
+observatories, and he will himself attend to the preparation of the
+mirrors."</p></div>
+
+<p><span class='pagenum'><a name="Page_285" id="Page_285">[Pg 285]</a></span></p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_84" id="Fig_84"></a>
+<img src="images/fig84.jpg" width="400" height="488" alt="Fig. 84." title="" />
+<span class="caption"><span class="smcap">Fig. 84.</span>&mdash;William Herschel.<br />
+<i>From an Original Picture in the Possession of</i> <span class="smcap">Wm. Watson</span>, M.D.,
+F.R.S.<br /><i>Painted by Abbott.</i> <i>Engraved by Ryder.</i></span>
+</div>
+
+<p>In 1783, Herschel married an estimable lady who sympathized with his
+pursuits. She was the only daughter of a City magnate, so his pecuniary
+difficulties, such as they were (they were never very troublesome to
+him), came to an end. They moved now into a more commodious house at
+Slough. Their one son, afterwards the famous Sir John Herschel,<span class='pagenum'><a name="Page_286" id="Page_286">[Pg 286]</a></span> was
+born some nine years later. But the marriage was rather a blow to his
+devoted sister: henceforth she lived in lodgings, and went over at
+night-time to help him observe. For it must be remarked that this family
+literally turned night into day. Whatever sleep they got was in the
+day-time. Every fine night without exception was spent in observing: and
+the quite incredible fierceness of the pursuit is illustrated, as
+strongly as it can be, by the following sentence out of Caroline's
+diary, at the time of the move from Datchet to Slough: "The last night
+at Datchet was spent in sweeping till daylight, and by the next evening
+the telescope stood ready for observation at Slough."</p>
+
+<p>Caroline was now often allowed to sweep with a small telescope on her
+own account. In this way she picked up a good many nebul&aelig; in the course
+of her life, and eight comets, four of which were quite new, and one of
+which, known since as Encke's comet, has become very famous.</p>
+
+<p>The work they got through between them is something astonishing. He made
+with his own hands 430 parabolic mirrors for reflecting telescopes,
+besides a great number of complete instruments. He was forty-two when he
+began contributing to the Royal Society; yet before he died he had sent
+them sixty-nine long and elaborate treatises. One of these memoirs is a
+catalogue of 1000 nebul&aelig;. Fifteen years after he sends in another 1000;
+and some years later another 500. He also discovered 806 double stars,
+which he proved were really corrected from the fact that they revolved
+round each other (<a href="#Page_309">p. 309</a>). He lived to see some of them perform half a
+revolution. For him the stars were not fixed: they moved slowly among
+themselves. He detected their proper motions. He passed the whole
+northern firmament in review four distinct times; counted the stars in
+3,400 gauge-fields, and estimated the brightness of hundreds of stars.
+He also measured as accurately as<span class='pagenum'><a name="Page_287" id="Page_287">[Pg 287]</a></span> he could their proper motions,
+devising for this purpose the method which still to this day remains in
+use.</p>
+
+<p>And what is the outcome of it all? It is not Uranus, nor the satellites,
+nor even the double stars and the nebul&aelig; considered as mere objects: it
+is the beginning of a science of the stars.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_85" id="Fig_85"></a>
+<img src="images/fig85.jpg" width="400" height="446" alt="Fig. 85." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 85.</span>&mdash;Caroline Herschel.<br />
+<i>From a Drawing from Life, by</i> <span class="smcap">George M&uuml;ller</span>, 1847.</span>
+</div>
+
+<p>Hitherto the stars had only been observed for nautical and practical
+purposes. Their times of rising and southing and setting had been noted;
+they had been treated as a clock or piece of dead mechanism, and as
+fixed points of reference. All the energies of astronomers had gone out
+towards the solar system. It was the planets that had been<span class='pagenum'><a name="Page_288" id="Page_288">[Pg 288]</a></span> observed.
+Tycho had observed and tabulated their positions. Kepler had found out
+some laws of their motion. Galileo had discovered their peculiarities
+and attendants. Newton and Laplace had perceived every detail of their
+laws.</p>
+
+<p>But for the stars&mdash;the old Ptolemaic system might still have been true.
+They might still be mere dots in a vast crystalline sphere, all set at
+about one distance, and subservient to the uses of the earth.</p>
+
+<p>Herschel changed all this. Instead of sameness, he found variety;
+instead of uniformity of distance, limitless and utterly limitless
+fields and boundless distances; instead of rest and quiescence, motion
+and activity; instead of stagnation, life.</p>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_86" id="Fig_86"></a>
+<img src="images/fig86.jpg" width="350" height="374" alt="Fig. 86." title="" />
+<span class="caption"><span class="smcap">Fig. 86.</span>&mdash;The double-double star &#949; Lyr&aelig; as seen
+under three different powers.</span>
+</div>
+
+<p>Yes, that is what Herschel discovered&mdash;the life and activity of the
+whole visible universe. No longer was our little solar system to be the
+one object of regard, no longer were its phenomena to be alone
+interesting to man. With Herschel every star was a solar system. And
+more than that: he found suns revolving round suns, at distances such as
+the mind reels at, still obeying the same law of gravitation as pulls an
+apple from a tree. He tried hard to estimate the distance of the stars
+from the earth, but there he failed: it was too hopeless a problem. It
+was solved some time after his death by Bessel, and the distances of<span class='pagenum'><a name="Page_289" id="Page_289">[Pg 289]</a></span>
+many stars are now known but these distances are awful and unspeakable.
+Our distance from the sun shrinks up into a mere speck&mdash;the whole solar
+system into a mere unit of measurement, to be repeated hundreds of
+thousands of times before we reach the stars.</p>
+
+<p>Yet their motion is visible&mdash;yes, to very accurate measurement quite
+plain. One star, known as 61 Cygni, was then and is now rushing along at
+the rate of 100 miles every second. Not that you must imagine that this
+makes any obvious and apparent change in its position. No, for all
+ordinary and practical purposes they are still fixed stars; thousands of
+years will show us no obvious change; "Adam" saw precisely the same
+constellations as we do: it is only by refined micrometric measurement
+with high magnifying power that their flight can be detected.</p>
+
+<p>But the sun is one of the stars&mdash;not by any means a specially large or
+bright one; Sirius we now know to be twenty times as big as the sun. The
+sun is one of the stars: then is it at rest? Herschel asked this
+question and endeavoured to answer it. He succeeded in the most
+astonishing manner. It is, perhaps, his most remarkable discovery, and
+savours of intuition. This is how it happened. With imperfect optical
+means and his own eyesight to guide him, he considered and pondered over
+the proper motion of the stars as he had observed it, till he discovered
+a kind of uniformity running through it all. Mixed up with
+irregularities and individualities, he found that in a certain part of
+the heavens the stars were on the whole opening out&mdash;separating slowly
+from each other; on the opposite side of the heavens they were on the
+average closing up&mdash;getting slightly nearer to each other; while in
+directions at right angles to this they were fairly preserving their
+customary distances asunder.</p>
+
+<p>Now, what is the moral to be drawn from such uniformity of behaviour
+among unconnected bodies? Surely that this part of their motion is only
+apparent&mdash;that it is we who are moving. Travelling over a prairie
+bounded by a belt of<span class='pagenum'><a name="Page_290" id="Page_290">[Pg 290]</a></span> trees, we should see the trees in our line of
+advance opening out, and those behind closing up; we should see in fact
+the same kind of apparent motion as Herschel was able to detect among
+the stars: the opening out being most marked near the constellation
+Hercules. The conclusion is obvious: the sun, with all its planets, must
+be steadily moving towards a point in the constellation Hercules. The
+most accurate modern research has been hardly able to improve upon this
+statement of Herschel's. Possibly the solar system may ultimately be
+found to revolve round some other body, but what that is no one knows.
+All one can tell is the present direction of the majestic motion: since
+it was discovered it has continued unchanged, and will probably so
+continue for thousands of years.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_87" id="Fig_87"></a>
+<img src="images/fig87.jpg" width="400" height="353" alt="Fig. 87." title="" />
+<span class="caption"><span class="smcap">Fig. 87.</span>&mdash;Old drawing of the cluster in Hercules.</span>
+</div>
+
+<p>And, finally, concerning the nebul&aelig;. These mysterious objects exercised
+a strong fascination for Herschel, and<span class='pagenum'><a name="Page_291" id="Page_291">[Pg 291]</a></span> many are the speculations he
+indulges in concerning them. At one time he regards them all as clusters
+of stars, and the Milky Way as our cluster; the others he regards as
+other universes almost infinitely distant; and he proceeds to gauge and
+estimate the shape of our own universe or galaxy of suns, the Milky Way.</p>
+
+<p>Later on, however, he pictures to himself the nebul&aelig; as nascent suns:
+solar systems before they are formed. Some he thinks have begun to
+aggregate, while some are still glowing gas.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_88" id="Fig_88"></a>
+<img src="images/fig88.jpg" width="400" height="271" alt="Fig. 88." title="" />
+<span class="caption"><span class="smcap">Fig. 88.</span>&mdash;Old drawing of the Andromeda nebula.</span>
+</div>
+
+<p>He likens the heavens to a garden in which there are plants growing in
+all manner of different stages: some shooting, some in leaf, some in
+flower, some bearing seed, some decaying; and thus at one inspection we
+have before us the whole life-history of the plant.</p>
+
+<p>Just so he thinks the heavens contain worlds, some old, some dead, some
+young and vigorous, and some in the act of being formed. The nebul&aelig; are
+these latter, and the nebulous stars are a further stage in the
+condensation towards a sun.</p>
+
+<p><span class='pagenum'><a name="Page_292" id="Page_292">[Pg 292]</a></span></p><p>And thus, by simple observation, he is led towards something very like
+the nebular hypothesis of Laplace; and his position, whether it be true
+or false, is substantially the same as is held to-day.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_89" id="Fig_89"></a>
+<img src="images/fig89.jpg" width="400" height="342" alt="Fig. 89." title="" />
+<span class="caption"><span class="smcap">Fig. 89.</span>&mdash;The great nebula in Orion.</span>
+</div>
+
+<p>We <i>know</i> now that many of the nebul&aelig; consist of innumerable isolated
+particles and may be spoken of as gas. We know that some are in a state
+of whirling motion. We know also that such gas left to itself will
+slowly as it cools condense and shrink, so as to form a central solid
+nucleus; and also, if it were in whirling motion, that it would send off
+rings from itself, and that these rings could break up into planets. In
+two familiar cases the ring has<span class='pagenum'><a name="Page_293" id="Page_293">[Pg 293]</a></span> not yet thus aggregated into planet or
+satellite&mdash;the zone of asteroids, and Saturn's ring.</p>
+
+<p>The whole of this could not have been asserted in Herschel's time: for
+further information the world had to wait.</p>
+
+<p>These are the problems of modern astronomy&mdash;these and many others, which
+are the growth of this century, aye, and the growth of the last thirty
+or forty, and indeed of the last ten years. Even as I write, new and
+very confirmatory discoveries are being announced. The Milky Way <i>does</i>
+seem to have some affinity with our sun. And the chief stars of the
+constellation of Orion constitute another family, and are enveloped in
+the great nebula, now by photography perceived to be far greater than
+had ever been imagined.</p>
+
+<p>What is to be the outcome of it all I know not; but sure I am of this,
+that the largest views of the universe that we are able to frame, and
+the grandest manner of its construction that we can conceive, are
+certain to pale and shrink and become inadequate when confronted with
+the truth.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_294" id="Page_294">[Pg 294]</a></span></p>
+<h4><a name="NOTES_TO_LECTURE_XIII" id="NOTES_TO_LECTURE_XIII"></a>NOTES TO LECTURE XIII</h4>
+
+
+<p><span class="smcap">Bode's Law.</span>&mdash;Write down the series 0, 3, 6, 12, 24, 48, &amp;c.; add 4 to
+each, and divide by 10; you get the series:</p>
+
+
+<div class='center'>
+<table border="0" width="100%" cellpadding="4" cellspacing="2" summary="Bode's Law Series">
+<tr>
+ <td align='center'>&middot;4</td>
+ <td align='center'>&middot;7</td>
+ <td align='center'>1&middot;0</td>
+ <td align='center'>1&middot;6</td>
+ <td align='center'>2&middot;8</td>
+ <td align='center'>5&middot;2</td>
+ <td align='center'>10&middot;0</td>
+ <td align='center'>19&middot;6</td>
+ <td align='center'>38&middot;8</td>
+</tr>
+<tr>
+ <td align='center'>Mercury</td>
+ <td align='center'>Venus</td>
+ <td align='center'>Earth</td>
+ <td align='center'>Mars</td>
+ <td align='center'>&mdash;&mdash;</td>
+ <td align='center'>Jupiter</td>
+ <td align='center'>Saturn</td>
+ <td align='center'>Uranus</td>
+ <td align='center'>&mdash;&mdash;</td>
+</tr>
+</table></div>
+
+<p class="noin">numbers which very fairly represent the distances of the then known
+planets from the sun in the order specified.</p>
+
+<p>Ceres was discovered on the 1st of January, 1801, by Piazzi; Pallas in
+March, 1802, by Olbers; Juno in 1804, by Harding; and Vesta in 1807, by
+Olbers. No more asteroids were discovered till 1845, but there are now
+several hundreds known. Their diameters range from 500 to 20 miles.</p>
+
+<p>Neptune was discovered from the perturbations of Uranus by sheer
+calculation, carried on simultaneously and independently by Leverrier in
+Paris, and Adams in Cambridge. It was first knowingly seen by Galle, of
+Berlin, on the 23rd of September, 1846.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_295" id="Page_295">[Pg 295]</a></span></p>
+<h3><a name="LECTURE_XIII" id="LECTURE_XIII"></a>LECTURE XIII</h3>
+
+<h5>THE DISCOVERY OF THE ASTEROIDS</h5>
+
+
+<p><span class="smcap">Up</span> to the time of Herschel, astronomical interest centred on the solar
+system. Since that time it has been divided, and a great part of our
+attention has been given to the more distant celestial bodies. The solar
+system has by no means lost its interest&mdash;it has indeed gained in
+interest continually, as we gain in knowledge concerning it; but in
+order to follow the course of science it will be necessary for us to
+oscillate to and fro, sometimes attending to the solar system&mdash;the
+planets and their satellites&mdash;sometimes extending our vision to the
+enormously more distant stellar spaces.</p>
+
+<p>Those who have read the third lecture in Part I. will remember the
+speculation in which Kepler indulged respecting the arrangements of the
+planets, the order in which they succeeded one another in space, and the
+law of their respective distances from the sun; and his fanciful guess
+about the five regular solids inscribed and circumscribed about their
+orbits.</p>
+
+<p>The rude coincidences were, however, accidental, and he failed to
+discover any true law. No thoroughly satisfactory law is known at the
+present day. And yet, if the nebular hypothesis or anything like it be
+true, there must be some law to be discovered hereafter, though it may
+be a very complicated one.</p>
+
+<p><span class='pagenum'><a name="Page_296" id="Page_296">[Pg 296]</a></span></p><p>An empirical relation is, however, known: it was suggested by Tatius,
+and published by Bode, of Berlin, in 1772. It is always known as Bode's
+law.</p>
+
+<div class="blockquot"><p>Bode's law asserts that the distance of each planet is
+approximately double the distance of the inner adjacent planet from
+the sun, but that the rate of increase is distinctly slower than
+this for the inner ones; consequently a better approximation will
+be obtained by adding a constant to each term of an appropriate
+geometrical progression. Thus, form a doubling series like this,
+1&frac12;, 3, 6, 12, 24, &amp;c. doubling each time; then add 4 to each,
+and you get a series which expresses very fairly the relative
+distances of the successive planets from the sun, except that the
+number for Mercury is rather erroneous, and we now know that at the
+other extreme the number for Neptune is erroneous too.</p>
+
+<p>I have stated it in the notes above in a form calculated to give
+the law every chance, and a form that was probably fashionable
+after the discovery of Uranus; but to call the first term of the
+doubling series 0 is evidently not quite fair, though it puts
+Mercury's distance right. Neptune's distance, however, turns out to
+be more nearly 30 times the earth's distance than 38&middot;8. The others
+are very nearly right: compare column D of the table preceding
+Lecture III. on <a href="#Page_57">p. 57</a>, with the numbers in the notes on <a href="#Page_294">p. 294</a>. </p></div>
+
+<p>The discovery of Uranus a few years afterwards, in 1781, at 19&middot;2 times
+the earth's distance from the sun, lent great <i>&eacute;cl&acirc;t</i> to the law, and
+seemed to establish its right to be regarded as at least a close
+approximation to the truth.</p>
+
+<p>The gap between Mars and Jupiter, which had often been noticed, and
+which Kepler filled with a hypothetical planet too small to see, comes
+into great prominence by this law of Bode. So much so, that towards the
+end of last century an enthusiastic German, von Zach, after some search
+himself for the expected planet, arranged a committee of observing
+astronomers, or, as he termed it, a body of astronomical detective
+police, to begin a systematic search for this missing subject of the
+sun.</p>
+
+<p><span class='pagenum'><a name="Page_297" id="Page_297">[Pg 297]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_90" id="Fig_90"></a>
+<img src="images/fig90.jpg" width="400" height="677" alt="Fig. 90." title="" />
+<span class="caption"><span class="smcap">Fig. 90.</span>&mdash;Planetary orbits to scale; showing the
+Asteroidal region between Jupiter and Mars. (The orbits of satellites
+are exaggerated.)</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_298" id="Page_298">[Pg 298]</a></span></p><p>In 1800 the preliminaries were settled: the heavens near the zodiac
+were divided into twenty-four regions, each of which was intrusted to
+one observer to be swept. Meanwhile, however, quite independently of
+these arrangements in Germany, and entirely unknown to this committee, a
+quiet astronomer in Sicily, Piazzi, was engaged in making a catalogue of
+the stars. His attention was directed to a certain region in Taurus by
+an error in a previous catalogue, which contained a star really
+non-existent.</p>
+
+<p>In the course of his scrutiny, on the 1st of January, 1801, he noticed a
+small star which next evening appeared to have shifted. He watched it
+anxiously for successive evenings, and by the 24th of January he was
+quite sure he had got hold of some moving body, not a star: probably, he
+thought, a comet. It was very small, only of the eighth magnitude; and
+he wrote to two astronomers (one of them Bode himself) saying what he
+had observed. He continued to observe till the 11th of February, when he
+was attacked by illness and compelled to cease.</p>
+
+<p>His letters did not reach their destination till the end of March.
+Directly Bode opened his letter he jumped to the conclusion that this
+must be the missing planet. But unfortunately he was unable to verify
+the guess, for the object, whatever it was, had now got too near the sun
+to be seen. It would not be likely to be out again before September, and
+by that time it would be hopelessly lost again, and have just as much to
+be rediscovered as if it had never been seen.</p>
+
+<p>Mathematical astronomers tried to calculate a possible orbit for the
+body from the observations of Piazzi, but the observed places were so
+desperately few and close together. It was like having to determine a
+curve from three points close together. Three observations ought to
+serve,<a name="FNanchor_27_27" id="FNanchor_27_27"></a><a href="#Footnote_27_27" class="fnanchor">[27]</a><span class='pagenum'><a name="Page_299" id="Page_299">[Pg 299]</a></span> but if they are taken with insufficient interval between
+them it is extremely difficult to construct the whole circumstances of
+the orbit from them. All the calculations gave different results, and
+none were of the slightest use.</p>
+
+<p>The difficulty as it turned out was most fortunate. It resulted in the
+discovery of one of the greatest mathematicians, perhaps the greatest,
+that Germany has ever produced&mdash;Gauss. He was then a young man of
+twenty-five, eking out a living by tuition. He had invented but not
+published several powerful mathematical methods (one of them now known
+as "the method of least squares"), and he applied them to Piazzi's
+observations. He was thus able to calculate an orbit, and to predict a
+place where, by the end of the year, the planet should be visible. On
+the 31st of December of that same year, very near the place predicted by
+Gauss, von Zach rediscovered it, and Olbers discovered it also the next
+evening. Piazzi called it Ceres, after the tutelary goddess of Sicily.</p>
+
+<p>Its distance from the sun as determined by Gauss was 2&middot;767 times the
+earth's distance. Bode's law made it 2&middot;8. It was undoubtedly the missing
+planet. But it was only one hundred and fifty or two hundred miles in
+diameter&mdash;the smallest heavenly body known at the time of its discovery.
+It revolves the same way as other planets, but the plane of its orbit is
+tilted 10&deg; to the plane of the ecliptic, which was an exceptionally
+large amount.</p>
+
+<p>Very soon, a more surprising discovery followed. Olbers, while searching
+for Ceres, had carefully mapped the part of the heavens where it was
+expected; and in March, 1802, he saw in this place a star he had not
+previously noticed. In two hours he detected its motion, and in a month
+he sent his observations to Gauss, who returned as answer the calculated
+orbit. It was distant 2&middot;67, like Ceres, and was a little smaller, but it
+had a very excentric orbit: its plane being tilted 34&frac12;&deg;, an
+extraordinary inclination. This was called Pallas.</p>
+
+<p><span class='pagenum'><a name="Page_300" id="Page_300">[Pg 300]</a></span></p><p>Olbers at once surmised that these two planets were fragments of a
+larger one, and kept an eager look out for other fragments.</p>
+
+<p>In two years another was seen, in the course of charting the region of
+the heavens traversed by Ceres and Pallas. It was smaller than either,
+and was called Juno.</p>
+
+<p>In 1807 the persevering search of Olbers resulted in the discovery of
+another, with a very oblique orbit, which Gauss named Vesta. Vesta is
+bigger than any of the others, being five hundred miles in diameter, and
+shines like a star of the sixth magnitude. Gauss by this time had become
+so practised in the difficult computations that he worked out the
+complete orbit of Vesta within ten hours of receiving the observational
+data from Olbers.</p>
+
+<p>For many weary years Olbers kept up a patient and unremitting search for
+more of these small bodies, or fragments of the large planet as he
+thought them; but his patience went unrewarded, and he died in 1840
+without seeing or knowing of any more. In 1845 another was found,
+however, in Germany, and a few weeks later two others by Mr. Hind in
+England. Since then there seems no end to them; numbers have been
+discovered in America, where Professors Peters and Watson have made a
+specialty of them, and have themselves found something like a hundred.</p>
+
+<p>Vesta is the largest&mdash;its area being about the same as that of Central
+Europe, without Russia or Spain&mdash;and the smallest known is about twenty
+miles in diameter, or with a surface about the size of Kent. The whole
+of them together do not nearly equal the earth in bulk.</p>
+
+<p>The main interest of these bodies to us lies in the question, What is
+their history? Can they have been once a single planet broken up? or are
+they rather an abortive attempt at a planet never yet formed into one?</p>
+
+<p>The question is not <i>entirely</i> settled, but I can tell you which way
+opinion strongly tends at the present time.</p>
+
+<p>Imagine a shell travelling in an elliptic orbit round the<span class='pagenum'><a name="Page_301" id="Page_301">[Pg 301]</a></span> earth to
+suddenly explode: the centre of gravity of all its fragments would
+continue moving along precisely the same path as had been traversed by
+the centre of the shell before explosion, and would complete its orbit
+quite undisturbed. Each fragment would describe an orbit of its own,
+because it would be affected by a different initial velocity; but every
+orbit would be a simple ellipse, and consequently every piece would in
+time return through its starting-point&mdash;viz. the place at which the
+explosion occurred. If the zone of asteroids had a common point through
+which they all successively passed, they could be unhesitatingly
+asserted to be the remains of an exploded planet. But they have nothing
+of the kind; their orbits are scattered within a certain broad zone&mdash;a
+zone everywhere as broad as the earth's distance from the sun,
+92,000,000 miles&mdash;with no sort of law indicating an origin of this kind.</p>
+
+<p>It must be admitted, however, that the fragments of our supposed shell
+might in the course of ages, if left to themselves, mutually perturb
+each other into a different arrangement of orbits from that with which
+they began. But their perturbations would be very minute, and moreover,
+on Laplace's theory, would only result in periodic changes, provided
+each mass were rigid. It is probable that the asteroids were at one time
+not rigid, and hence it is difficult to say what may have happened to
+them; but there is not the least reason to believe that their present
+arrangement is derivable in any way from an explosion, and it is certain
+that an enormous time must have elapsed since such an event if it ever
+occurred.</p>
+
+<p>It is far more probable that they never constituted one body at all, but
+are the remains of a cloudy ring thrown off by the solar system in
+shrinking past that point: a small ring after the immense effort which
+produced Jupiter and his satellites: a ring which has aggregated into a
+multitude of little lumps instead of a few big ones. Such an event is
+not unique in the solar system;<span class='pagenum'><a name="Page_302" id="Page_302">[Pg 302]</a></span> there is a similar ring round Saturn.
+At first sight, and to ordinary careful inspection, this differs from
+the zone of asteroids in being a solid lump of matter, like a quoit. But
+it is easy to show from the theory of gravitation, that a solid ring
+could not possibly be stable, but would before long get precipitated
+excentrically upon the body of the planet. Devices have been invented,
+such as artfully distributed irregularities calculated to act as
+satellites and maintain stability; but none of these things really work.
+Nor will it do to imagine the rings fluid; they too would destroy each
+other. The mechanical behaviour of a system of rings, on different
+hypotheses as to their constitution, has been worked out with consummate
+skill by Clerk Maxwell; who finds that the only possible constitution
+for Saturn's assemblage of rings is a multitude of discrete particles
+each pursuing its independent orbit. Saturn's ring is, in fact, a very
+concentrated zone of minor asteroids, and there is every reason to
+conclude that the origin of the solar asteroids cannot be very unlike
+the origin of the Saturnian ones. The nebular hypothesis lends itself
+readily to both.</p>
+
+<p>The interlockings and motions of the particles in Saturn's rings are
+most beautiful, and have been worked out and stated by Maxwell with
+marvellous completeness. His paper constituted what is called "The Adams
+Prize Essay" for 1856. Sir George Airy, one of the adjudicators
+(recently Astronomer-Royal), characterized it as "one of the most
+remarkable applications of mathematics to physics that I have ever
+seen."</p>
+
+<p>There are several distinct constituent rings in the entire Saturnian
+zone, and each perturbs the other, with the result that they ripple and
+pulse in concord. The waves thus formed absorb the effect of the mutual
+perturbations, and prevent an accumulation which would be dangerous to
+the persistence of the whole.</p>
+
+<p>The only effect of gravitational perturbation and of collisions is
+gradually to broaden out the whole ring, enlarging<span class='pagenum'><a name="Page_303" id="Page_303">[Pg 303]</a></span> its outer and
+diminishing its inner diameter. But if there were any frictional
+resistance in the medium through which the rings spin, then other
+effects would slowly occur, which ought to be looked for with interest.
+So complete and intimate is the way Maxwell works out and describes the
+whole circumstances of the motion of such an assemblage of particles,
+and so cogent his argument as to the necessity that they must move
+precisely so, and no otherwise, else the rings would not be stable, that
+it was a Cambridge joke concerning him that he paid a visit to Saturn
+one evening, and made his observations on the spot.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_304" id="Page_304">[Pg 304]</a></span></p>
+<h4><a name="NOTES_TO_LECTURE_XIV" id="NOTES_TO_LECTURE_XIV"></a>NOTES TO LECTURE XIV</h4>
+
+
+<p>The total number of stars in the heavens visible to a good eye is about
+5,000. The total number at present seen by telescope is about
+50,000,000. The number able to impress a photographic plate has not yet
+been estimated; but it is enormously greater still. Of those which we
+can see in these latitudes, about 14 are of the first magnitude, 48 of
+the second, 152 of the third, 313 of the fourth, 854 of the fifth, and
+2,010 of the sixth; total, 3,391.</p>
+
+<p>The quickest-moving stars known are a double star of the sixth
+magnitude, called 61 Cygni, and one of the seventh magnitude, called
+Groombridge 1830. The velocity of the latter is 200 miles a second. The
+nearest known stars are 61 Cygni and &#945; Centauri. The distance
+of these from us is about 400,000 times the distance of the sun. Their
+parallax is accordingly half a second of arc. Sirius is more than a
+million times further from us than our sun is, and twenty times as big;
+many of the brightest stars are at more than double this distance. The
+distance of Arcturus is too great to measure even now. Stellar parallax
+was first securely detected in 1838, by Bessel, for 61 Cygni. Bessel was
+born in 1784, and died in 1846, shortly before the discovery of Neptune.</p>
+
+<p>The stars are suns, and are most likely surrounded by planets. One
+planet belonging to Sirius has been discovered. It was predicted by
+Bessel, its position calculated by Peters, and seen by Alvan Clark in
+1862. Another predicted one, belonging to Procyon, has not yet been
+seen.</p>
+
+<p>A velocity of 5 miles a second could carry a projectile right round the
+earth. A velocity of 7 miles a second would carry it away from the
+earth, and round the sun. A velocity of 27 miles a second would carry a
+projectile right out of the solar system never to return.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_305" id="Page_305">[Pg 305]</a></span></p>
+<h3><a name="LECTURE_XIV" id="LECTURE_XIV"></a>LECTURE XIV</h3>
+
+<h5>BESSEL&mdash;THE DISTANCES OF THE STARS, AND THE DISCOVERY OF STELLAR PLANETS</h5>
+
+
+<p><span class="smcap">We</span> will now leave the solar system for a time, and hastily sketch the
+history of stellar astronomy from the time of Sir William Herschel.</p>
+
+<p>You remember how greatly Herschel had changed the aspect of the heavens
+for man,&mdash;how he had found that none of the stars were really fixed, but
+were moving in all manner of ways: some of this motion only apparent,
+much of it real. Nevertheless, so enormously distant are they, that if
+we could be transported back to the days of the old Chald&aelig;an
+astronomers, or to the days of Noah, we should still see the heavens
+with precisely the same aspect as they wear now. Only by refined
+apparatus could any change be discoverable in all those centuries. For
+all practical purposes, therefore, the stars may still be well called
+fixed.</p>
+
+<p>Another thing one may notice, as showing their enormous distances, is
+that from every planet of the solar system the aspect of the heavens
+will be precisely the same. Inhabitants of Mars, or Jupiter, or Saturn,
+or Uranus, will see exactly the same constellations as we do. The whole
+dimensions of the solar system shrink up into a speck when so
+contemplated. And from the stars none of the planetary orbs of our
+system are visible at all; nothing but the sun is visible, and that
+merely as a twinkling star, brighter than some, but fainter than many
+others.</p>
+
+<p><span class='pagenum'><a name="Page_306" id="Page_306">[Pg 306]</a></span></p><p>The sun and the stars are one. Try to realize this distinctly, and keep
+it in mind. I find it often difficult to drive this idea home. After
+some talk on the subject a friendly auditor will report, "the lecturer
+then described the stars, including that greatest and most magnificent
+of all stars, the sun." It would be difficult more completely to
+misapprehend the entire statement. When I say the sun is one of the
+stars, I mean one among the others; we are a long way from them, they
+are a long way from each other. They need be no more closely packed
+among each other than we are closely packed among them; except that some
+of them are double or multiple, and we are not double.</p>
+
+<div class="blockquot"><p>It is highly desirable to acquire an intimate knowledge of the
+constellations and a nodding acquaintance with their principal
+stars. A description of their peculiarities is dull and
+uninteresting unless they are at least familiar by name. A little
+<i>viv&acirc; voce</i> help to begin with, supplemented by patient night
+scrutiny with a celestial globe or star maps under a tent or shed,
+is perhaps the easiest way: a very convenient instrument for the
+purpose of learning the constellations is the form of map called a
+"planisphere," because it can be made to show all the
+constellations visible at a given time at a given date, and no
+others. The Greek alphabet also is a thing that should be learnt by
+everybody. The increased difficulty in teaching science owing to
+the modern ignorance of even a smattering of Greek is becoming
+grotesque. The stars are named from their ancient grouping into
+constellations, and by the prefix of a Greek letter to the larger
+ones, and of numerals to the smaller ones. The biggest of all have
+special Arabic names as well. The brightest stars are called of
+"the first magnitude," the next are of "the second magnitude," and
+so on. But this arrangement into magnitudes has become technical
+and precise, and intermediate or fractional magnitudes are
+inserted. Those brighter than the ordinary first magnitude are
+therefore now spoken of as of magnitude &frac12;, for instance, or &middot;6,
+which is rather confusing. Small telescopic stars are often only
+named by their numbers in some specified catalogue&mdash;a dull but
+sufficient method.</p>
+
+<p>Here is a list of the stars visible from these latitudes, which are
+popularly considered as of the first magnitude. All of them should
+be familiarly recognized in the heavens, whenever seen.</p>
+
+<p>
+<span class='pagenum'><a name="Page_307" id="Page_307">[Pg 307]</a></span></p>
+
+
+<div class='center'>
+<table border="0" width="30%" cellpadding="4" cellspacing="0" summary="First Magnitude Stars">
+<tr>
+ <td align='left'><small>&nbsp;&nbsp;Star.</small></td>
+ <td align='left'><small>&nbsp;&nbsp;Constellation.</small></td></tr>
+<tr>
+ <td align='left'>Sirius</td>
+ <td align='left'>Canis major</td>
+</tr>
+<tr>
+ <td align='left'>Procyon</td>
+ <td align='left'>Canis minor</td>
+</tr>
+<tr>
+ <td align='left'>Rigel</td>
+ <td align='left'>Orion</td>
+</tr>
+<tr>
+ <td align='left'>Betelgeux</td>
+ <td align='left'>Orion</td>
+</tr>
+<tr>
+ <td align='left'>Castor</td>
+ <td align='left'>Gemini</td>
+</tr>
+<tr>
+ <td align='left'>Pollux</td>
+ <td align='left'>Gemini</td>
+</tr>
+<tr>
+ <td align='left'>Aldebaran</td>
+ <td align='left'>Taurus</td>
+</tr>
+<tr>
+ <td align='left'>Arcturus</td>
+ <td align='left'>Bo&ouml;tes</td>
+</tr>
+<tr>
+ <td align='left'>Vega</td>
+ <td align='left'>Lyra</td>
+</tr>
+<tr>
+ <td align='left'>Capella</td>
+ <td align='left'>Auriga</td>
+</tr>
+<tr>
+ <td align='left'>Regulus</td>
+ <td align='left'>Leo</td>
+</tr>
+<tr>
+ <td align='left'>Altair</td>
+ <td align='left'>Aquila</td>
+</tr>
+<tr>
+ <td align='left'>Fomalhaut</td>
+ <td align='left'>Southern Fish</td>
+</tr>
+<tr>
+ <td align='left'>Spica</td>
+ <td align='left'>Virgo</td>
+</tr>
+</table></div>
+
+<p class="noin">&#945; Cygni is a little below the first magnitude. So,
+perhaps, is Castor. In the southern heavens, Canopus and &#945;
+Centauri rank next after Sirius in brightness. </p></div>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_91" id="Fig_91"></a>
+<img src="images/fig91.jpg" width="400" height="116" alt="Fig. 91." title="" />
+<span class="caption"><span class="smcap">Fig. 91.</span>&mdash;Diagram illustrating Parallax.</span>
+</div>
+
+<p>The distances of the fixed stars had, we know, been a perennial problem,
+and many had been the attempts to solve it. All the methods of any
+precision have depended on the Copernican fact that the earth in June
+was 184 million miles away from its position in December, and that
+accordingly the grouping and aspect of the heavens should be somewhat
+different when seen from so different a point of view. An apparent
+change of this sort is called generally parallax; <i>the</i> parallax of a
+star being technically defined as the angle subtended at the star by the
+radius of the earth's orbit: that is to say, the angle E&#963;S;
+where E is the earth, S the sun, and &#963; a star (<a href="#Fig_91">Fig. 91</a>).</p>
+
+<p>Plainly, the further off &#963; is, the more nearly parallel will<span class='pagenum'><a name="Page_308" id="Page_308">[Pg 308]</a></span>
+the two lines to it become. And the difficulty of determining the
+parallax was just this, that the more accurately the observations were
+made, the more nearly parallel did those lines become. The angle was, in
+fact, just as likely to turn out negative as positive&mdash;an absurd result,
+of course, to be attributed to unavoidable very minute inaccuracies.</p>
+
+<p>For a long time absolute methods of determining parallax were attempted;
+for instance, by observing the position of the star with respect to the
+zenith at different seasons of the year. And many of these
+determinations appeared to result in success. Hooke fancied he had
+measured a parallax for Vega in this way, amounting to 30" of arc.
+Flamsteed obtained 40" for &#947; Draconis. Roemer made a serious
+attempt by comparing observations of Vega and Sirius, stars almost the
+antipodes of each other in the celestial vault; hoping to detect some
+effect due to the size of the earth's orbit, which should apparently
+displace them with the season of the year. All these fancied results
+however, were shown to be spurious, and their real cause assigned, by
+the great discovery of the aberration of light by Bradley.</p>
+
+<p>After this discovery it was possible to watch for still outstanding very
+minute discrepancies; and so the problem of stellar parallax was
+attacked with fresh vigour by Piazzi, by Brinkley, and by Struve. But
+when results were obtained, they were traced after long discussion to
+age and gradual wear of the instrument, or to some other minute
+inaccuracy. The more carefully the observation was made, the more nearly
+zero became the parallax&mdash;the more nearly infinite the distance of the
+stars. The brightest stars were the ones commonly chosen for the
+investigation, and Vega was a favourite, because, going near the zenith,
+it was far removed from the fluctuating and tiresome disturbances of
+atmospheric refraction. The reason bright stars were chosen was because
+they were presumably nearer than the others; and indeed a rough guess at
+their probable<span class='pagenum'><a name="Page_309" id="Page_309">[Pg 309]</a></span> distance was made by supposing them to be of the same
+size as the sun, and estimating their light in comparison with sunlight.
+By this confessedly unsatisfactory method it had been estimated that
+Sirius must be 140,000 times further away than the sun is, if he be
+equally big. We now know that Sirius is much further off than this; and
+accordingly that he is much brighter, perhaps sixty times as bright,
+though not necessarily sixty times as big, as our sun. But even
+supposing him of the same light-giving power as the sun, his parallax
+was estimated as 1"&middot;8, a quantity very difficult to be sure of in any
+absolute determination.</p>
+
+<p>Relative methods were, however, also employed, and the advantages of one
+of these (which seems to have been suggested by Galileo) so impressed
+themselves upon William Herschel that he made a serious attempt to
+compass the problem by its means. The method was to take two stars in
+the same telescopic field and carefully to estimate their apparent
+angular distance from each other at different seasons of the year. All
+such disturbances as precession, aberration, nutation, refraction, and
+the like, would affect them both equally, and could thus be eliminated.
+If they were at the same distance from the solar system, relative
+parallax would, indeed, also be eliminated; but if, as was probable,
+they were at different distances, then they would apparently shift
+relatively to one another, and the amount of shift, if it could be
+observed, would measure, not indeed the distance of either from the
+earth, but their distance from each other. And this at any rate would be
+a step. It might be completed by similarly treating other stars in the
+same field, taking them in pairs together. A bright and a faint star
+would naturally be suitable, because their distances were likely to be
+unequal; and so Herschel fixed upon a number of doublets which he knew
+of, containing one bright and one faint component. For up to that time
+it had been supposed that such grouping in occasional pairs or triplets
+was chance coincidence, the two being optically<span class='pagenum'><a name="Page_310" id="Page_310">[Pg 310]</a></span> foreshortened together,
+but having no real connection or proximity. Herschel failed in what he
+was looking for, but instead of that he discovered the real connection
+of a number of these doublets, for he found that they were slowly
+revolving round each other. There are a certain number of merely optical
+or accidental doublets, but the majority of them are real pairs of suns
+revolving round each other.</p>
+
+<p>This relative method of mapping micrometrically a field of neighbouring
+stars, and comparing their configuration now and six months hence, was,
+however, the method ultimately destined to succeed; and it is, I
+believe, the only method which has succeeded down to the present day.
+Certainly it is the method regularly employed, at Dunsink, at the Cape
+of Good Hope, and everywhere else where stellar parallax is part of the
+work.</p>
+
+<p>Between 1830 and 1840 the question was ripe for settlement, and, as
+frequently happens with a long-matured difficulty, it gave way in three
+places at once. Bessel, Henderson, and Struve almost simultaneously
+announced a stellar parallax which could reasonably be accepted. Bessel
+was a little the earliest, and by far the most accurate. His, indeed,
+was the result which commanded confidence, and to him the palm must be
+awarded.</p>
+
+<p>He was largely a self-taught student, having begun life in a
+counting-house, and having abandoned business for astronomy. But
+notwithstanding these disadvantages, he became a highly competent
+mathematician as well as a skilful practical astronomer. He was
+appointed to superintend the construction of Germany's first great
+astronomical observatory, that of K&ouml;nigsberg, which, by his system,
+zeal, and genius, he rapidly made a place of the first importance.</p>
+
+<p>Struve at Dorpat, Bessel at K&ouml;nigsberg, and Henderson at the Cape of
+Good Hope&mdash;all of them at newly-equipped observatories&mdash;were severally
+engaged at the same problem.</p>
+
+<p>But the Russian and German observers had the advantage<span class='pagenum'><a name="Page_311" id="Page_311">[Pg 311]</a></span> of the work of
+one of the most brilliant opticians&mdash;I suppose the most brilliant&mdash;that
+has yet appeared: Fraunhofer, of Munich. An orphan lad, apprenticed to a
+maker of looking-glasses, and subject to hard struggles and privations
+in early life, he struggled upwards, and ultimately became head of the
+optical department of a Munich firm of telescope-makers. Here he
+constructed the famous "Dorpat refractor" for Struve, which is still at
+work; and designed the "K&ouml;nigsberg heliometer" for Bessel. He also made
+a long and most skilful research into the solar spectrum, which has
+immortalized his name. But his health was broken by early trials, and he
+died at the age of thirty-nine, while planning new and still more
+important optical achievements.</p>
+
+<p>A heliometer is the most accurate astronomical instrument for relative
+measurements of position, as a transit circle is the most accurate for
+absolute determinations. It consists of an equatorial telescope with
+object-glass cut right across, and each half movable by a sliding
+movement one past the other, the amount by which the two halves are
+dislocated being read off by a refined method, and the whole instrument
+having a multitude of appendages conducive to convenience and accuracy.
+Its use is to act as a micrometer or measurer of small distances.<a name="FNanchor_28_28" id="FNanchor_28_28"></a><a href="#Footnote_28_28" class="fnanchor">[28]</a>
+Each half of the object-glass gives a distinct image, which may be
+allowed to coincide or may be separated as occasion requires. If it be
+the components of a double star that are being examined, each component
+will in general be seen double, so that four images will be seen
+altogether; but by careful adjustment it will be possible to arrange
+that one image of each pair shall be superposed on or coincide with each
+other, in which case only three images are visible; the amount of
+dislocation of the halves of the object-glass necessary to accomplish<span class='pagenum'><a name="Page_312" id="Page_312">[Pg 312]</a></span>
+this is what is read off. The adjustment is one that can be performed
+with extreme accuracy, and by performing it again and again with all
+possible modifications, an extremely accurate determination of the
+angular distance between the two components is obtained.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_92" id="Fig_92"></a>
+<img src="images/fig92.jpg" width="400" height="517" alt="Fig. 92." title="" />
+<span class="caption"><span class="smcap">Fig. 92.</span>&mdash;Heliometer.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_313" id="Page_313">[Pg 313]</a></span></p><p>Bessel determined to apply this beautiful instrument to the problem of
+stellar parallax; and he began by considering carefully the kind of star
+for which success was most likely. Hitherto the brightest had been most
+attended to, but Bessel thought that quickness of proper motion would be
+a still better test of nearness. Not that either criterion is conclusive
+as to distance, but there was a presumption in favour of either a very
+bright or an obviously moving star being nearer than a faint or a
+stationary one; and as the "bright" criterion had already been often
+applied without result, he decided to try the other. He had already
+called attention to a record by Piazzi in 1792 of a double star in
+Cygnus whose proper motion was five seconds of arc every year&mdash;a motion
+which caused this telescopic object, 61 Cygni, to be known as "the
+flying star." Its motion is not really very perceptible, for it will
+only have traversed one-third of a lunar diameter in the course of a
+century; still it was the quickest moving star then known. The position
+of this interesting double he compared with two other stars which were
+seen simultaneously in the field of the heliometer, by the method I have
+described, throughout the whole year 1838; and in the last month of that
+year he was able to announce with confidence a distinct though very
+small parallax; substantiating it with a mass of detailed evidence which
+commanded the assent of astronomers. The amount of it he gave as
+one-third of a second. We know now that he was very nearly right, though
+modern research makes it more like half a second.<a name="FNanchor_29_29" id="FNanchor_29_29"></a><a href="#Footnote_29_29" class="fnanchor">[29]</a></p>
+
+<p>Soon afterwards, Struve announced a quarter of a second as the parallax
+of Vega, but that is distinctly too great; and<span class='pagenum'><a name="Page_314" id="Page_314">[Pg 314]</a></span> Henderson announced for
+&#945; Centauri (then thought to be a double) a parallax of one
+second, which, if correct, would make it quite the nearest of all the
+stars, but the result is now believed to be about twice too big.</p>
+
+<p>Knowing the distance of 61 Cygni, we can at once tell its real rate of
+travel&mdash;at least, its rate across our line of sight: it is rather over
+three million miles a day.</p>
+
+<p>Now just consider the smallness of the half second of arc, thus
+triumphantly though only approximately measured. It is the angle
+subtended by twenty-six feet at a distance of 2,000 miles. If a
+telescope planted at New York could be directed to a house in England,
+and be then turned so as to set its cross-wire first on one end of an
+ordinary room and then on the other end of the same room, it would have
+turned through half a second, the angle of greatest stellar parallax.
+Or, putting it another way. If the star were as near us as New York is,
+the sun, on the same scale, would be nine paces off. As twenty-six feet
+is to the distance of New York, so is ninety-two million miles to the
+distance of the nearest fixed star.</p>
+
+<p>Suppose you could arrange some sort of telegraphic vehicle able to carry
+you from here to New York in the tenth part of a second&mdash;<i>i.e.</i> in the
+time required to drop two inches&mdash;such a vehicle would carry you to the
+moon in twelve seconds, to the sun in an hour and a quarter. Travelling
+thus continually, in twenty-four hours you would leave the last member
+of the solar system behind you, and begin your plunge into the depths of
+space. How long would it be before you encountered another object? A
+month, should you guess? Twenty years you must journey with that
+prodigious speed before you reach the nearest star, and then another
+twenty years before you reach another. At these awful distances from one
+another the stars are scattered in space, and were they not brilliantly
+self-luminous and glowing like our sun, they would be hopelessly
+invisible.</p>
+
+<p><span class='pagenum'><a name="Page_315" id="Page_315">[Pg 315]</a></span></p><p>I have spoken of 61 Cygni as a flying star, but there is another which
+goes still quicker, a faint star, 1830 in Groombridge's Catalogue. Its
+distance is far greater than that of 61 Cygni, and yet it is seen to
+move almost as quickly. Its actual speed is about 200 miles a
+second&mdash;greater than the whole visible firmament of fifty million stars
+can control; and unless the universe is immensely larger than anything
+we can see with the most powerful telescopes, or unless there are crowds
+of invisible non-luminous stars mixed up with the others, it can only be
+a temporary visitor to this frame of things; it is rushing from an
+infinite distance to an infinite distance; it is passing through our
+visible universe for the first and only time&mdash;it will never return. But
+so gigantic is the extent of visible space, that even with its amazing
+speed of 200 miles every second, this star will take two or three
+million years to get out of sight of our present telescopes, and several
+thousand years before it gets perceptibly fainter than it is now.</p>
+
+<p>Have we any reason for supposing that the stars we see are all there
+are? In other words, have we any reason for supposing all celestial
+objects to be sufficiently luminous to be visible? We have every ground
+for believing the contrary. Every body in the solar system is dull and
+dark except the sun, though probably Jupiter is still red-hot. Why may
+not some of the stars be dark too? The genius of Bessel surmised this,
+and consistently upheld the doctrine that the astronomy of the future
+would have to concern itself with dark and invisible bodies; he preached
+"an astronomy of the invisible." Moreover he predicted the presence of
+two such dark bodies&mdash;one a companion of Sirius, the other of Procyon.
+He noticed certain irregularities in the motions of these stars which he
+asserted must be caused by their revolving round other bodies in a
+period of half a century. He announced in 1844 that both Sirius and
+Procyon were double stars, but that their companions, though large, were
+dark, and therefore invisible.</p>
+
+<p><span class='pagenum'><a name="Page_316" id="Page_316">[Pg 316]</a></span></p><p>No one accepted this view, till Peters, in America, found in 1851 that
+the hypothesis accurately explained the anomalous motion of Sirius, and,
+in fact, indicated an exact place where the companion ought to be. The
+obscure companion of Sirius became now a recognized celestial object,
+although it had never been seen, and it was held to revolve round Sirius
+in fifty years, and to be about half as big.</p>
+
+<p>In 1862, the firm of Alvan Clark and Sons, of New York, were completing
+a magnificent 18-inch refractor, and the younger Clark was trying it on
+Sirius, when he said: "Why, father, the star has a companion!" The elder
+Clark also looked, and sure enough there was a faint companion due east
+of the bright star, and in just the position required by theory. Not
+that the Clarks knew anything about the theory. They were keen-sighted
+and most skilful instrument-makers, and they made the discovery by
+accident. After it had once been seen, it was found that several of the
+large telescopes of the world were able to show it. It is half as big,
+but it only gives <span class="above">1</span>&#8260;<span class="below">10000</span>th part of the light that Sirius gives. No
+doubt it shines partly with a borrowed light and partly with a dull heat
+of its own. It is a real planet, but as yet too hot to live on. It will
+cool down in time, as our earth has cooled and as Jupiter is cooling,
+and no doubt become habitable enough. It does revolve round Sirius in a
+period of 49&middot;4 years&mdash;almost exactly what Bessel assigned to it.</p>
+
+<p>But Bessel also assigned a dark companion to Procyon. It and its
+luminous neighbour are considered to revolve round each other in a
+period of forty years, and astronomers feel perfectly assured of its
+existence, though at present it has not been seen by man.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_317" id="Page_317">[Pg 317]</a></span></p>
+<h3><a name="LECTURE_XV" id="LECTURE_XV"></a>LECTURE XV</h3>
+
+<h5>THE DISCOVERY OF NEPTUNE</h5>
+
+
+<p><span class="smcap">We</span> approach to-night perhaps the greatest, certainly the most
+conspicuous, triumphs of the theory of gravitation. The explanation by
+Newton of the observed facts of the motion of the moon, the way he
+accounted for precession and nutation and for the tides, the way in
+which Laplace explained every detail of the planetary motions&mdash;these
+achievements may seem to the professional astronomer equally, if not
+more, striking and wonderful; but of the facts to be explained in these
+cases the general public are necessarily more or less ignorant, and so
+no beauty or thoroughness of treatment appeals to them, nor can excite
+their imaginations. But to predict in the solitude of the study, with no
+weapons other than pen, ink, and paper, an unknown and enormously
+distant world, to calculate its orbit when as yet it had never been
+seen, and to be able to say to a practical astronomer, "Point your
+telescope in such a direction at such a time, and you will see a new
+planet hitherto unknown to man"&mdash;this must always appeal to the
+imagination with dramatic intensity, and must awaken some interest in
+almost the dullest.</p>
+
+<p>Prediction is no novelty in science; and in astronomy least of all is it
+a novelty. Thousands of years ago, Thales, and others whose very names
+we have forgotten, could<span class='pagenum'><a name="Page_318" id="Page_318">[Pg 318]</a></span> predict eclipses with some certainty, though
+with only rough accuracy. And many other phenomena were capable of
+prediction by accumulated experience. We have seen, for instance (coming
+to later times), how a gap between Mars and Jupiter caused a missing
+planet to be suspected and looked for, and to be found in a hundred
+pieces. We have seen, also, how the abnormal proper-motion of Sirius
+suggested to Bessel the existence of an unseen companion. And these last
+instances seem to approach very near the same class of prediction as
+that of the discovery of Neptune. Wherein, then, lies the difference?
+How comes it that some classes of prediction&mdash;such as that if you put
+your finger in fire it will get burnt&mdash;are childishly easy and
+commonplace, while others excite in the keenest intellects the highest
+feelings of admiration? Mainly, the difference lies, first, in the
+grounds on which the prediction is based; second, on the difficulty of
+the investigation whereby it is accomplished; third, in the completeness
+and the accuracy with which it can be verified. In all these points, the
+discovery of Neptune stands out pre-eminently among the verified
+predictions of science, and the circumstances surrounding it are of
+singular interest.</p>
+
+<hr style='width: 15%;' />
+
+<p>In 1781, Sir William Herschel discovered the planet Uranus. Now you know
+that three distinct observations suffice to determine the orbit of a
+planet completely, and that it is well to have the three observations as
+far apart as possible so as to minimize the effects of minute but
+necessary errors of observation. (<a href="#Page_298">See p. 298.</a>) Directly Uranus was
+found, therefore, old records of stellar observations were ransacked,
+with the object of discovering whether it had ever been unwittingly seen
+before. If seen, it had been thought of course to be a star (for it
+shines like a star of the sixth magnitude, and can therefore be just
+seen without a telescope if one knows precisely where to look for it,
+and<span class='pagenum'><a name="Page_319" id="Page_319">[Pg 319]</a></span> if one has good sight), but if it had been seen and catalogued as a
+star it would have moved from its place, and the catalogue would by that
+entry be wrong. The thing to detect, therefore, was errors in the
+catalogues: to examine all entries, and see if the stars entered
+actually existed, or were any of them missing. If a wrong entry were
+discovered, it might of course have been due to some clerical error,
+though that is hardly probable considering the care taken over these
+things, or it might have been some tailless comet or other, or it might
+have been the newly found planet.</p>
+
+<p>So the next thing was to calculate backwards, and see if by any
+possibility the planet could have been in that place at that time.
+Examined in this way the tabulated observations of Flamsteed showed that
+he had unwittingly observed Uranus five distinct times, the first time
+in 1690, nearly a century before Herschel discovered its true nature.
+But more remarkable still, Le Monnier, of Paris, had observed it eight
+times in one month, cataloguing it each time as a different star. If
+only he had reduced and compared his observations, he would have
+anticipated Herschel by twelve years. As it was, he missed it
+altogether. It was seen once by Bradley also. Altogether it had been
+seen twenty times.</p>
+
+<p>These old observations of Flamsteed and those of Le Monnier, combined
+with those made after Herschel's discovery, were very useful in
+determining an exact orbit for the new planet, and its motion was
+considered thoroughly known. It was not an <i>exact</i> ellipse, of course:
+none of the planets describe <i>exact</i> ellipses&mdash;each perturbs all the
+rest, and these small perturbations must be taken into account, those of
+Jupiter and Saturn being by far the most important.</p>
+
+<p>For a time Uranus seemed to travel regularly and as expected, in the
+orbit which had been calculated for it; but early in the present century
+it began to be slightly refractory,<span class='pagenum'><a name="Page_320" id="Page_320">[Pg 320]</a></span> and by 1820 its actual place showed
+quite a distinct discrepancy from its position as calculated with the
+aid of the old observations. It was at first thought that this
+discrepancy must be due to inaccuracies in the older observations, and
+they were accordingly rejected, and tables prepared for the planet based
+on the newer and more accurate observations only. But by 1830 it became
+apparent that it would not accurately obey even these. The error
+amounted to some 20". By 1840 it was as much as 90', or a minute and a
+half. This discrepancy is quite distinct, but still it is very small,
+and had two objects been in the heavens at once, the actual Uranus and
+the theoretical Uranus, no unaided eye could possibly have distinguished
+them or detected that they were other than a single star.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="Fig_93" id="Fig_93"></a>
+<img src="images/fig93.jpg" width="400" height="239" alt="Fig. 93." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 93.</span>&mdash;Perturbations of Uranus.</span>
+<p><small>The chance observations by Flamsteed, by Le Monnier, and others, are
+plotted in this diagram, as well as the modern determinations made after
+Herschel had discovered the nature of the planet. The decades are laid
+off horizontally. Vertical distance represents the difference between
+observed and subsequently calculated longitudes&mdash;in other words, the
+principal perturbations caused by Neptune. To show the scale, a number
+of standard things are represented too by lengths measured upwards from
+the line of time, viz: the smallest quantity perceptible to the naked
+eye,&mdash;the maximum angle of aberration, of nutation, and of stellar
+parallax; though this last is too small to be properly indicated. The
+perturbations are much bigger than these; but compared with what can be
+seen without a telescope they are small&mdash;the distance between the
+component pairs of &#949; Lyr&aelig; (210") (<a href="#Fig_86">see fig. 86</a>, page 288), which
+a few keen-eyed persons can see as a simple double star, being about
+twice the greatest perturbation.</small></p>
+</div>
+
+
+
+<p><span class='pagenum'><a name="Page_321" id="Page_321">[Pg 321]</a></span></p><p>The diagram shows all the irregularities plotted in the light of our
+present knowledge; and, to compare with their amounts, a few standard
+things are placed on the same scale, such as the smallest interval
+capable of being detected with the unaided eye, the distance of the
+component stars in &#949; Lyr&aelig;, the constants of aberration, of
+nutation, and of stellar parallax.</p>
+
+<p>The errors of Uranus therefore, though small, were enormously greater
+than things which had certainly been observed; there was an unmistakable
+discrepancy between theory and observation. Some cause was evidently at
+work on this distant planet, causing it to disagree with its motion as
+calculated according to the law of gravitation. Some thought that the
+exact law of gravitation did not apply to so distant a body. Others
+surmised the presence of some foreign and unknown body, some comet, or
+some still more distant planet perhaps, whose gravitative attraction for
+Uranus was the cause of the whole difficulty&mdash;some perturbations, in
+fact, which had not been taken into account because of our ignorance of
+the existence of the body which caused them.</p>
+
+<p>But though such an idea was mentioned among astronomers, it was not
+regarded with any special favour, and was considered merely as one among
+a number of hypotheses which could be suggested as fairly probable.</p>
+
+<p>It is perfectly right not to attach much importance to unelaborated
+guesses. Not until the consequences of an hypothesis have been
+laboriously worked out&mdash;not until it can be shown capable of producing
+the effect quantitatively as well as qualitatively&mdash;does its statement
+rise above the level of a guess, and attain the dignity of a theory. A
+later stage still occurs when the theory has been actually and
+completely verified by agreement with observation.</p>
+
+<div class="blockquot"><p>Now the errors in the motion of Uranus, <i>i.e.</i> the discrepancy
+between its observed and calculated longitudes&mdash;all known
+disturbing causes, such as Jupiter and Saturn, being allowed
+for&mdash;are as follows (as quoted by Dr. Haughton) in seconds of
+arc:&mdash;</p>
+
+<p>
+<span class='pagenum'><a name="Page_322" id="Page_322">[Pg 322]</a></span></p>
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Ancient Star Observations">
+<tr>
+ <td align='center' colspan='3'><span class="smcap">Ancient Observations</span> (casually made, as of a star).</td>
+</tr>
+<tr class='tr1'>
+ <td align='left'>Flamsteed</td>
+ <td align='center'>1690</td>
+ <td align='right'>+61&middot;2</td>
+</tr>
+<tr>
+ <td class='tdlpl2'>"</td>
+ <td align='center'>1712</td>
+ <td align='right'>+92&middot;7</td>
+</tr>
+<tr>
+ <td class='tdlpl2'>"</td>
+ <td align='center'>1715</td>
+ <td align='right'>+73&middot;8</td>
+</tr>
+<tr>
+ <td align='left'>Le Monnier</td>
+ <td align='center'>1750</td>
+ <td align='right'>-47&middot;6</td>
+</tr>
+<tr>
+ <td align='left'>Bradley</td>
+ <td align='center'>1753</td>
+ <td align='right'>-39&middot;5</td>
+</tr>
+<tr>
+ <td align='left'>Mayer</td>
+ <td align='center'>1756</td>
+ <td align='right'>-45&middot;7</td>
+</tr>
+<tr>
+ <td align='left'>Le Monnier</td>
+ <td align='center'>1764</td>
+ <td align='right'>-34&middot;9</td>
+</tr>
+<tr>
+ <td class='tdlpl2'>"</td>
+ <td align='center'>1769</td>
+ <td align='right'>-19&middot;3</td>
+</tr>
+<tr>
+ <td class='tdlpl2'>"</td>
+ <td align='center'>1771</td>
+ <td align='right'>-2&middot;3</td>
+</tr>
+</table></div>
+
+
+<div class='center'><br />
+<table border="0" cellpadding="4" cellspacing="0" summary="Modern Observations">
+<tr>
+ <td align='center' colspan='2'><span class="smcap">Modern Observations.</span></td>
+</tr>
+<tr><td align='left'>1780</td><td align='right'>+3&middot;46</td></tr>
+<tr><td align='left'>1783</td><td align='right'>+8&middot;45</td></tr>
+<tr><td align='left'>1786</td><td align='right'>+12&middot;36</td></tr>
+<tr><td align='left'>1789</td><td align='right'>+19&middot;02</td></tr>
+<tr><td align='left'>1801</td><td align='right'>+22&middot;21</td></tr>
+<tr><td align='left'>1810</td><td align='right'>+23&middot;16</td></tr>
+<tr><td align='left'>1822</td><td align='right'>+20&middot;97</td></tr>
+<tr><td align='left'>1825</td><td align='right'>+18&middot;16</td></tr>
+<tr><td align='left'>1828</td><td align='right'>+10&middot;82</td></tr>
+<tr><td align='left'>1831</td><td align='right'>-3&middot;98</td></tr>
+<tr><td align='left'>1834</td><td align='right'>-20&middot;80</td></tr>
+<tr><td align='left'>1837</td><td align='right'>-42&middot;66</td></tr>
+<tr><td align='left'>1840</td><td align='right'>-66&middot;64</td></tr>
+</table></div>
+
+<p>These are the numbers plotted in the above diagram (<a href="#Fig_92">Fig. 92</a>), where
+H marks the discovery of the planet and the beginning of its
+regular observation. </p></div>
+
+<p>Something was evidently the matter with the planet. If the law of
+gravitation held exactly at so great a distance from the sun, there must
+be some perturbing force acting on it besides all those known ones which
+had been fully taken into account. Could it be an outer planet? The
+question occurred to several, and one or two tried if they could solve
+the problem, but were soon stopped by the tremendous difficulties of
+calculation.</p>
+
+<p>The ordinary problem of perturbation is difficult enough:<span class='pagenum'><a name="Page_323" id="Page_323">[Pg 323]</a></span> Given a
+disturbing planet in such and such a position, to find the perturbations
+it produces. This problem it was that Laplace worked out in the
+<i>M&eacute;canique C&eacute;leste</i>.</p>
+
+<p>But the inverse problem: Given the perturbations, to find the planet
+which causes them&mdash;such a problem had never yet been attacked, and by
+only a few had its possibility been conceived. Bessel made preparations
+for trying what he could do at it in 1840, but he was prevented by fatal
+illness.</p>
+
+<p>In 1841 the difficulties of the problem presented by these residual
+perturbations of Uranus excited the imagination of a young student, an
+undergraduate of St. John's College, Cambridge&mdash;John Couch Adams by
+name&mdash;and he determined to have a try at it as soon as he was through
+his Tripos. In January, 1843, he graduated as Senior Wrangler, and
+shortly afterwards he set to work. In less than two years he reached a
+definite conclusion; and in October, 1845, he wrote to the
+Astronomer-Royal, at Greenwich, Professor Airy, saying that the
+perturbations of Uranus would be explained by assuming the existence of
+an outer planet, which he reckoned was now situated in a specified
+latitude and longitude.</p>
+
+<p>We know now that had the Astronomer-Royal put sufficient faith in this
+result to point his big telescope to the spot indicated and commence
+sweeping for a planet, he would have detected it within 1&frac34;&deg; of the
+place assigned to it by Mr. Adams. But any one in the position of the
+Astronomer-Royal knows that almost every post brings an absurd letter
+from some ambitious correspondent or other, some of them having just
+discovered perpetual motion, or squared the circle, or proved the earth
+flat, or discovered the constitution of the moon, or of ether, or of
+electricity; and out of this mass of rubbish it requires great skill and
+patience to detect such gems of value as there may be.</p>
+
+<p>Now this letter of Mr. Adams's was indeed a jewel of the first water,
+and no doubt bore on its face a very different<span class='pagenum'><a name="Page_324" id="Page_324">[Pg 324]</a></span> appearance from the
+chaff of which I have spoken; but still Mr. Adams was an unknown man: he
+had graduated as Senior Wrangler it is true, but somebody must graduate
+as Senior Wrangler every year, and every year by no means produces a
+first-rate mathematician. Those behind the scenes, as Professor Airy of
+course was, having been a Senior Wrangler himself, knew perfectly well
+that the labelling of a young man on taking his degree is much more
+worthless as a testimony to his genius and ability than the general
+public are apt to suppose.</p>
+
+<p>Was it likely that a young and unknown man should have successfully
+solved so extremely difficult a problem? It was altogether unlikely.
+Still, he would test him: he would ask for further explanations
+concerning some of the perturbations which he himself had specially
+noticed, and see if Mr. Adams could explain these also by his
+hypothesis. If he could, there might be something in his theory. If he
+failed&mdash;well, there was an end of it. The questions were not difficult.
+They concerned the error of the radius vector. Mr. Adams could have
+answered them with perfect ease; but sad to say, though a brilliant
+mathematician, he was not a man of business. He did not answer Professor
+Airy's letter.</p>
+
+<p>It may to many seem a pity that the Greenwich Equatoreal was not pointed
+to the place, just to see whether any foreign object did happen to be in
+that neighbourhood; but it is no light matter to derange the work of an
+Observatory, and alter the work mapped out for the staff into a sudden
+sweep for a new planet, on the strength of a mathematical investigation
+just received by post. If observatories were conducted on these
+unsystematic and spasmodic principles, they would not be the calm,
+accurate, satisfactory places they are.</p>
+
+<p>Of course, if any one could have known that a new planet was to be had
+for the looking, <i>any</i> course would have been justified; but no one
+could know this. I do not suppose<span class='pagenum'><a name="Page_325" id="Page_325">[Pg 325]</a></span> that Mr. Adams himself could feel all
+that confidence in his attempted prediction. So there the matter
+dropped. Mr. Adams's communication was pigeon-holed, and remained in
+seclusion for eight or nine months.</p>
+
+<p>Meanwhile, and quite independently, something of the same sort was going
+on in France. A brilliant young mathematician, born in Normandy in 1811,
+had accepted the post of Astronomical Professor at the &Eacute;cole
+Polytechnique, then recently founded by Napoleon. His first published
+papers directed attention to his wonderful powers; and the official head
+of astronomy in France, the famous Arago, suggested to him the
+unexplained perturbations of Uranus as a worthy object for his fresh and
+well-armed vigour.</p>
+
+<p>At once he set to work in a thorough and systematic way. He first
+considered whether the discrepancies could be due to errors in the
+tables or errors in the old observations. He discussed them with minute
+care, and came to the conclusion that they were not thus to be explained
+away. This part of the work he published in November, 1845.</p>
+
+<p>He then set to work to consider the perturbations produced by Jupiter
+and Saturn, to see if they had been with perfect accuracy allowed for,
+or whether some minute improvements could be made sufficient to destroy
+the irregularities. He introduced several fresh terms into these
+perturbations, but none of them of sufficient magnitude to do more than
+slightly lessen the unexplained perturbations.</p>
+
+<p>He next examined the various hypotheses that had been suggested to
+account for them:&mdash;Was it a failure in the law of gravitation? Was it
+due to the presence of a resisting medium? Was it due to some unseen but
+large satellite? Or was it due to a collision with some comet?</p>
+
+<p>All these he examined and dismissed for various reasons one after the
+other. It was due to some steady continuous cause&mdash;for instance, some
+unknown planet. Could this planet be inside the orbit of Uranus? No, for
+then it<span class='pagenum'><a name="Page_326" id="Page_326">[Pg 326]</a></span> would perturb Saturn and Jupiter also, and they were not
+perturbed by it. It must, therefore, be some planet outside the orbit of
+Uranus, and in all probability, according to Bode's empirical law, at
+nearly double the distance from the sun that Uranus is. Lastly he
+proceeded to examine where this planet was, and what its orbit must be
+to produce the observed disturbances.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="Fig_94" id="Fig_94"></a>
+<img src="images/fig94.jpg" width="400" height="407" alt="Fig. 94." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 94.</span>&mdash;Uranus&#39;s and Neptune&#39;s relative positions.</span>
+<p><small>The above diagram, drawn to scale by Dr. Haughton, shows the paths of
+Uranus and Neptune, and their positions from 1781 to 1840, and
+illustrates the <i>direction</i> of their mutual perturbing force. In 1822
+the planets were in conjunction, and the force would then perturb the
+radius vector (or distance from the sun), but not the longitude (or
+place in orbit). Before that date Uranus had been hurried along, and
+after that date it had been retarded, by the pull of Neptune, and thus
+the observed discrepancies from its computed place were produced. The
+problem was first to disentangle the outstanding perturbations from
+those which would be caused by Jupiter and Saturn and all other known
+causes, and then to assign the place of an outer planet able to produce
+precisely those perturbations in Uranus.</small></p>
+</div>
+
+<p>Not without failures and disheartening complications was this part of
+the process completed. This was, after all, the real tug of war. So many
+unknown quantities: its mass,<span class='pagenum'><a name="Page_327" id="Page_327">[Pg 327]</a></span> its distance, its excentricity, the
+obliquity of its orbit, its position at any time&mdash;nothing known, in
+fact, about the planet except the microscopic disturbance it caused in
+Uranus, some thousand million miles away from it.</p>
+
+<p>Without going into further detail, suffice it to say that in June, 1846,
+he published his last paper, and in it announced to the world his
+theoretical position for the planet.</p>
+
+<p>Professor Airy received a copy of this paper before the end of the
+month, and was astonished to find that Leverrier's theoretical place for
+the planet was within 1&deg; of the place Mr. Adams had assigned to it eight
+months before. So striking a coincidence seemed sufficient to justify a
+Herschelian "sweep" for a week or two.</p>
+
+<p>But a sweep for so distant a planet would be no easy matter. When seen
+in a large telescope it would still only look like a star, and it would
+require considerable labour and watching to sift it out from the other
+stars surrounding it. We know that Uranus had been seen twenty times,
+and thought to be a star, before its true nature was by Herschel
+discovered; and Uranus is only about half as far away as Neptune is.</p>
+
+<p>Neither in Paris nor yet at Greenwich was any optical search undertaken;
+but Professor Airy wrote to ask M. Leverrier the same old question as he
+had fruitlessly put to Mr. Adams: Did the new theory explain the errors
+of the radius vector or not? The reply of Leverrier was both prompt and
+satisfactory&mdash;these errors were explained, as well as all the others.
+The existence of the object was then for the first time officially
+believed in.</p>
+
+<p>The British Association met that year at Southampton, and Sir John
+Herschel was one of its Sectional Presidents. In his inaugural address,
+on September 10th, 1846, he called attention to the researches of
+Leverrier and Adams in these memorable words:&mdash;</p>
+
+<div class="blockquot"><p>"The past year has given to us the new [minor] planet Astr&aelig;a; it
+has done more&mdash;it has given us the probable<span class='pagenum'><a name="Page_328" id="Page_328">[Pg 328]</a></span> prospect of another.
+We see it as Columbus saw America from the shores of Spain. Its
+movements have been felt trembling along the far-reaching line of
+our analysis with a certainty hardly inferior to ocular
+demonstration." </p></div>
+
+<p>It was about time to begin to look for it. So the Astronomer-Royal
+thought on reading Leverrier's paper. But as the national telescope at
+Greenwich was otherwise occupied, he wrote to Professor Challis, at
+Cambridge, to know if he would permit a search to be made for it with
+the Northumberland Equatoreal, the large telescope of Cambridge
+University, presented to it by one of the Dukes of Northumberland.</p>
+
+<p>Professor Challis said he would conduct the search himself; and shortly
+commenced a leisurely and dignified series of sweeps round about the
+place assigned by theory, cataloguing all the stars which he observed,
+intending afterwards to sort out his observations, compare one with
+another, and find out whether any one star had changed its position;
+because if it had it must be the planet. He thus, without giving an
+excessive time to the business, accumulated a host of observations,
+which he intended afterwards to reduce and sift at his leisure.</p>
+
+<p>The wretched man thus actually saw the planet twice&mdash;on August 4th and
+August 12th, 1846&mdash;without knowing it. If only he had had a map of the
+heavens containing telescopic stars down to the tenth magnitude, and if
+he had compared his observations with this map as they were made, the
+process would have been easy, and the discovery quick. But he had no
+such map. Nevertheless one was in existence: it had just been completed
+in that country of enlightened method and industry&mdash;Germany. Dr.
+Bremiker had not, indeed, completed his great work&mdash;a chart of the whole
+zodiac down to stars of the tenth magnitude&mdash;but portions of it were
+completed, and the special region where the new planet was expected
+happened to be among<span class='pagenum'><a name="Page_329" id="Page_329">[Pg 329]</a></span> the portions already just done. But in England
+this was not known.</p>
+
+<p>Meanwhile, Mr. Adams wrote to the Astronomer-Royal several additional
+communications, making improvements in his theory, and giving what he
+considered nearer and nearer approximations for the place of the planet.
+He also now answered quite satisfactorily, but too late, the question
+about the radius vector sent to him months before.</p>
+
+<p>Let us return to Leverrier. This great man was likewise engaged in
+improving his theory and in considering how best the optical search
+could be conducted. Actuated, probably, by the knowledge that in such
+matters as cataloguing and mapping Germany was then, as now, far ahead
+of all the other nations of the world, he wrote in September (the same
+September as Sir John Herschel delivered his eloquent address at
+Southampton) to Berlin. Leverrier wrote, I say, to Dr. Galle, head of
+the Observatory at Berlin, saying to him, clearly and decidedly, that
+the new planet was now in or close to such and such a position, and that
+if he would point his telescope to that part of the heavens he would see
+it; and, moreover, that he would be able to tell it from a star by its
+having a sensible magnitude, or disk, instead of being a mere point.</p>
+
+<p>Galle got the letter on the 23rd of September, 1846. That same evening
+he did point his telescope to the place Leverrier told him, and he saw
+the planet that very night. He recognized it first by its appearance. To
+his practised eye it did seem to have a small disk, and not quite the
+same aspect as an ordinary star. He then consulted Bremiker's great star
+chart, the part just engraved and finished, and sure enough on that
+chart there was no such star there. Undoubtedly it was the planet.</p>
+
+<p>The news flashed over Europe at the maximum speed with which news could
+travel at that date (which was not very fast); and by the 1st of October
+Professor Challis and Mr. Adams heard it at Cambridge, and had the
+pleasure of<span class='pagenum'><a name="Page_330" id="Page_330">[Pg 330]</a></span> knowing that they were forestalled, and that England was
+out of the race.</p>
+
+<p>It was an unconscious race to all concerned, however. Those in France
+knew nothing of the search going on in England. Mr. Adams's papers had
+never been published; and very annoyed the French were when a claim was
+set up on his behalf to a share in this magnificent discovery.
+Controversies and recriminations, excuses and justifications, followed;
+but the discussion has now settled down. All the world honours the
+bright genius and mathematical skill of Mr. Adams, and recognizes that
+he first solved the problem by calculation. All the world, too,
+perceives clearly the no less eminent mathematical talents of M.
+Leverrier, but it recognizes in him something more than the mere
+mathematician&mdash;the man of energy, decision, and character.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_331" id="Page_331">[Pg 331]</a></span></p>
+<h3><a name="LECTURE_XVI" id="LECTURE_XVI"></a>LECTURE XVI</h3>
+
+<h5>COMETS AND METEORS</h5>
+
+
+<p><span class="smcap">We</span> have now considered the solar system in several aspects, and we have
+passed in review something of what is known about the stars. We have
+seen how each star is itself, in all probability, the centre of another
+and distinct solar system, the constituents of which are too dark and
+far off to be visible to us; nothing visible here but the central sun
+alone, and that only as a twinkling speck.</p>
+
+<p>But between our solar system and these other suns&mdash;between each of these
+suns and all the rest&mdash;there exist vast empty spaces, apparently devoid
+of matter.</p>
+
+<p>We have now to ask, Are these spaces really empty? Is there really
+nothing in space but the nebul&aelig;, the suns, their planets, and their
+satellites? Are all the bodies in space of this gigantic size? May there
+not be an infinitude of small bodies as well?</p>
+
+<p>The answer to this question is in the affirmative. There appears to be
+no special size suited to the vastness of space; we find, as a matter of
+fact, bodies of all manner of sizes, ranging by gradations from the most
+tremendous suns, like Sirius, down through ordinary suns to smaller
+ones, then to planets of all sizes, satellites still smaller, then the
+asteroids, till we come to the smallest satellite of Mars, only about
+ten miles in diameter, and weighing only some billion tons&mdash;the smallest
+of the regular bodies belonging to the solar system known.</p>
+
+<p><span class='pagenum'><a name="Page_332" id="Page_332">[Pg 332]</a></span></p><p>But, besides all these, there are found to occur other masses, not much
+bigger and some probably smaller, and these we call comets when we see
+them. Below these, again, we find masses varying from a few tons in
+weight down to only a few pounds or ounces, and these when we see them,
+which is not often, we call meteors or shooting-stars; and to the size
+of these meteorites there would appear to be no limit: some may be
+literal grains of dust. There seems to be a regular gradation of size,
+therefore, ranging from Sirius to dust; and apparently we must regard
+all space as full of these cosmic particles&mdash;stray fragments, as it
+were, perhaps of some older world, perhaps going to help to form a new
+one some day. As Kepler said, there are more "comets" in the sky than
+fish in the sea. Not that they are at all crowded together, else they
+would make a cosmic haze. The transparency of space shows that there
+must be an enormous proportion of clear space between each, and they are
+probably much more concentrated near one of the big bodies than they are
+in interstellar space.<a name="FNanchor_30_30" id="FNanchor_30_30"></a><a href="#Footnote_30_30" class="fnanchor">[30]</a> Even during the furious hail of meteors in
+November 1866 it was estimated that their average distance apart in the
+thickest of the shower was 35 miles.</p>
+
+<p>Consider the nature of a meteor or shooting-star. We ordinarily see them
+as a mere streak of light; sometimes they leave a luminous tail behind
+them; occasionally they appear as an actual fire-ball, accompanied by an
+explosion; sometimes, but very seldom, they are seen to drop, and may
+subsequently be dug up as a lump of iron or rock, showing signs of rough
+treatment by excoriation and heat. These last are the meteorites, or
+siderites, or a&euml;rolites, or bolides,<span class='pagenum'><a name="Page_333" id="Page_333">[Pg 333]</a></span> of our museums. They are popularly
+spoken of as thunderbolts, though they have nothing whatever to do with
+atmospheric electricity.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_95" id="Fig_95"></a>
+<img src="images/fig95.jpg" width="400" height="447" alt="Fig. 95." title="" />
+<span class="caption"><span class="smcap">Fig. 95.</span>&mdash;Meteorite.</span>
+</div>
+
+<p>They appear to be travelling rocky or metallic fragments which in their
+journey through space are caught in the earth's atmosphere and
+instantaneously ignited by the friction. Far away in the depths of space
+one of these bodies felt the attracting power of the sun, and began
+moving towards him. As it approached, its speed grew<span class='pagenum'><a name="Page_334" id="Page_334">[Pg 334]</a></span> gradually quicker
+and quicker continually, until by the time it has approached to within
+the distance of the earth, it whizzes past with the velocity of
+twenty-six miles a second. The earth is moving on its own account
+nineteen miles every second. If the two bodies happened to be moving in
+opposite directions, the combined speed would be terrific; and the
+faintest trace of atmosphere, miles above the earth's surface, would
+exert a furious grinding action on the stone. A stream of particles
+would be torn off; if of iron, they would burn like a shower of filings
+from a firework, thus forming a trail; and the mass itself would be
+dissipated, shattered to fragments in an instant.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_96" id="Fig_96"></a>
+<img src="images/fig96.jpg" width="350" height="546" alt="Fig. 96." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 96.</span>&mdash;Meteor stream crossing field of telescope.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_335" id="Page_335">[Pg 335]</a></span></p>
+<div class="figcenter" style="width: 500px;"><br /><a name="Fig_97" id="Fig_97"></a>
+<img src="images/fig97.jpg" width="400" height="613" alt="Fig. 97." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 97.</span>&mdash;Diagram of direction of earth's orbital
+motion, showing that after midnight, <i>i.e.</i> between midnight and noon,
+more asteroids are likely to be swept up by any locality than between
+noon and midnight. [From Sir R.S. Ball.]</div>
+</div>
+
+<p>Even if the earth were moving laterally, the same thing would occur. But
+if earth and stone happened to be moving in the same direction, there
+would be only the differential velocity of seven miles a second; and
+though this is in all conscience great enough, yet there might be a
+chance for a residue of the nucleus to escape entire destruction, though
+it would be scraped, heated, and superficially molten by the friction;
+but so much of its<span class='pagenum'><a name="Page_336" id="Page_336">[Pg 336]</a></span> speed would be rubbed out of it, that on striking
+the earth it might bury itself only a few feet or yards in the soil, so
+that it could be dug out. The number of those which thus reach the earth
+is comparatively infinitesimal. Nearly all get ground up and dissipated
+by the atmosphere; and fortunate it is for us that they are so. This
+bombardment of the exposed face of the moon must be something
+terrible.<a name="FNanchor_31_31" id="FNanchor_31_31"></a><a href="#Footnote_31_31" class="fnanchor">[31]</a></p>
+
+<p>Thus, then, every shooting-star we see, and all the myriads that we do
+not and cannot see because they occur in the day-time, all these bright
+flashes or streaks, represent the death and burial of one of these
+flying stones. It had been careering on its own account through space
+for untold ages, till it meets a planet. It cannot strike the actual
+body of the planet&mdash;the atmosphere is a sufficient screen; the
+tremendous friction reduces it to dust in an instant, and this dust then
+quietly and leisurely settles down on to the surface.</p>
+
+<p>Evidence of the settlement of meteoric dust is not easy to obtain in
+such a place as England, where the dust which accumulates is seldom of a
+celestial character; but on the snow-fields of Greenland or the
+Himalayas dust can be found; and by a Committee of the British
+Association distinct evidence of molten globules of iron and other
+materials appropriate to a&euml;rolites has been obtained, by the simple
+process of collecting, melting, and filtering long exposed snow.
+Volcanic ash may be mingled with it, but under the microscope the
+volcanic and the meteoric constituents have each a distinctive
+character.</p>
+
+<p>The quantity of meteoric material which reaches the earth as dust must
+be immensely in excess of the minute quantity which arrives in the form
+of lumps. Hundreds or thousands of tons per annum must be received; and
+the accretion must, one would think, in the course of ages be able to
+exert some influence on the period of the earth's rotation&mdash;the<span class='pagenum'><a name="Page_337" id="Page_337">[Pg 337]</a></span> length
+of the day. It is too small, however, to have been yet certainly
+detected. Possibly, it is altogether negligible.</p>
+
+<p>It has been suggested that those stones which actually fall are not the
+true cosmic wanderers, but are merely fragments of our own earth, cast
+up by powerful volcanoes long ago when the igneous power of the earth
+was more vigorous than now&mdash;cast up with a speed of close upon seven
+miles a second; and now in these quiet times gradually being swept up by
+the earth, and so returning whence they came.</p>
+
+<p>I confess I am unable to draw a clear distinction between one set and
+the other. Some falling stars may have had an origin of this sort, but
+certainly others have not; and it would seem very unlikely that one set
+only should fall bodily upon the earth, while the others should always
+be rubbed to powder. Still, it is a possibility to be borne in mind.</p>
+
+<p>We have spoken of these cosmic visitors as wandering masses of stone or
+iron; but we should be wrong if we associated with the term "wandering"
+any ideas of lawlessness and irregularity of path. These small lumps of
+matter are as obedient to the law of gravity as any large ones can be.
+They must all, therefore, have definite orbits, and these orbits will
+have reference to the main attracting power of our system&mdash;they will, in
+fact, be nearly all careering round the sun.</p>
+
+<p>Each planet may, in truth, have a certain following of its own. Within
+the limited sphere of the earth's predominant attraction, for instance,
+extending some way beyond the moon, we may have a number of satellites
+that we never see, all revolving regularly in elliptic orbits round the
+earth. But, comparatively speaking, these satellite meteorites are few.
+The great bulk of them will be of a planetary character&mdash;they will be
+attendant upon the sun.</p>
+
+<p>It may seem strange that such minute bodies should<span class='pagenum'><a name="Page_338" id="Page_338">[Pg 338]</a></span> have regular orbits
+and obey Kepler's laws, but they must. All three laws must be as
+rigorously obeyed by them as by the planets themselves. There is nothing
+in the smallness of a particle to excuse it from implicit obedience to
+law. The only consequence of their smallness is their inability to
+perturb others. They cannot appreciably perturb either the planets they
+approach or each other. The attracting power of a lump one million tons
+in weight is very minute. A pound, on the surface of such a body of the
+same density as the earth, would be only pulled to it with a force equal
+to that with which the earth pulls a grain. So the perturbing power of
+such a mass on distant bodies is imperceptible. It is a good thing it is
+so: accurate astronomy would be impossible if we had to take into
+account the perturbations caused by a crowd of invisible bodies.
+Astronomy would then approach in complexity some of the problems of
+physics.</p>
+
+<p>But though we may be convinced from the facts of gravitation that these
+meteoric stones, and all other bodies flying through space near our
+solar system, must be constrained by the sun to obey Kepler's laws, and
+fly round it in some regular elliptic or hyperbolic orbit, what chance
+have we of determining that orbit? At first sight, a very poor chance,
+for we never see them except for the instant when they splash into our
+atmosphere; and for them that instant is instant death. It is unlikely
+that any escape that ordeal, and even if they do, their career and orbit
+are effectually changed. Henceforward they must become attendants on the
+earth. They may drop on to its surface, or they may duck out of our
+atmosphere again, and revolve round us unseen in the clear space between
+earth and moon.</p>
+
+<p>Nevertheless, although the problem of determining the original orbit of
+any given set of shooting-stars before it struck us would seem nearly
+insoluble, it has been solved, and solved with some approach to
+accuracy; being done by the help of observations of certain other
+bodies. The bodies<span class='pagenum'><a name="Page_339" id="Page_339">[Pg 339]</a></span> by whose help this difficult problem has been
+attacked and resolved are comets. What are comets?</p>
+
+<p>I must tell you that the scientific world is not entirely and completely
+decided on the structure of comets. There are many floating ideas on the
+subject, and some certain knowledge. But the subject is still, in many
+respects, an open one, and the ideas I propose to advocate you will
+accept for no more than they are worth, viz. as worthy to be compared
+with other and different views.</p>
+
+<p>Up to the time of Newton, the nature of comets was entirely unknown.
+They were regarded with superstitious awe as fiery portents, and were
+supposed to be connected with the death of some king, or with some
+national catastrophe.</p>
+
+<p>Even so late as the first edition of the <i>Principia</i> the problem of
+comets was unsolved, and their theory is not given; but between the
+first and the second editions a large comet appeared, in 1680, and
+Newton speculated on its appearance and behaviour. It rushed down very
+close to the sun, spun half round him very quickly, and then receded
+from him again. If it were a material substance, to which the law of
+gravitation applied, it must be moving in a conic section with the sun
+in one focus, and its radius vector must sweep out equal areas in equal
+times. Examining the record of its positions made at observatories, he
+found its observed path quite accordant with theory; and the motion of
+comets was from that time understood. Up to that time no one had
+attempted to calculate an orbit for a comet. They had been thought
+irregular and lawless bodies. Now they were recognized as perfectly
+obedient to the law of gravitation, and revolving round the sun like
+everything else&mdash;as members, in fact, of our solar system, though not
+necessarily permanent members.</p>
+
+<p>But the orbit of a comet is very different from a planetary one. The
+excentricity of its orbit is enormous&mdash;in other words, it is either a
+very elongated ellipse or a parabola.<span class='pagenum'><a name="Page_340" id="Page_340">[Pg 340]</a></span> The comet of 1680, Newton found
+to move in an orbit so nearly a parabola that the time of describing it
+must be reckoned in hundreds of years at the least. It is now thought
+possible that it may not be quite a parabola, but an ellipse so
+elongated that it will not return till 2255. Until that date arrives,
+however, uncertainty will prevail as to whether it is a periodic comet,
+or one of those that only visit our system once. If it be periodic, as
+suspected, it is the same as appeared when Julius C&aelig;sar was killed, and
+which likewise appeared in the years 531 and 1106 <span class="ampm">A.D.</span> Should it appear
+in 2255, our posterity will probably regard it as a memorial of Newton.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_98" id="Fig_98"></a>
+<img src="images/fig98.jpg" width="400" height="431" alt="Fig. 98." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 98.</span>&mdash;Parabolic and elliptic orbits. The <i>a b</i>
+(visible) portions are indistinguishable.</span>
+</div>
+
+<p>The next comet discussed in the light of the theory of gravitation was
+the famous one of Halley. You know<span class='pagenum'><a name="Page_341" id="Page_341">[Pg 341]</a></span> something of the history of this.
+Its period is 75&frac12; years. Halley saw it in 1682, and predicted its
+return in 1758 or 1759&mdash;the first cometary prediction. Clairaut
+calculated its return right within a month (<a href="#Page_219">p. 219</a>). It has been back
+once more, in 1835; and this time its date was correctly predicted
+within three days, because Uranus was now known. It was away at its
+furthest point in 1873. It will be back again in 1911.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_99" id="Fig_99"></a>
+<img src="images/fig99.jpg" width="400" height="356" alt="Fig. 99." title="" />
+<span class="caption"><span class="smcap">Fig. 99.</span>&mdash;Orbit of Halley&#39;s comet.</span>
+</div>
+
+<p>Coming to recent times, we have the great comets of 1843 and of 1858,
+the history of neither being known. Quite possibly they arrived then for
+the first time. Possibly the second will appear again in 3808. But
+besides these great comets, there are a multitude of telescopic ones,
+which do not show these striking features, and have no gigantic tail.
+Some have no tail at all, others have at best a few insignificant
+streamers, and others show a faint haze looking like a microscopic
+nebula.</p>
+
+<p>All these comets are of considerable extent&mdash;some millions of miles
+thick usually, and yet stars are clearly visible through them. Hence
+they must be matter of very small density; their tails can be nothing
+more dense than a filmy<span class='pagenum'><a name="Page_342" id="Page_342">[Pg 342]</a></span> mist, but their nucleus must be something more
+solid and substantial.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_100" id="Fig_100"></a>
+<img src="images/fig100.jpg" width="400" height="236" alt="Fig. 100." title="" />
+<span class="caption"><span class="smcap">Fig. 100.</span>&mdash;Various appearances of Halley&#39;s comet when
+last seen.</span>
+</div>
+
+<p>I have said that comets arrive from the depths of space, rush towards
+and round the sun, whizzing past the earth with a speed of twenty-six
+miles a second, on round the sun with a far greater velocity than that,
+and then rush off again. Now, all the time they are away from the sun
+they are invisible. It is only as they get near him that they begin to
+expand and throw off tails and other appendages. The sun's heat is
+evidently evaporating them, and driving away a cloud of mist and
+volatile matter. This is when they can be seen. The comet is most
+gorgeous when it is near the sun, and as soon as it gets a reasonable
+distance away from him it is perfectly invisible.</p>
+
+<p>The matter evaporated from the comet by the sun's heat does not
+return&mdash;it is lost to the comet; and hence, after a few such journeys,
+its volatile matter gets appreciably diminished, and so old-established
+periodic comets have no tails to speak of. But the new visitants, coming
+from the depths of space for the first time&mdash;these have great supplies<span class='pagenum'><a name="Page_343" id="Page_343">[Pg 343]</a></span>
+of volatile matter, and these are they which show the most magnificent
+tails.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_101" id="Fig_101"></a>
+<img src="images/fig101.jpg" width="400" height="675" alt="Fig. 101." title="" />
+<span class="caption"><span class="smcap">Fig. 101.</span>&mdash;Head of Donati&#39;s comet of 1858.</span>
+</div>
+
+<p>The tail of a comet is always directed away from the sun as if it were
+repelled. To this rule there is no exception. It is suggested, and held
+as most probable, that the tail and sun are similarly electrified, and
+that the repulsion of the tail is electrical repulsion. Some great force
+is obviously at work to account for the enormous distance to which the
+tail is shot in a few hours. The pressure of the sun's light can do
+something, and is a force that must not be ignored when small particles
+are being dealt with. (Cf. <i>Modern Views of Electricity</i>, 2nd edition,
+p. 363.)</p>
+
+<p>Now just think what analogies there are between comets and meteors. Both
+are bodies travelling in orbits round the sun, and both are mostly
+invisible, but both become visible to us under certain circumstances.
+Meteors become visible when they plunge into the extreme limits of our<span class='pagenum'><a name="Page_344" id="Page_344">[Pg 344]</a></span>
+atmosphere. Comets become visible when they approach the sun. Is it
+possible that comets are large meteors which dip into the solar
+atmosphere, and are thus rendered conspicuously luminous? Certainly they
+do not dip into the actual main atmosphere of the sun, else they would
+be utterly destroyed; but it is possible that the sun has a faint trace
+of atmosphere extending far beyond this, and into this perhaps these
+meteors dip, and glow with the friction. The particles thrown off might
+be, also by friction, electrified; and the vaporous tail might be thus
+accounted for.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_102" id="Fig_102"></a>
+<img src="images/fig102.jpg" width="400" height="246" alt="Fig. 102." title="" />
+<span class="caption"><span class="smcap">Fig. 102.</span>&mdash;Halley&#39;s Comet.</span>
+</div>
+
+<p>Let us make this hypothesis provisionally&mdash;that comets are large
+meteors, or a compact swarm of meteors, which, coming near the sun, find
+a highly rarefied sort of atmosphere, in which they get heated and
+partly vaporized, just as ordinary meteorites do when they dip into the
+atmosphere of the earth. And let us see whether any facts bear out the
+analogy and justify the hypothesis.</p>
+
+<p>I must tell you now the history of three bodies, and you will see that
+some intimate connection between comets and<span class='pagenum'><a name="Page_345" id="Page_345">[Pg 345]</a></span> meteors is proved. The
+three bodies are known as, first, Encke's comet; second, Biela's comet;
+third, the November swarm of meteors.</p>
+
+<p>Encke's comet (one of those discovered by Miss Herschel) is an
+insignificant-looking telescopic comet of small period, the orbit of
+which was well known, and which was carefully observed at each
+reappearance after Encke had calculated its orbit. It was the quickest
+of the comets, returning every 3&frac12; years.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_103" id="Fig_103"></a>
+<img src="images/fig103.jpg" width="400" height="511" alt="Fig. 103." title="" />
+<span class="caption"><span class="smcap">Fig. 103.</span>&mdash;Encke&#39;s comet.</span>
+</div>
+
+<p>It was found, however, that its period was not quite constant; it kept
+on getting slightly shorter. The comet, in fact, returned to the sun
+slightly before its time. Now this effect is exactly what friction
+against a solar atmosphere would bring about. Every time it passed near
+the sun a little velocity would be rubbed out of it. But the velocity is
+that which carries it away, hence it would not go quite so far, and
+therefore would return a little sooner. Any revolving body subject to
+friction must revolve quicker<span class='pagenum'><a name="Page_346" id="Page_346">[Pg 346]</a></span> and quicker, and get nearer and nearer
+its central body, until, if the process goes on long enough, it must
+drop upon its surface. This seems the kind of thing happening to Encke's
+comet. The effect is very small, and not thoroughly proved; but, so far
+as it goes, the evidence points to a greatly extended rare solar
+atmosphere, which rubs some energy out of it at every perihelion
+passage.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_104" id="Fig_104"></a>
+<img src="images/fig104.jpg" width="400" height="283" alt="Fig. 104." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 104.</span>&mdash;Biela&#39;s comet as last seen, in two portions.</span>
+</div>
+
+<p>Next, Biela's comet. This also was a well known and carefully observed
+telescopic comet, with a period of six years. In one of its distant
+excursions, it was calculated that it must pass very near Jupiter, and
+much curiosity was excited as to what would happen to it in consequence
+of the perturbation it must experience. As I have said, comets are only
+visible as they approach the sun, and a watch was kept for it about its
+appointed time. It was late, but it did ultimately arrive.</p>
+
+<p>The singular thing about it, however, was that it was now double. It had
+apparently separated into two. This was in 1846. It was looked for again
+in 1852, and this time the components were further separated. Sometimes
+one was brighter, sometimes the other. Next time it ought to have come
+round no one could find either portion. The<span class='pagenum'><a name="Page_347" id="Page_347">[Pg 347]</a></span> comet seemed to have wholly
+disappeared. It has never been seen since. It was then recorded and
+advertised as the missing comet.</p>
+
+<p>But now comes the interesting part of the story. The orbit of this Biela
+comet was well known, and it was found that on a certain night in 1872
+the earth would cross the orbit, and had some chance of encountering the
+comet. Not a very likely chance, because it need not be in that part of
+its orbit at the time; but it was suspected not to be far off&mdash;if still
+existent. Well, the night arrived, the earth did cross the orbit, and
+there was seen, not the comet, but a number of shooting-stars. Not one
+body, nor yet two, but a multitude of bodies&mdash;in fact, a swarm of
+meteors. Not a very great swarm, such as sometimes occurs, but still a
+quite noticeable one; and this shower of meteors is definitely
+recognized as flying along the track of Biela's comet. They are known as
+the Andromedes.</p>
+
+<p>This observation has been generalized. Every cometary orbit is marked by
+a ring of meteoric stones travelling round it, and whenever a number of
+shooting-stars are seen quickly one after the other, it is an evidence
+that we are crossing the track of some comet. But suppose instead of
+only crossing the track of a comet we were to pass close to the comet
+itself, we should then expect to see an extraordinary swarm&mdash;a multitude
+of shooting-stars. Such phenomena have occurred. The most famous are
+those known as the November meteors, or Leonids.</p>
+
+<p>This is the third of those bodies whose history I had to tell you.
+Professor H.A. Newton, of America, by examining ancient records arrived
+at the conclusion that the earth passed through a certain definite
+meteor shoal every thirty-three years. He found, in fact, that every
+thirty-three years an unusual flight of shooting-stars was witnessed in
+November, the earliest record being 599 <span class="ampm">A.D.</span> Their last appearance had
+been in 1833, and he therefore predicted their return in 1866 or 1867.
+Sure enough, in November,<span class='pagenum'><a name="Page_348" id="Page_348">[Pg 348]</a></span> 1866, they appeared; and many must remember
+seeing that glorious display. Although their hail was almost continuous,
+it is estimated that their average distance apart was thirty-five miles!
+Their radiant point was and always is in the constellation Leo, and
+hence their name Leonids.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_105" id="Fig_105"></a>
+<img src="images/fig105.jpg" width="500" height="449" alt="Fig. 105." title="" />
+<span class="caption"><span class="smcap">Fig. 105.</span>&mdash;Radiant point perspective. The arrows
+represent a number of approximately parallel meteor-streaks
+foreshortened from a common vanishing-point.</span>
+</div>
+
+<div class="blockquot"><p>A parallel stream fixed in space necessarily exhibits a definite
+aspect with reference to the fixed stars. Its aspect with respect
+to the earth will be very changeable, because of the rotation and
+revolution of that body, but its position with respect to
+constellations will be steady. Hence each meteor swarm, being a
+steady parallel stream of rushing<span class='pagenum'><a name="Page_349" id="Page_349">[Pg 349]</a></span> masses, always strikes us from
+the same point in stellar space, and by this point (or radiant) it
+is identified and named.</p>
+
+<p>The paths do not appear to us to be parallel, because of
+perspective: they seem to radiate and spread in all directions from
+a fixed centre like spokes, but all these diverging streaks are
+really parallel lines optically foreshortened by different amounts
+so as to produce the radiant impression.</p>
+
+<p>The annexed diagram (<a href="#Fig_105">Fig. 105</a>) clearly illustrates the fact that
+the "radiant" is the vanishing point of a number of parallel lines. </p></div>
+
+<div class="figcenter" style="width: 350px;"><a name="Fig_106" id="Fig_106"></a>
+<img src="images/fig106.jpg" width="350" height="486" alt="Fig. 106." title="" />
+<span class="caption"><span class="smcap">Fig. 106.</span>&mdash;Orbit of November meteors.</span>
+</div>
+
+<p>This swarm is specially interesting to us from the fact that we cross
+its orbit every year. Its orbit and the earth's<span class='pagenum'><a name="Page_350" id="Page_350">[Pg 350]</a></span> intersect. Every
+November we go through it, and hence every November we see a few
+stragglers of this immense swarm. The swarm itself takes thirty-three
+years on its revolution round the sun, and hence we only encounter it
+every thirty-three years.</p>
+
+<p>The swarm is of immense size. In breadth it is such that the earth,
+flying nineteen miles a second, takes four or five hours to cross it,
+and this is therefore the time the display lasts. But in length it is
+far more enormous. The speed with which it travels is twenty-five miles
+a second, (for its orbit extends as far as Uranus, although by no means
+parabolic), and yet it takes more than a year to pass. Imagine a
+procession 200,000 miles broad, every individual rushing along at the
+rate of twenty-five miles every second, and the whole procession so long
+that it takes more than a year to pass. It is like a gigantic shoal of
+herrings swimming round and round the sun every thirty-three years, and
+travelling past the earth with that tremendous velocity of twenty-five
+miles a second. The earth dashes through the swarm and sweeps up
+myriads. Think of the countless numbers swept up by the whole earth in
+crossing such a shoal as that! But heaps more remain, and probably the
+millions which are destroyed every thirty-three years have not yet made
+any very important difference to the numbers still remaining.</p>
+
+<p>The earth never misses this swarm. Every thirty-three years it is bound
+to pass through some part of them, for the shoal is so long that if the
+head is just missed one November the tail will be encountered next
+November. This is a plain and obvious result of its enormous length. It
+may be likened to a two-foot length of sewing silk swimming round and
+round an oval sixty feet in circumference. But, you will say, although
+the numbers are so great that destroying a few millions or so every
+thirty-three years makes but little difference to them, yet, if this
+process has been going on from all eternity, they ought to be all swept<span class='pagenum'><a name="Page_351" id="Page_351">[Pg 351]</a></span>
+up. Granted; and no doubt the most ancient swarms have already all or
+nearly all been swept up.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_107" id="Fig_107"></a>
+<img src="images/fig107.jpg" width="400" height="598" alt="Fig. 107." title="" /><br />
+<div class="caption1"><span class="smcap">Fig. 107.</span>&mdash;Orbit of November meteors; showing their
+probable parabolic orbit previous to 126 <span class="ampm">A.D.</span>, and its sudden conversion
+into an elliptic orbit by the violent perturbation caused by Uranus,
+which at that date occupied the position shown.</div>
+</div>
+
+<p>The August meteors, or Perseids, are an example. Every August we cross
+their path, and we have a small meteoric<span class='pagenum'><a name="Page_352" id="Page_352">[Pg 352]</a></span> display radiating from the
+sword-hand of Perseus, but never specially more in one August than
+another. It would seem as if the main shoal has disappeared, and nothing
+is now left but the stragglers; or perhaps it is that the shoal has
+gradually become uniformly distributed all along the path. Anyhow, these
+August meteors are reckoned much more ancient members of the solar
+system than are the November meteors. The November meteors are believed
+to have entered the solar system in the year 126 <span class="ampm">A.D.</span></p>
+
+<p>This may seem an extraordinary statement. It is not final, but it is
+based on the calculations of Leverrier&mdash;confirmed recently by Mr. Adams.
+A few moments will suffice to make the grounds of it clear. Leverrier
+calculated the orbit of the November meteors, and found them to be an
+oval extending beyond Uranus. It was perturbed by the outer planets near
+which it went, so that in past times it must have moved in a slightly
+different orbit. Calculating back to their past positions, it was found
+that in a certain year it must have gone very near to Uranus, and that
+by the perturbation of this planet its path had been completely changed.
+Originally it had in all probability been a comet, flying in a parabolic
+orbit towards the sun like many others. This one, encountering Uranus,
+was pulled to pieces as it were, and its orbit made elliptical as shown
+in <a href="#Fig_107">Fig. 107</a>. It was no longer free to escape and go away into the depths
+of space: it was enchained and made a member of the solar system. It
+also ceased to be a comet; it was degraded into a shoal of meteors.</p>
+
+<p>This is believed to be the past history of this splendid swarm. Since
+its introduction to the solar system it has made 52 revolutions: its
+next return is due in November, 1899, and I hope that it may occur in
+the English dusk, and (<a href="#Fig_97">see Fig. 97</a>) in a cloudless after-midnight sky,
+as it did in 1866.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_353" id="Page_353">[Pg 353]</a></span></p>
+<h4><a name="NOTES_FOR_LECTURE_XVII" id="NOTES_FOR_LECTURE_XVII"></a>NOTES FOR LECTURE XVII</h4>
+
+
+<p>The tide-generating force of one body on another is directly as the mass
+of the one body and inversely as the cube of the distance between them.
+Hence the moon is more effective in producing terrestrial tides than the
+sun.</p>
+
+<p>The tidal wave directly produced by the moon in the open ocean is about
+5 feet high, that produced by the sun is about 2 feet. Hence the average
+spring tide is to the average neap as about 7 to 3. The lunar tide
+varies between apogee and perigee from 4&middot;3 to 5&middot;9.</p>
+
+<p>The solar tide varies between aphelion and perihelion from 1&middot;9 to 2&middot;1.
+Hence the highest spring tide is to the lowest neap as 5&middot;9 + 2&middot;1 is to
+4&middot;3 -2&middot;1, or as 8 to 2&middot;2.</p>
+
+<p>The semi-synchronous oscillation of the Southern Ocean raises the
+magnitude of oceanic tides somewhat above these directly generated
+values.</p>
+
+<p>Oceanic tides are true waves, not currents. Coast tides are currents.
+The momentum of the water, when the tidal wave breaks upon a continent
+and rushes up channels, raises coast tides to a much greater height&mdash;in
+some places up to 50 or 60 feet, or even more.</p>
+
+<p>Early observed connections between moon and tides would be these:&mdash;</p>
+
+<div class="blockquot"><p class="hang">1st. Spring tides at new and full moon.</p>
+
+<p class="hang">2nd. Average interval between tide and tide is half a lunar, not a
+solar, day&mdash;a lunar day being the interval between two successive
+returns of the moon to the meridian: 24 hours and 50 minutes.</p>
+
+<p class="hang">3rd. The tides of a given place at new and full moon occur always
+at the same time of day whatever the season of the year.</p></div>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_354" id="Page_354">[Pg 354]</a></span></p>
+<h3><a name="LECTURE_XVII" id="LECTURE_XVII"></a>LECTURE XVII</h3>
+
+<h5>THE TIDES</h5>
+
+
+<p><span class="smcap">Persons</span> accustomed to make use of the Mersey landing-stages can hardly
+fail to have been struck with two obvious phenomena. One is that the
+gangways thereto are sometimes almost level, and at other times very
+steep; another is that the water often rushes past the stage rather
+violently, sometimes south towards Garston, sometimes north towards the
+sea. They observe, in fact, that the water has two periodic motions&mdash;one
+up and down, the other to and fro&mdash;a vertical and a horizontal motion.
+They may further observe, if they take the trouble, that a complete
+swing of the water, up and down, or to and fro, takes place about every
+twelve and a half hours; moreover, that soon after high and low water
+there is no current&mdash;the water is stationary, whereas about half-way
+between high and low it is rushing with maximum speed either up or down
+the river.</p>
+
+<p>To both these motions of the water the name <i>tide</i> is given, and both
+are extremely important. Sailors usually pay most attention to the
+horizontal motion, and on charts you find the tide-races marked; and the
+places where there is but a small horizontal rush of the water are
+labelled "very little tide here." Landsmen, or, at any rate, such of the
+more philosophic sort as pay any attention to the matter at all, think
+most of the vertical motion of the water&mdash;its amount of rise and fall.</p>
+
+<p><span class='pagenum'><a name="Page_355" id="Page_355">[Pg 355]</a></span></p><p>Dwellers in some low-lying districts in London are compelled to pay
+attention to the extra high tides of the Thames, because it is, or was,
+very liable to overflow its banks and inundate their basements.</p>
+
+<p>Sailors, however, on nearing a port are also greatly affected by the
+time and amount of high water there, especially when they are in a big
+ship; and we know well enough how frequently Atlantic liners, after
+having accomplished their voyage with good speed, have to hang around
+for hours waiting till there is enough water to lift them over the
+Bar&mdash;that standing obstruction, one feels inclined to say disgrace, to
+the Liverpool harbour.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_108" id="Fig_108"></a>
+<img src="images/fig108.jpg" width="400" height="249" alt="Fig. 108." title="" />
+<span class="caption"><span class="smcap">Fig. 108.</span>&mdash;The Mersey</span>
+</div>
+
+<p>To us in Liverpool the tides are of supreme importance&mdash;upon them the
+very existence of the city depends&mdash;for without them Liverpool would not
+be a port. It may be familiar to many of you how this is, and yet it is
+a matter that cannot be passed over in silence. I will therefore call
+your attention to the Ordnance Survey of the estuaries of the Mersey and
+the Dee. You see first that there is a great tendency for sand-banks to
+accumulate all about this coast, from North Wales right away round to
+Southport. You see next that the port of Chester has been practically
+silted<span class='pagenum'><a name="Page_356" id="Page_356">[Pg 356]</a></span> up by the deposits of sand in the wide-mouthed Dee, while the
+port of Liverpool remains open owing to the scouring action of the tide
+in its peculiarly shaped channel. Without the tides the Mersey would be
+a wretched dribble not much bigger than it is at Warrington. With them,
+this splendid basin is kept open, and a channel is cut of such depth
+that the <i>Great Eastern</i> easily rode in it in all states of the water.</p>
+
+<p>The basin is filled with water every twelve hours through its narrow
+neck. The amount of water stored up in this basin at high tide I
+estimate as 600 million tons. All this quantity flows through the neck
+in six hours, and flows out again in the next six, scouring and
+cleansing and carrying mud and sand far out to sea. Just at present the
+currents set strongest on the Birkenhead side of the river, and
+accordingly a "Pluckington bank" unfortunately grows under the Liverpool
+stage. Should this tendency to silt up the gates of our docks increase,
+land can be reclaimed on the other side of the river between Tranmere
+and Rock Ferry, and an embankment made so as to deflect the water over
+Liverpool way, and give us a fairer proportion of the current. After
+passing New Brighton the water spreads out again to the left; its
+velocity forward diminishes; and after a few miles it has no power to
+cut away that sandbank known as the Bar. Should it be thought desirable
+to make it accomplish this, and sweep the Bar further out to sea into
+deeper water, it is probable that a rude training wall (say of old
+hulks, or other removable partial obstruction) on the west of Queen's
+Channel, arranged so as to check the spreading out over all this useless
+area, may be quite sufficient to retain the needed extra impetus in the
+water, perhaps even without choking up the useful old Rock Channel,
+through which smaller ships still find convenient exit.</p>
+
+<p>Now, although the horizontal rush of the tide is necessary to our
+existence as a port, it does not follow that the accompanying rise and
+fall of the water is an unmixed blessing.<span class='pagenum'><a name="Page_357" id="Page_357">[Pg 357]</a></span> To it is due the need for all
+the expensive arrangements of docks and gates wherewith to store up the
+high-level water. Quebec and New York are cities on such magnificent
+rivers that the current required to keep open channel is supplied
+without any tidal action, although Quebec is nearly 1,000 miles from the
+open ocean; and accordingly, Atlantic liners do not hover in mid-river
+and discharge passengers by tender, but they proceed straight to the
+side of the quays lining the river, or, as at New York, they dive into
+one of the pockets belonging to the company running the ship, and there
+discharge passengers and cargo without further trouble, and with no need
+for docks or gates. However, rivers like the St. Lawrence and the Hudson
+are the natural property of a gigantic continent; and we in England may
+be well contented with the possession of such tidal estuaries as the
+Mersey, the Thames, and the Humber. That by pertinacious dredging the
+citizens of Glasgow manage to get large ships right up their small
+river, the Clyde, to the quays of the town, is a remarkable fact, and
+redounds very highly to their credit.</p>
+
+<p>We will now proceed to consider the connection existing between the
+horizontal rush of water and its vertical elevation, and ask, Which is
+cause and which is effect? Does the elevation of the ocean cause the
+tidal flow, or does the tidal flow cause the elevation? The answer is
+twofold: both statements are in some sense true. The prime cause of the
+tide is undoubtedly a vertical elevation of the ocean, a tidal wave or
+hump produced by the attraction of the moon. This hump as it passes the
+various channels opening into the ocean raises their level, and causes
+water to flow up them. But this simple oceanic tide, although the cause
+of all tide, is itself but a small affair. It seldom rises above six or
+seven feet, and tides on islands in mid-ocean have about this value or
+less. But the tides on our coasts are far greater than this&mdash;they rise
+twenty or thirty feet, or even fifty feet occasionally, at some places,
+as at<span class='pagenum'><a name="Page_358" id="Page_358">[Pg 358]</a></span> Bristol. Why is this? The horizontal motion of the water gives it
+such an impetus or momentum that its motion far transcends that of the
+original impulse given to it, just as a push given to a pendulum may
+cause it to swing over a much greater arc than that through which the
+force acts. The inrushing water flowing up the English Channel or the
+Bristol Channel or St. George's Channel has such an impetus that it
+propels itself some twenty or thirty feet high before it has exhausted
+its momentum and begins to descend. In the Bristol Channel the gradual
+narrowing of the opening so much assists this action that the tides
+often rise forty feet, occasionally fifty feet, and rush still further
+up the Severn in a precipitous and extraordinary hill of water called
+"the bore."</p>
+
+<p>Some places are subject to considerable rise and fall of water with very
+little horizontal flow; others possess strong tidal races, but very
+little elevation and depression. The effect observed at any given place
+entirely depends on whether the place has the general character of a
+terminus, or whether it lies <i>en route</i> to some great basin.</p>
+
+<p>You must understand, then, that all tide takes its rise in the free and
+open ocean under the action of the moon. No ordinary-sized sea like the
+North Sea, or even the Mediterranean, is big enough for more than a just
+appreciable tide to be generated in it. The Pacific, the Atlantic, and
+the Southern Oceans are the great tidal reservoirs, and in them the
+tides of the earth are generated as low flat humps of gigantic area,
+though only a few feet high, oscillating up and down in the period of
+approximately twelve hours. The tides we, and other coast-possessing
+nations, experience are the overflow or back-wash of these oceanic
+humps, and I will now show you in what manner the great Atlantic
+tide-wave reaches the British Isles twice a day.</p>
+
+<p><span class='pagenum'><a name="Page_359" id="Page_359">[Pg 359]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_109" id="Fig_109"></a>
+<img src="images/fig109.jpg" width="400" height="403" alt="Fig. 109." title="" />
+<span class="caption"><span class="smcap">Fig. 109.</span>&mdash;Co-tidal lines.</span>
+</div>
+
+<p><a href="#Fig_109">Fig. 109</a> shows the contour lines of the great wave as it rolls in east
+from the Atlantic, getting split by the Land's End and by Ireland into
+three portions; one of which rushes up the English Channel and through
+the Straits of Dover. Another rolls up the Irish Sea, with a minor
+offshoot up the Bristol Channel, and, curling round Anglesey, flows
+along the North Wales coast and fills Liverpool Bay and the Mersey. The
+third branch streams round the north coast of Ireland, past the Mull of
+Cantyre and Rathlin Island; part fills up the Firth of Clyde, while the
+rest flows south, and, swirling round the west side of the Isle of Man,
+helps the southern current to fill the Bay of Liverpool. The rest of the
+great wave impinges on the coast of Scotland, and, curling round it,
+fills up the North Sea right away to the Norway coast, and then flows
+down below Denmark, joining the southern and earlier arriving stream.
+The diagram I show you is a rough chart of cotidal<span class='pagenum'><a name="Page_360" id="Page_360">[Pg 360]</a></span> lines, which I made
+out of the information contained in <i>Whitaker's Almanac</i>.</p>
+
+<p>A place may thus be fed with tide by two distinct channels, and many
+curious phenomena occur in certain places from this cause. Thus it may
+happen that one channel is six hours longer than the other, in which
+case a flow will arrive by one at the same time as an ebb arrives by the
+other; and the result will be that the place will have hardly any tide
+at all, one tide interfering with and neutralizing the other. This is
+more markedly observed at other parts of the world than in the British
+Isles. Whenever a place is reached by two channels of different length,
+its tides are sure to be peculiar, and probably small.</p>
+
+<p>Another cause of small tide is the way the wave surges to and fro in a
+channel. The tidal wave surging up the English Channel, for instance,
+gets largely reflected by the constriction at Dover, and so a crest
+surges back again, as we may see waves reflected in a long trough or
+tilted bath. The result is that Southampton has two high tides rapidly
+succeeding one another, and for three hours the high-water level varies
+but slightly&mdash;a fact of evident convenience to the port.</p>
+
+<p>Places on a nodal line, so to speak, about the middle of the length of
+the channel, have a minimum of rise and fall, though the water rushes
+past them first violently up towards Dover, where the rise is
+considerable, and then back again towards the ocean. At Portland, for
+instance, the total rise and fall is very small: it is practically on a
+node. Yarmouth, again, is near a less marked node in the North Sea,
+where stationary waves likewise surge to and fro, and accordingly the
+tidal rise and fall at Yarmouth is only about five feet (varying from
+four and a half to six), whereas at London it is twenty or thirty feet,
+and at Flamborough Head or Leith it is from twelve to sixteen feet.</p>
+
+<p>It is generally supposed that water never flows up-hill, but in these
+cases of oscillation it flows up-hill for three<span class='pagenum'><a name="Page_361" id="Page_361">[Pg 361]</a></span> hours together. The
+water is rushing up the English Channel towards Dover long after it is
+highest at the Dover end; it goes on piling itself up, until its
+momentum is checked by the pressure, and then it surges back. It
+behaves, in fact, very like the bob of a pendulum, which rises against
+gravity at every quarter swing.</p>
+
+<p>To get a very large tide, the place ought to be directly accessible by a
+long sweep of a channel to the open ocean, and if it is situate on a
+gradually converging opening, the ebb and flow may be enormous. The
+Severn is the best example of this on the British Isles; but the largest
+tides in the world are found, I believe, in the Bay of Fundy, on the
+coast of North America, where they sometimes rise one hundred and twenty
+feet. Excessive or extra tides may be produced occasionally in any place
+by the propelling force of a high wind driving the water towards the
+shore; also by a low barometer, <i>i.e.</i> by a local decrease in the
+pressure of the air.</p>
+
+<p>Well, now, leaving these topographical details concerning tides, which
+we see to be due to great oceanic humps (great in area that is, though
+small in height), let us proceed to ask what causes these humps; and if
+it be the moon that does it, how does it do it?</p>
+
+<p>The statement that the moon causes the tides sounds at first rather an
+absurdity, and a mere popular superstition. Galileo chaffed Kepler for
+believing it. Who it was that discovered the connection between moon and
+tides we know not&mdash;probably it is a thing which has been several times
+rediscovered by observant sailors or coast-dwellers&mdash;and it is certainly
+a very ancient piece of information.</p>
+
+<p>Probably the first connection observed was that about full moon and
+about new moon the tides are extra high, being called spring tides,
+whereas about half-moon the tides are much less, and are called neap
+tides. The word spring in this connection has no reference to the season
+of the year; except that both words probably represent the same idea of
+energetic<span class='pagenum'><a name="Page_362" id="Page_362">[Pg 362]</a></span> uprising or upspringing, while the word neap comes from nip,
+and means pinched, scanty, nipped tide.</p>
+
+<p>The next connection likely to be observed would be that the interval
+between two day tides was not exactly a solar day of twenty-four hours,
+but a lunar day of fifty minutes longer. For by reason of the moon's
+monthly motion it lags behind the sun about fifty minutes a day, and the
+tides do the same, and so perpetually occur later and later, about fifty
+minutes a day later, or 12 hours and 25 minutes on the average between
+tide and tide.</p>
+
+<p>A third and still more striking connection was also discovered by some
+of the ancient great navigators and philosophers&mdash;viz. that the time of
+high water at a given place at full moon is always the same, or very
+nearly so. In other words, the highest or spring tides always occur
+nearly at the same time of day at a given place. For instance, at
+Liverpool this time is noon and midnight. London is about two hours and
+a half later. Each port has its own time for receiving a given tide, and
+the time is called the "establishment" of the port. Look out a day when
+the moon is full, and you will find the Liverpool high tide occurs at
+half-past eleven, or close upon it. The same happens when the moon is
+new. A day after full or new moon the spring tides rise to their
+highest, and these extra high tides always occur in Liverpool at noon
+and at midnight, whatever the season of the year. About the equinoxes
+they are liable to be extraordinarily high. The extra low tides here are
+therefore at 6 a.m. and 6 p.m., and the 6 p.m. low tide is a nuisance to
+the river steamers. The spring tides at London are highest about
+half-past two.</p>
+
+<hr style='width: 15%;' />
+
+<p>It is, therefore, quite clear that the moon has to do with the tides. It
+and the sun together are, in fact, the whole cause of them; and the mode
+in which these bodies act by gravitative attraction was first made out
+and explained in remarkably full detail by Sir Isaac Newton. You will
+find<span class='pagenum'><a name="Page_363" id="Page_363">[Pg 363]</a></span> his account of the tides in the second and third books of the
+<i>Principia</i>; and though the theory does not occupy more than a few pages
+of that immortal work, he succeeds not only in explaining the local
+tidal peculiarities, much as I have done to-night, but also in
+calculating the approximate height of mid-ocean solar tide; and from the
+observed lunar tide he shows how to determine the then quite unknown
+mass of the moon. This was a quite extraordinary achievement, the
+difficulty of which it is not easy for a person unused to similar
+discussions fully to appreciate. It is, indeed, but a small part of what
+Newton accomplished, but by itself it is sufficient to confer
+immortality upon any ordinary philosopher, and to place him in a front
+rank.</p>
+
+<div class="figcenter" style="width: 400px;"><a name="Fig_110" id="Fig_110"></a>
+<img src="images/fig110.jpg" width="400" height="217" alt="Fig. 110." title="" />
+<span class="caption"><span class="smcap">Fig. 110.</span>&mdash;Whirling earth model.</span>
+</div>
+
+<p>To make intelligible Newton's theory of the tides, I must not attempt to
+go into too great detail. I will consider only the salient points.
+First, you know that every mass of matter attracts every other piece of
+matter; second, that the moon revolves round the earth, or rather that
+the earth and moon revolve round their common centre of gravity once a
+month; third, that the earth spins on its own axis once a day; fourth,
+that when a thing is whirled round, it tends to fly out from the centre
+and requires a force to hold it in. These are the principles involved.
+You can whirl a bucket full of water vertically round without spilling<span class='pagenum'><a name="Page_364" id="Page_364">[Pg 364]</a></span>
+it. Make an elastic globe rotate, and it bulges out into an oblate or
+orange shape; as illustrated by the model shown in <a href="#Fig_110">Fig. 110</a>. This is
+exactly what the earth does, and Newton calculated the bulging of it as
+fourteen miles all round the equator. Make an elastic globe revolve
+round a fixed centre outside itself, and it gets pulled into a prolate
+or lemon shape; the simplest illustrative experiment is to attach a
+string to an elastic bag or football full of water, and whirl it round
+and round. Its prolateness is readily visible.</p>
+
+<p>Now consider the earth and moon revolving round each other like a man
+whirling a child round. The child travels furthest, but the man cannot
+merely rotate, he leans back and thus also describes a small circle: so
+does the earth; it revolves round the common centre of gravity of earth
+and moon (<i>cf.</i> <a href="#Page_212">p. 212</a>). This is a vital point in the comprehension of
+the tides: the earth's centre is not at rest, but is being whirled round
+by the moon, in a circle about <span class="above">1</span>&#8260;<span class="below">80</span> as big as the circle which the moon
+describes, because the earth weighs eighty times as much as the moon.
+The effect of the revolution is to make both bodies slightly protrude in
+the direction of the line joining them; they become slightly "prolate"
+as it is called&mdash;that is, lemon-shaped. Illustrating still by the man
+and child, the child's legs fly outwards so that he is elongated in the
+direction of a radius; the man's coat-tails fly out too, so that he too
+is similarly though less elongated. These elongations or protuberances
+constitute the tides.</p>
+
+<p><span class='pagenum'><a name="Page_365" id="Page_365">[Pg 365]</a></span></p>
+<div class="figcenter" style="width: 550px;"><a name="Fig_111" id="Fig_111"></a>
+<img src="images/fig111.jpg" width="400" height="110" alt="Fig. 111." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 111.</span>&mdash;Earth and moon model, illustrating the
+production of statical or &quot;equilibrium&quot; tides when the whole is whirled
+about the point G.</span>
+</div>
+
+<p><a href="#Fig_111">Fig. 111</a> shows a model to illustrate the mechanism. A couple of
+cardboard disks (to represent globes of course), one four times the
+diameter of the other, and each loaded so as to have about the correct
+earth-moon ratio of weights, are fixed at either end of a long stick,
+and they balance about a certain point, which is their common centre of
+gravity. For convenience this point is taken a trifle too far out from
+the centre of the earth&mdash;that is, just beyond its surface. Through the
+balancing point G a bradawl is stuck, and on that as pivot the whole
+readily revolves. Now, behind the circular disks, you see, are four
+pieces of card of appropriate shape, which are able to slide out under
+proper forces. They are shown dotted in the figure, and are lettered A,
+B, C, D. The inner pair, B and C, are attached to each other by a bit of
+string, which has to typify the attraction of gravitation; the outer
+pair, A and D, are not attached to anything, but have a certain amount
+of play against friction in slots parallel to the length of the stick.
+The moon-disk is also slotted, so a small amount of motion is possible
+to it along the stick or bar. These things being so arranged, and the
+protuberant pieces of card being all pushed home, so that they are
+hidden behind their respective disks, the whole is spun rapidly round
+the centre of gravity, G. The result of a brief spin is to make A and D
+fly out by centrifugal force and show, as in the figure; while the moon,
+flying out too in its slot, tightens up the string, which causes B and C
+to be pulled out too. Thus all four high tides are produced, two on the
+earth and two on the moon, A and D being caused by centrifugal force, B
+and C by the attraction of gravitation. Each disk has become prolate in
+the same sort of fashion as yielding globes do. Of course the fluid
+ocean takes this shape more easily and more completely than the solid
+earth can, and so here are the very oceanic humps we have been talking
+about, and about three feet high (<a href="#Fig_112">Fig. 112</a>). If there were a sea on the
+<i>moon</i>, its humps would be a good deal bigger; but there probably is no
+sea<span class='pagenum'><a name="Page_366" id="Page_366">[Pg 366]</a></span> there, and if there were, the earth's tides are more interesting to
+us, at any rate to begin with.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_112" id="Fig_112"></a>
+<img src="images/fig112.jpg" width="400" height="205" alt="Fig. 112." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 112.</span>&mdash;Earth and moon (earth&#39;s rotation neglected).</span>
+</div>
+
+<p>The humps as so far treated are always protruding in the earth-moon
+line, and are stationary. But now we have to remember that the earth is
+spinning inside them. It is not easy to see what precise effect this
+spin will have upon the humps, even if the world were covered with a
+uniform ocean; but we can see at any rate that however much they may get
+displaced, and they do get displaced a good deal, they cannot possibly
+be carried round and round. The whole explanation we have given of their
+causes shows that they must maintain some steady aspect with respect to
+the moon&mdash;in other words, they must remain stationary as the earth spins
+round. Not that the same identical water remains stationary, for in that
+case it would have to be dragged over the earth's equator at the rate of
+1,000 miles an hour, but the hump or wave-crest remains stationary. It
+is a true wave, or form only, and consists of continuously changing
+individual particles. The same is true of all waves, except breaking
+ones.</p>
+
+<p>Given, then, these stationary humps and the earth spinning on its axis,
+we see that a given place on the earth will be carried round and round,
+now past a hump, and six<span class='pagenum'><a name="Page_367" id="Page_367">[Pg 367]</a></span> hours later past a depression: another six
+hours and it will be at the antipodal hump, and so on. Thus every six
+hours we shall travel from the region in space where the water is high
+to the region where it is low; and ignoring our own motion we shall say
+that the sea first rises and then falls; and so, with respect to the
+place, it does. Thus the succession of high and low water, and the two
+high tides every twenty-four hours, are easily understood in their
+easiest and most elementary aspect. A more complete account of the
+matter it will be wisest not to attempt: suffice it to say that the
+difficulties soon become formidable when the inertia of the water, its
+natural time of oscillation, the varying obliquity of the moon to the
+ecliptic, its varying distance, and the disturbing action of the sun are
+taken into consideration. When all these things are included, the
+problem becomes to ordinary minds overwhelming. A great many of these
+difficulties were successfully attacked by Laplace. Others remained for
+modern philosophers, among whom are Sir George Airy, Sir William
+Thomson, and Professor George Darwin.</p>
+
+<div class="blockquot"><p>I may just mention that the main and simplest effect of including
+the inertia or momentum of the water is to dislocate the obvious
+and simple connexion between high water and high moon; inertia
+always tends to make an effect differ in phase by a quarter period
+from the cause producing it, as may be illustrated by a swinging
+pendulum. Hence high water is not to be expected when the
+tide-raising force is a maximum, but six hours later; so that,
+considering inertia and neglecting friction, there would be low
+water under the moon. Including friction, something nearer the
+equilibrium state of things occurs. With <i>sufficient</i> friction the
+motion becomes dead-beat again, <i>i.e.</i> follows closely the force
+that causes it.</p></div>
+
+<p>Returning to the elementary discussion, we see that the rotation of the
+earth with respect to the humps will not be performed in exactly
+twenty-four hours, because the humps are travelling slowly after the
+moon, and will complete a revolution in a month in the same direction as
+the earth<span class='pagenum'><a name="Page_368" id="Page_368">[Pg 368]</a></span> is rotating. Hence a place on the earth has to catch them up,
+and so each high tide arrives later and later each day&mdash;roughly
+speaking, an hour later for each day tide; not by any means a constant
+interval, because of superposed disturbances not here mentioned, but on
+the average about fifty minutes.</p>
+
+<p>We see, then, that as a result of all this we get a pair of humps
+travelling all over the surface of the earth, about once a day. If the
+earth were all ocean (and in the southern hemisphere it is nearly all
+ocean), then they would go travelling across the earth, tidal waves
+three feet high, and constituting the mid-ocean tides. But in the
+northern hemisphere they can only thus journey a little way without
+striking land. As the moon rises at a place on the east shores of the
+Atlantic, for instance, the waters begin to flow in towards this place,
+or the tide begins to rise. This goes on till the moon is overhead and
+for some time afterwards, when the tide is at its highest. The hump then
+follows the moon in its apparent journey across to America, and there
+precipitates itself upon the coast, rushing up all the channels, and
+constituting the land tide. At the same time, the water is dragged away
+from the east shores, and so <i>our</i> tide is at its lowest. The same thing
+repeats itself in a little more than twelve hours again, when the other
+hump passes over the Atlantic, as the moon journeys beneath the earth,
+and so on every day.</p>
+
+<div class="blockquot"><p>In the free Southern Ocean, where land obstruction is comparatively
+absent, the water gets up a considerable swing by reason of its
+accumulated momentum, and this modifies and increases the open
+ocean tides there. Also for some reason, I suppose because of the
+natural time of swing of the water, one of the humps is there
+usually much larger than the other; and so places in the Indian and
+other offshoots of the Southern Ocean get their really high tide
+only once every twenty-four hours. These southern tides are in fact
+much more complicated than those the British Isles receive. Ours
+are singularly simple. No doubt some trace of the influence of the
+Southern Ocean is felt in the North Atlantic, but any ocean
+extending over<span class='pagenum'><a name="Page_369" id="Page_369">[Pg 369]</a></span> 90&deg; of longitude is big enough to have its own
+tides generated; and I imagine that the main tides we feel are thus
+produced on the spot, and that they are simple because the
+damping-out being vigorous, and accumulated effects small, we feel
+the tide-producing forces more directly. But for authoritative
+statements on tides, other books must be read. I have thought, and
+still think, it best in an elementary exposition to begin by a
+consideration of the tide-generating forces as if they acted on a
+non-rotating earth. It is the tide generating forces, and not the
+tides themselves, that are really represented in Figs. 112 and 114.
+The rotation of the earth then comes in as a disturbing cause. A
+more complete exposition would begin with the rotating earth, and
+would superpose the attraction of the moon as a disturbing cause,
+treating it as a problem in planetary perturbation, the ocean being
+a sort of satellite of the earth. This treatment, introducing
+inertia but ignoring friction and land obstruction, gives low water
+in the line of pull, and high water at right angles, or where the
+pull is zero; in the same sort of way as a pendulum bob is highest
+where most force is pulling it down, and lowest where no force is
+acting on it. For a clear treatment of the tides as due to the
+perturbing forces of sun and moon, see a little book by Mr. T.K.
+Abbott of Trinity College, Dublin. (Longman.)</p></div>
+
+<div class="figcenter" style="width: 500px;"><a name="Fig_113" id="Fig_113"></a>
+<img src="images/fig113.jpg" width="400" height="192" alt="Fig. 113." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 113.</span>&mdash;Maps showing how comparatively free from land
+obstruction the ocean in the Southern Hemisphere is.</span>
+</div>
+
+<p>If the moon were the only body that swung the earth round, this is all
+that need be said in an elementary treatment; but it is not the only
+one. The moon swings the earth round once a month, the sun swings it
+round once a year. The circle of swing is bigger, but the speed is so
+much slower that the protuberance produced is only one-third of that
+caused by the monthly whirl; <i>i.e.</i> the simple<span class='pagenum'><a name="Page_370" id="Page_370">[Pg 370]</a></span> solar tide in the open
+sea, without taking momentum into account, is but a little more than a
+foot high, while the simple lunar tide is about three feet. When the two
+agree, we get a spring tide of four feet; when they oppose each other,
+we get a neap tide of only two feet. They assist each other at full moon
+and at new moon. At half-moon they oppose each other. So we have spring
+tides regularly once a fortnight, with neap tides in between.</p>
+
+<div class="figcenter" style="width: 450px;"><a name="Fig_114" id="Fig_114"></a>
+<img src="images/fig114.jpg" width="450" height="582" alt="Fig. 114." title="" />
+<span class="caption"><span class="smcap">Fig. 114.</span>&mdash;Spring and neap tides.</span>
+</div>
+
+<p><a href="#Fig_114">Fig. 114</a> gives the customary diagrams to illustrate these simple things.
+You see that when the moon and sun act at<span class='pagenum'><a name="Page_371" id="Page_371">[Pg 371]</a></span> right angles (<i>i.e.</i> at every
+half-moon), the high tides of one coincide with the low tides of the
+other; and so, as a place is carried round by the earth's rotation, it
+always finds either solar or else lunar high water, and only experiences
+the difference of their two effects. Whereas, when the sun and moon act
+in the same line (as they do at new and full moon), their high and low
+tides coincide, and a place feels their effects added together. The tide
+then rises extra high and falls extra low.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="Fig_115" id="Fig_115"></a>
+<img src="images/fig115.jpg" width="450" height="400" alt="Fig. 115." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 115.</span>&mdash;Tidal clock. The position of the disk B shows
+the height of the tide. The tide represented is a nearly high tide eight
+feet above mean level.</span>
+</div>
+
+<p>Utilizing these principles, a very elementary form of tidal-clock, or
+tide-predicter, can be made, and for an open coast station it really
+would not give the tides so very badly. It consists of a sort of clock
+face with two hands, one nearly three times as long as the other. The
+short hand,<span class='pagenum'><a name="Page_372" id="Page_372">[Pg 372]</a></span> CA, should revolve round C once in twelve hours, and the
+vertical height of its end A represents the height of the solar tide on
+the scale of horizontal lines ruled across the face of the clock. The
+long hand, AB, should revolve round A once in twelve hours and
+twenty-five minutes, and the height of its end B (if A were fixed on the
+zero line) would represent the lunar tide. The two revolutions are made
+to occur together, either by means of a link-work parallelogram, or,
+what is better in practice, by a string and pulleys, as shown; and the
+height of the end point, B, of the third side or resultant, CB, read off
+on a scale of horizontal parallel lines behind, represents the
+combination or actual tide at the place. Every fortnight the two will
+agree, and you will get spring tides of maximum height CA + AB; every
+other fortnight the two will oppose, and you will get neap tides of
+maximum height CA-AB.</p>
+
+<p>Such a clock, if set properly and driven in the ordinary way, would then
+roughly indicate the state of the tide whenever you chose to look at it
+and read the height of its indicating point. It would not indeed be very
+accurate, especially for such an inclosed station as Liverpool is, and
+that is probably why they are not made. A great number of disturbances,
+some astronomical, some terrestrial, have to be taken into account in
+the complete theory. It is not an easy matter to do this, but it can be,
+and has been, done; and a tide-predicter has not only been constructed,
+but two of them are in regular work, predicting the tides for years
+hence&mdash;one, the property of the Indian Government, for coast stations of
+India; the other for various British and foreign stations, wherever the
+necessary preliminary observations have been made. These machines are
+the invention of Sir William Thomson. The tide-tables for Indian ports
+are now always made by means of them.</p>
+
+<p><span class='pagenum'><a name="Page_373" id="Page_373">[Pg 373]</a></span></p>
+<div class="figcenter" style="width: 400px;"><a name="Fig_116" id="Fig_116"></a>
+<img src="images/fig116.jpg" width="400" height="469" alt="Fig. 116." title="" />
+<span class="caption"><span class="smcap">Fig. 116.</span>&mdash;Sir William Thomson (Lord Kelvin).</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_374" id="Page_374">[Pg 374]</a></span></p>
+<div class="figcenter" style="width: 600px;"><a name="Fig_117" id="Fig_117"></a>
+<img src="images/fig117.jpg" width="450" height="610" alt="Fig. 117." title="" /><br />
+<span class="caption"><span class="smcap">Fig. 117.</span>&mdash;Tide-gauge for recording local tides, a
+pencil moved up and down by a float writes on a drum driven by
+clockwork.</span>
+</div>
+
+<p>The first thing to be done by any port which wishes its tides to be
+predicted is to set up a tide-gauge, or automatic recorder, and keep it
+working for a year or two. The tide-gauge is easy enough to understand:
+it marks the height of the tide at every instant by an irregular curved
+line like a barometer chart (<a href="#Fig_117">Fig. 117</a>). These observational curves so
+obtained have next to be fed into a fearfully complex machine, which it
+would take a whole lecture to make even partially intelligible, but <a href="#Fig_118">Fig.
+118</a> shows its aspect. It consists of ten integrating machines in a row,
+coupled up and working together. This is the "harmonic analyzer," and
+the result of passing the curve through this machine is to give you all
+the constituents of which it is built up, viz. the lunar tide, the solar
+tide, and eight of the sub-tides or disturbances. These ten values are
+then set off into a third machine, the tide-predicter proper. The
+general mode of action of this machine is not difficult to understand.
+It consists of a string wound over and under a set of pulleys, which are
+each set on an excentric, so as to have an up-and-down<span class='pagenum'><a name="Page_375" id="Page_375">[Pg 375]</a></span> motion. These
+up-and-down motions are all different, and there are ten of these
+movable pulleys, which by their respective excursions represent the
+lunar tide, the solar tide, and the eight disturbances already analyzed
+out of the tide-gauge curve by the harmonic analyzer. One end of the
+string is fixed, the other carries a pencil which writes a trace on a
+revolving drum of paper&mdash;a trace which represents the combined motion of
+all the pulleys, and so predicts the exact height of the tide at the
+place, at any future time you like. The machine can be turned quite
+quickly, so that a year's tides can be run off with every detail in
+about half-an-hour. This is the easiest part of the operation. Nothing
+has to be done but to keep it supplied with paper and pencil, and turn a
+handle as if it were a coffee-mill instead of a tide-mill. (Figs. 119
+and 120.)</p>
+
+<div class="figcenter" style="width: 450px;"><a name="Fig_118" id="Fig_118"></a>
+<img src="images/fig118.jpg" width="450" height="105" alt="Fig. 118." title="" />
+<span class="caption"><span class="smcap">Fig. 118.</span>&mdash;Harmonic analyzer; for analyzing out the
+constituents from a set of observational curves.</span>
+</div>
+
+<p>My subject is not half exhausted. I might go on to discuss the question
+of tidal energy&mdash;whether it can be ever utilized for industrial
+purposes; and also the very interesting question whence it comes. Tidal
+energy is almost the only terrestrial form of energy that does not
+directly or indirectly come from<span class='pagenum'><a name="Page_376" id="Page_376">[Pg 376]</a></span> the sun. The energy of tides is now
+known to be obtained at the expense of the earth's rotation; and
+accordingly our day must be slowly, very slowly, lengthening. The tides
+of past ages have destroyed the moon's rotation, and so it always turns
+the same face to us. There is every reason to believe that in geologic
+ages the moon was nearer to us than it is now, and that accordingly our
+tides were then far more violent, rising some hundreds of feet instead
+of twenty or thirty, and sweeping every six hours right over the face of
+a country, ploughing down hills, denuding rocks, and producing a copious
+sedimentary deposit.</p>
+
+<div class="figcenter" style="width: 450px;"><a name="Fig_119" id="Fig_119"></a>
+<img src="images/fig119.jpg" width="450" height="352" alt="Fig. 119." title="" />
+<span class="caption"><span class="smcap">Fig. 119.</span>&mdash;Tide-predicter, for combining the ascertained
+constituents into a tidal curve for the future.</span>
+</div>
+
+<p>In thus discovering the probable violent tides of past<span class='pagenum'><a name="Page_377" id="Page_377">[Pg 377]</a></span> ages, astronomy
+has, within the last few years, presented geology with the most powerful
+denuding agent known; and the study of the earth's past history cannot
+fail to be greatly affected by the modern study of the intricate and
+refined conditions attending prolonged tidal action on incompletely
+rigid bodies. [Read on this point the last chapter of Sir R. Ball's
+<i><a href="http://www.gutenberg.org/files/27378/27378-h/27378-h.htm#Page_531">Story of the Heavens</a></i>.]</p>
+
+<div class="figcenter" style="width: 600px;"><a name="Fig_120" id="Fig_120"></a>
+<img src="images/fig120.jpg" width="600" height="217" alt="Fig. 120." title="" />
+<span class="caption"><span class="smcap">Fig. 120.</span>&mdash;Weekly sheet of curves. Tides for successive
+days are predicted on the same sheet of paper, to economise space.</span>
+</div>
+
+<p>I might also point out that the magnitude of our terrestrial tides
+enables us to answer the question as to the internal fluidity of the
+earth. It used to be thought that the earth's crust was comparatively
+thin, and that it contained a molten interior. We now know that this is
+not the case. The interior of the earth is hot indeed, but it is not
+fluid. Or at least, if it be fluid, the amount of fluid is but very
+small compared with the thickness of the unyielding crust. All these,
+and a number of other most interesting questions, fringe the<span class='pagenum'><a name="Page_378" id="Page_378">[Pg 378]</a></span> subject of
+the tides; the theoretical study of which, started by Newton, has
+developed, and is destined in the future to further develop, into one of
+the most gigantic and absorbing investigations&mdash;having to do with the
+stability or instability of solar systems, and with the construction and
+decay of universes.</p>
+
+<p>These theories are the work of pioneers now living, whose biographies it
+is therefore unsuitable for us to discuss, nor shall I constantly
+mention their names. But Helmholtz, and Thomson, are household words,
+and you well know that in them and their disciples the race of Pioneers
+maintains its ancient glory.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_379" id="Page_379">[Pg 379]</a></span></p>
+<h4><a name="NOTES_FOR_LECTURE_XVIII" id="NOTES_FOR_LECTURE_XVIII"></a>NOTES FOR LECTURE XVIII</h4>
+
+
+<p>Tides are due to incomplete rigidity of bodies revolving round each
+other under the action of gravitation, and at the same time spinning on
+their axes.</p>
+
+<p>Two spheres revolving round each other can only remain spherical if
+rigid; if at all plastic they become prolate. If either rotate on its
+axis, in the same or nearly the same plane as it revolves, that one is
+necessarily subject to tides.</p>
+
+<p>The axial rotation tends to carry the humps with it, but the pull of the
+other body keeps them from moving much. Hence the rotation takes place
+against a pull, and is therefore more or less checked and retarded. This
+is the theory of Von Helmholtz.</p>
+
+<p>The attracting force between two such bodies is no longer <i>exactly</i>
+towards the centre of revolution, and therefore Kepler's second law is
+no longer precisely obeyed: the rate of description of areas is subject
+to slight acceleration. The effect of this tangential force acting on
+the tide-compelling body is gradually to increase its distance from the
+other body.</p>
+
+<p>Applying these statements to the earth and moon, we see that tidal
+energy is produced at the expense of the earth's rotation, and that the
+length of the day is thereby slowly increasing. Also that the moon's
+rotation relative to the earth has been destroyed by past tidal action
+in it (the only residue of ancient lunar rotation now being a scarcely
+perceptible libration), so that it turns always the same face towards
+us. Moreover, that its distance from the earth is steadily increasing.
+This last is the theory of Professor G.H. Darwin.</p>
+
+<p>Long ago the moon must therefore have been much nearer the earth, and
+the day was much shorter. The tides were then far more violent.</p>
+
+<p>Halving the distance would make them eight times as high; quartering it
+would increase them sixty-four-fold. A most powerful geological denuding
+agent. Trade winds and storms were also more violent.</p>
+
+<p>If ever the moon were close to the earth, it would have to revolve round
+it in about three hours. If the earth rotated on its axis in three
+hours, when fluid or pasty, it would be unstable, and begin to separate
+a portion of itself as a kind of bud, which might then get detached and
+gradually<span class='pagenum'><a name="Page_380" id="Page_380">[Pg 380]</a></span> pushed away by the violent tidal action. Hence it is possible
+that this is the history of the moon. If so, it is probably an
+exceptional history. The planets were not formed from the sun in this
+way.</p>
+
+<p>Mars' moons revolve round him more quickly than the planet rotates:
+hence with them the process is inverted, and they must be approaching
+him and may some day crash along his surface. The inner moon is now
+about 4,000 miles away, and revolves in 7&frac12;. It appears to be
+about 20 miles in diameter, and weighs therefore, if composed of rock,
+40 billion tons. Mars rotates in 24&frac12; hours.</p>
+
+<p>A similar fate may <i>possibly</i> await our moon ages hence&mdash;by reason of
+the action of terrestrial tides produced by the sun.</p>
+
+
+
+<hr /><p><span class='pagenum'><a name="Page_381" id="Page_381">[Pg 381]</a></span></p>
+<h3><a name="LECTURE_XVIII" id="LECTURE_XVIII"></a>LECTURE XVIII</h3>
+
+<h5>THE TIDES, AND PLANETARY EVOLUTION</h5>
+
+
+<p><span class="smcap">In</span> the last lecture we considered the local peculiarities of the tides,
+the way in which they were formed in open ocean under the action of the
+moon and the sun, and also the means by which their heights and times
+could be calculated and predicted years beforehand. Towards the end I
+stated that the subject was very far from being exhausted, and
+enumerated some of the large and interesting questions which had been
+left untouched. It is with some of these questions that I propose now to
+deal.</p>
+
+<p>I must begin by reminding you of certain well-known facts, a knowledge
+of which I may safely assume.</p>
+
+<p>And first we must remind ourselves of the fact that almost all the rocks
+which form the accessible crust of the earth were deposited by the
+agency of water. Nearly all are arranged in regular strata, and are
+composed of pulverized materials&mdash;materials ground down from
+pre-existing rocks by some denuding and grinding action. They nearly all
+contain vestiges of ancient life embedded in them, and these vestiges
+are mainly of marine origin. The strata which were once horizontal are
+now so no longer&mdash;they have been tilted and upheaved, bent and
+distorted, in many places. Some of them again have been metamorphosed by
+fire, so that their organic remains have been destroyed, and the traces
+of their aqueous origin almost obliterated. But still, to the eye of the
+geologist, all are of aqueous or sedimentary<span class='pagenum'><a name="Page_382" id="Page_382">[Pg 382]</a></span> origin: roughly speaking,
+one may say they were all deposited at the bottom of some ancient sea.</p>
+
+<p>The date of their formation no man yet can tell, but that it was vastly
+distant is certain. For the geological era is not over. Aqueous action
+still goes on: still does frost chip the rocks into fragments; still do
+mountain torrents sweep stone and mud and <i>d&eacute;bris</i> down the gulleys and
+watercourses; still do rivers erode their channels, and carry mud and
+silt far out to sea. And, more powerful than any of these agents of
+denudation, the waves and the tides are still at work along every
+coast-line, eating away into the cliffs, undermining gradually and
+submerging acre after acre, and making with the refuse a shingly, or a
+sandy, or a muddy beach&mdash;the nucleus of a new geological formation.</p>
+
+<p>Of all denuding agents, there can be no doubt that, to the land exposed
+to them, the waves of the sea are by far the most powerful. Think how
+they beat and tear, and drive and drag, until even the hardest rock,
+like basalt, becomes honeycombed into strange galleries and
+passages&mdash;Fingal's Cave, for instance&mdash;and the softer parts are crumbled
+away. But the area now exposed to the teeth of the waves is not great.
+The fury of a winter storm may dash them a little higher than usual, but
+they cannot reach cliffs 100 feet high. They can undermine such cliffs
+indeed, and then grind the fragments to powder, but their direct action
+is limited. Not so limited, however, as they would be without the tides.
+Consider for a moment the denudation import of the tides: how does the
+existence of tidal rise and fall affect the geological problem?</p>
+
+<p>The scouring action of the tidal currents themselves is not to be
+despised. It is the tidal ebb and flow which keeps open channel in the
+Mersey, for instance. But few places are so favourably situated as
+Liverpool in this respect, and the direct scouring action of the tides
+in general is not very great. Their geological import mainly consists in
+this&mdash;that they raise and lower the surface waves at regular intervals,<span class='pagenum'><a name="Page_383" id="Page_383">[Pg 383]</a></span>
+so as to apply them to a considerable stretch of coast. The waves are a
+great planing machine attacking the land, and the tides raise and lower
+this planing machine, so that its denuding tooth is applied, now twenty
+feet vertically above mean level, now twenty feet below.</p>
+
+<p>Making all allowance for the power of winds and waves, currents, tides,
+and watercourses, assisted by glacial ice and frost, it must be apparent
+how slowly the work of forming the rocks is being carried on. It goes on
+steadily, but so slowly that it is estimated to take 6000 years to wear
+away one foot of the American continent by all the denuding causes
+combined. To erode a stratum 5000 feet thick will require at this rate
+thirty million years.</p>
+
+<p>The age of the earth is not at all accurately known, but there are many
+grounds for believing it not to be much older than some thirty million
+years. That is to say, not greatly more than this period of time has
+elapsed since it was in a molten condition. It may be as old as a
+hundred million years, but its age is believed by those most competent
+to judge to be more likely within this limit than beyond it. But if we
+ask what is the thickness of the rocks which in past times have been
+formed, and denuded, and re-formed, over and over again, we get an
+answer, not in feet, but in miles. The Laurentian and Huronian rocks of
+Canada constitute a stratum ten miles thick; and everywhere the rocks at
+the base of our stratified system are of the most stupendous volume and
+thickness.</p>
+
+<p>It has always been a puzzle how known agents could have formed these
+mighty masses, and the only solution offered by geologists was,
+unlimited time. Given unlimited time, they could, of course, be formed,
+no matter how slowly the process went on. But inasmuch as the time
+allowable since the earth was cool enough for water to exist on it
+except as steam is not by any means unlimited, it becomes necessary to
+look for a far more powerful engine than any now existing; there must
+have been some denuding agent<span class='pagenum'><a name="Page_384" id="Page_384">[Pg 384]</a></span> in those remote ages&mdash;ages far more
+distant from us than the Carboniferous period, far older than any forms
+of life, fossil or otherwise, ages among the oldest known to geology&mdash;a
+denuding agent must have then existed, far more powerful than any we now
+know.</p>
+
+<p>Such an agent it has been the privilege of astronomy and physics, within
+the last ten years, to discover. To this discovery I now proceed to lead
+up.</p>
+
+<p>Our fundamental standard of time is the period of the earth's
+rotation&mdash;the length of the day. The earth is our one standard clock:
+all time is expressed in terms of it, and if it began to go wrong, or if
+it did not go with perfect uniformity, it would seem a most difficult
+thing to discover its error, and a most puzzling piece of knowledge to
+utilize when found.</p>
+
+<p>That it does not go much wrong is proved by the fact that we can
+calculate back to past astronomical events&mdash;ancient eclipses and the
+like&mdash;and we find that the record of their occurrence, as made by the
+old magi of Chald&aelig;a, is in very close accordance with the result of
+calculation. One of these famous old eclipses was observed in Babylon
+about thirty-six centuries ago, and the Chald&aelig;an astronomers have put on
+record the time of its occurrence. Modern astronomers have calculated
+back when it should have occurred, and the observed time agrees very
+closely with the actual, but not exactly. Why not exactly?</p>
+
+<p>Partly because of the acceleration of the moon's mean motion, as
+explained in the lecture on Laplace (<a href="#Page_262">p. 262</a>). The orbit of the earth was
+at that time getting rounder, and so, as a secondary result, the speed
+of the moon was slightly increasing. It is of the nature of a
+perturbation, and is therefore a periodic not a progressive or
+continuous change, and in a sufficiently long time it will be reversed.
+Still, for the last few thousand years the moon's motion has been, on
+the whole, accelerated (though there seems to be a very slight retarding
+force in action too).</p>
+
+<p><span class='pagenum'><a name="Page_385" id="Page_385">[Pg 385]</a></span></p><p>Laplace thought that this fact accounted for the whole of the
+discrepancy; but recently, in 1853, Professor Adams re-examined the
+matter, and made a correction in the details of the theory which
+diminishes its effect by about one-half, leaving the other half to be
+accounted for in some other way. His calculations have been confirmed by
+Professor Cayley. This residual discrepancy, when every known cause has
+been allowed for, amounts to about one hour.</p>
+
+<div class="blockquot"><p>The eclipse occurred later than calculation warrants. Now this
+would have happened from either of two causes, either an
+acceleration of the moon in her orbit, or a retardation of the
+earth in her diurnal rotation&mdash;a shortening of the month or a
+lengthening of the day, or both. The total discrepancy being, say,
+two hours, an acceleration of six seconds-per-century per century
+will in thirty-six centuries amount to one hour; and this,
+according to the corrected Laplacian theory, is what has occurred.
+But to account for the other hour some other cause must be sought,
+and at present it is considered most probably due to a steady
+retardation of the earth's rotation&mdash;a slow, very slow, lengthening
+of the day.</p>
+
+<p>The statement that a solar eclipse thirty-six centuries ago was an
+hour late, means that a place on the earth's surface came into the
+shadow one hour behind time&mdash;that is, had lagged one twenty-fourth
+part of a revolution. The earth, therefore, had lost this amount in
+the course of 3600 &times; 365&frac14; revolutions. The loss per revolution
+is exceedingly small, but it accumulates, and at any era the total
+loss is the sum of all the losses preceding it. It may be worth
+while just to explain this point further.</p>
+
+<p>Suppose the earth loses a small piece of time, which I will call an
+instant, per day; a locality on the earth will come up to a given
+position one instant late on the first day after an event. On the
+next day it would come up two instants late by reason of the
+previous loss; but it also loses another instant during the course
+of the second day, and so the total lateness by the end of that day
+amounts to three instants. The day after, it will be going slower
+from the beginning at the rate of two instants a day, it will lose
+another instant on the fresh day's own account, and it started
+three instants late; hence the aggregate loss by the end of the
+third day is 1 + 2 + 3 = 6. By the end of the fourth day the whole
+loss will be 1 + 2 + 3 + 4, and so on. Wherefore by merely losing
+one instant every day the total loss in <i>n</i> days is (1 + 2 + 3 +
+... + <i>n</i>)<span class='pagenum'><a name="Page_386" id="Page_386">[Pg 386]</a></span> instants, which amounts to &frac12;<i>n</i> (<i>n</i> + 1) instants;
+or practically, when <i>n</i> is big, to &frac12;<i>n</i><sup>2</sup>. Now in thirty-six
+centuries there have been 3600 &times; 365&frac14; days, and the total loss
+has amounted to an hour; hence the length of "an instant," the loss
+per diem, can be found from the equation &frac12;(3600 &times; 365)<sup>2</sup> instants
+= 1 hour; whence one "instant" equals the 240 millionth part of a
+second. This minute quantity represents the retardation of the
+earth per day. In a year the aggregate loss mounts up to <span class="above">1</span>&#8260;<span class="below">3600</span>th
+part of a second, in a century to about three seconds, and in
+thirty-six centuries to an hour. But even at the end of the
+thirty-six centuries the day is barely any longer; it is only 3600
+&times; 365 instants, that is <span class="above">1</span>&#8260;<span class="below">180</span>th of a second, longer than it was at
+the beginning. And even a million years ago, unless the rate of
+loss was different (as it probably was), the day would only be
+thirty-five minutes shorter, though by that time the aggregate
+loss, as measured by the apparent lateness of any perfectly
+punctual event reckoned now, would have amounted to nine years.
+(These numbers are to be taken as illustrative, not as precisely
+representing terrestrial fact.)</p></div>
+
+<p>What can have caused the slowing down? Swelling of the earth by reason
+of accumulation of meteoric dust might do something, but probably very
+little. Contraction of the earth as it goes on cooling would act in the
+opposite direction, and probably more than counterbalance the dust
+effect. The problem is thus not a simple one, for there are several
+disturbing causes, and for none of them are the data enough to base a
+quantitative estimate upon; but one certain agent in lengthening the
+day, and almost certainly the main agent, is to be found in the tides.</p>
+
+<p>Remember that the tidal humps were produced as the prolateness of a
+sphere whirled round and round a fixed centre, like a football whirled
+by a string. These humps are pulled at by the moon, and the earth
+rotates on its axis against this pull. Hence it tends to be constantly,
+though very slightly, dragged back.</p>
+
+<p>In so far as the tidal wave is allowed to oscillate freely, it will
+swing with barely any maintaining force, giving back at one
+quarter-swing what it has received at the previous quarter; but in so
+far as it encounters friction, which it<span class='pagenum'><a name="Page_387" id="Page_387">[Pg 387]</a></span> does in all channels where
+there is an actual ebb and flow of the water, it has to receive more
+than it gives back, and the balance of energy has to be made up to it,
+or the tides would cease. The energy of the tides is, in fact,
+continually being dissipated by friction, and all the energy so
+dissipated is taken from the rotation of the earth. If tidal energy were
+utilized by engineers, the machines driven would be really driven at the
+expense of the earth's rotation: it would be a mode of harnessing the
+earth and using the moon as fixed point or fulcrum; the moon pulling at
+the tidal protuberance, and holding it still as the earth rotates, is
+the mechanism whereby the energy is extracted, the handle whereby the
+friction brake is applied.</p>
+
+<div class="blockquot"><p>Winds and ocean currents have no such effect (as Mr. Fronde in
+<i>Oceania</i> supposes they have), because they are all accompanied by
+a precisely equal counter-current somewhere else, and no internal
+rearrangement of fluid can affect the motion of a mass as a whole;
+but the tides are in different case, being produced, not by
+internal inequalities of temperature, but by a straightforward pull
+from an external body. </p></div>
+
+<p>The ultimate effect of tidal friction and dissipation of energy will,
+therefore, be to gradually retard the earth till it does not rotate with
+reference to the moon, <i>i.e.</i> till it rotates once while the moon
+revolves once; in other words, to make the day and the month equal. The
+same cause must have been in operation, but with eighty-fold greater
+intensity, on the moon. It has ceased now, because the rotation has
+stopped, but if ever the moon rotated on its axis with respect to the
+earth, and if it were either fluid itself or possessed any liquid ocean,
+then the tides caused by the pull of the earth must have been
+prodigious, and would tend to stop its rotation. Have they not
+succeeded? Is it not probable that this is <i>why</i> the moon always now
+turns the same face towards us? It is believed to be almost certainly
+the cause. If so, there was a time when the moon behaved
+differently&mdash;when it rotated more quickly<span class='pagenum'><a name="Page_388" id="Page_388">[Pg 388]</a></span> than it revolved, and
+exhibited to us its whole surface. And at this era, too, the earth
+itself must have rotated a little faster, for it has been losing speed
+ever since.</p>
+
+<p>We have thus arrived at this fact, that a thousand years ago the day was
+a trifle shorter than it is now. A million years ago it was, perhaps, an
+hour shorter. Twenty million years ago it must have been much shorter.
+Fifty million years ago it may have been only a few hours long. The
+earth may have spun round then quite quickly. But there is a limit. If
+it spun too fast it would fly to pieces. Attach shot by means of wax to
+the whirling earth model, <a href="#Fig_110">Fig. 110</a>, and at a certain speed the cohesion
+of the wax cannot hold them, so they fly off. The earth is held together
+not by cohesion but by gravitation; it is not difficult to reckon how
+fast the earth must spin for gravity at its surface to be annulled, and
+for portions to fly off. We find it about one revolution in three hours.
+This is a critical speed. If ever the day was three hours long,
+something must have happened. The day can never have been shorter than
+that; for if it were, the earth would have a tendency to fly in pieces,
+or, at least, to separate into two pieces. Remember this, as a natural
+result of a three-hour day, which corresponds to an unstable state of
+things; remember also that in some past epoch a three-hour day is a
+probability.</p>
+
+<div class="blockquot"><p>If we think of the state of things going on in the earth's
+atmosphere, if it had an atmosphere at that remote date, we shall
+recognize the existence of the most fearful tornadoes. The trade
+winds, which are now peaceful agents of commerce, would then be
+perpetual hurricanes, and all the denudation agents of the
+geologist would be in a state of feverish activity. So, too, would
+the tides: instead of waiting six hours between low and high tide,
+we should have to wait only three-quarters of an hour. Every
+hour-and-a-half the water would execute a complete swing from high
+tide to high again. </p></div>
+
+<p>Very well, now leave the earth, and think what has been happening to the
+moon all this while.</p>
+
+<p><span class='pagenum'><a name="Page_389" id="Page_389">[Pg 389]</a></span></p><p>We have seen that the moon pulls the tidal hump nearest to it back; but
+action and reaction are always equal and opposite&mdash;it cannot do that
+without itself getting pulled forward. The pull of the earth on the moon
+will therefore not be quite central, but will be a little in advance of
+its centre; hence, by Kepler's second law, the rate of description of
+areas by its radius vector cannot be constant, but must increase (<a href="#Page_208">p.
+208</a>). And the way it increases will be for the radius vector to
+lengthen, so as to sweep out a bigger area. Or, to put it another way,
+the extra speed tending to be gained by the moon will fling it further
+away by extra centrifugal force. This last is not so good a way of
+regarding the matter; though it serves well enough for the case of a
+ball whirled at the end of an elastic string. After having got up the
+whirl, the hand holding the string may remain almost fixed at the centre
+of the circle, and the motion will continue steadily; but if the hand be
+moved so as always to pull the string a little in advance of the centre,
+the speed of whirl will increase, the elastic will be more and more
+stretched, until the whirling ball is describing a much larger circle.
+But in this case it will likewise be going faster&mdash;distance and speed
+increase together. This is because it obeys a different law from
+gravitation&mdash;the force is not inversely as the square, or any other
+single power, of the distance. It does not obey any of Kepler's laws,
+and so it does not obey the one which now concerns us, viz. the third;
+which practically states that the further a planet is from the centre
+the slower it goes; its velocity varies inversely with the square root
+of its distance (<a href="#Page_74">p. 74</a>).</p>
+
+<p>If, instead of a ball held by elastic, it were a satellite held by
+gravity, an increase in distance must be accompanied by a diminution in
+speed. The time of revolution varies as the square of the cube root of
+the distance (Kepler's third law). Hence, the tidal reaction on the
+moon, having as its primary effect, as we have seen, the pulling the
+moon a little forward, has also the secondary or indirect effect of
+making<span class='pagenum'><a name="Page_390" id="Page_390">[Pg 390]</a></span> it move slower and go further off. It may seem strange that an
+accelerating pull, directed in front of the centre, and therefore always
+pulling the moon the way it is going, should retard it; and that a
+retarding force like friction, if such a force acted, should hasten it,
+and make it complete its orbit sooner; but so it precisely is.</p>
+
+<p>Gradually, but very slowly, the moon is receding from us, and the month
+is becoming longer. The tides of the earth are pushing it away. This is
+not a periodic disturbance, like the temporary acceleration of its
+motion discovered by Laplace, which in a few centuries, more or less,
+will be reversed; it is a disturbance which always acts one way, and
+which is therefore cumulative. It is superposed upon all periodic
+changes, and, though it seems smaller than they, it is more inexorable.
+In a thousand years it makes scarcely an appreciable change, but in a
+million years its persistence tells very distinctly; and so, in the long
+run, the month is getting longer and the moon further off. Working
+backwards also, we see that in past ages the moon must have been nearer
+to us than it is now, and the month shorter.</p>
+
+<p>Now just note what the effect of the increased nearness of the moon was
+upon our tides. Remember that the tide-generating force varies inversely
+as the cube of distance, wherefore a small change of distance will
+produce a great difference in the tide-force.</p>
+
+<p>The moon's present distance is 240 thousand miles. At a time when it was
+only 190 thousand miles, the earth's tides would have been twice as high
+as they are now. The pushing away action was then a good deal more
+violent, and so the process went on quicker. The moon must at some time
+have been just half its present distance, and the tides would then have
+risen, not 20 or 30 feet, but 160 or 200 feet. A little further back
+still, we have the moon at one-third of its present distance from the
+earth, and the tides 600 feet high. Now just contemplate the effect of a
+600-foot tide. We are here only about 150 feet above the level<span class='pagenum'><a name="Page_391" id="Page_391">[Pg 391]</a></span> of the
+sea; hence, the tide would sweep right over us and rush far away inland.
+At high tide we should have some 200 feet of blue water over our heads.
+There would be nothing to stop such a tide as that in this neighbourhood
+till it reached the high lands of Derbyshire. Manchester would be a
+seaport then with a vengeance!</p>
+
+<p>The day was shorter then, and so the interval between tide and tide was
+more like ten than twelve hours. Accordingly, in about five hours, all
+that mass of water would have swept back again, and great tracts of sand
+between here and Ireland would be left dry. Another five hours, and the
+water would come tearing and driving over the country, applying its
+furious waves and currents to the work of denudation, which would
+proceed apace. These high tides of enormously distant past ages
+constitute the denuding agent which the geologist required. They are
+very ancient&mdash;more ancient than the Carboniferous period, for instance,
+for no trees could stand the furious storms that must have been
+prevalent at this time. It is doubtful whether any but the very lowest
+forms of life then existed. It is the strata at the bottom of the
+geological scale that are of the most portentous thickness, and the only
+organism suspected in them is the doubtful <i>Eozoon Canadense</i>. Sir
+Robert Ball believes, and several geologists agree with him, that the
+mighty tides we are contemplating may have been co&aelig;val with this ancient
+Laurentian formation, and others of like nature with it.</p>
+
+<p>But let us leave geology now, and trace the inverted progress of events
+as we recede in imagination back through the geological era, beyond,
+into the dim vista of the past, when the moon was still closer and
+closer to the earth, and was revolving round it quicker and quicker,
+before life or water existed on it, and when the rocks were still
+molten.</p>
+
+<p>Suppose the moon once touched the earth's surface, it is easy to
+calculate, according to the principles of gravitation,<span class='pagenum'><a name="Page_392" id="Page_392">[Pg 392]</a></span> and with a
+reasonable estimate of its size as then expanded by heat, how fast it
+must then have revolved round the earth, so as just to save itself from
+falling in. It must have gone round once every three hours. The month
+was only three hours long at this initial epoch.</p>
+
+<p>Remember, however, the initial length of the day. We found that it was
+just possible for the earth to rotate on its axis in three hours, and
+that when it did so, something was liable to separate from it. Here we
+find the moon in contact with it, and going round it in this same
+three-hour period. Surely the two are connected. Surely the moon was a
+part of the earth, and was separating from it.</p>
+
+<p>That is the great discovery&mdash;the origin of the moon.</p>
+
+<p>Once, long ages back, at date unknown, but believed to be certainly as
+much as fifty million years ago, and quite possibly one hundred million,
+there was no moon, only the earth as a molten globe, rapidly spinning on
+its axis&mdash;spinning in about three hours. Gradually, by reason of some
+disturbing causes, a protuberance, a sort of bud, forms at one side, and
+the great inchoate mass separates into two&mdash;one about eighty times as
+big as the other. The bigger one we now call earth, the smaller we now
+call moon. Round and round the two bodies went, pulling each other into
+tremendously elongated or prolate shapes, and so they might have gone on
+for a long time. But they are unstable, and cannot go on thus: they must
+either separate or collapse. Some disturbing cause acts again, and the
+smaller mass begins to revolve less rapidly. Tides at once
+begin&mdash;gigantic tides of molten lava hundreds of miles high; tides not
+in free ocean, for there was none then, but in the pasty mass of the
+entire earth. Immediately the series of changes I have described begins,
+the speed of rotation gets slackened, the moon's mass gets pushed
+further and further away, and its time of revolution grows rapidly
+longer. The changes went on rapidly at first, because the tides were so
+gigantic; but gradually, and by slow degrees, the bodies<span class='pagenum'><a name="Page_393" id="Page_393">[Pg 393]</a></span> get more
+distant, and the rate of change more moderate. Until, after the lapse of
+ages, we find the day twenty-four hours long, the moon 240,000 miles
+distant, revolving in 27&#8531; days, and the tides only existing in the
+water of the ocean, and only a few feet high. This is the era we call
+"to-day."</p>
+
+<p>The process does not stop here: still the stately march of events goes
+on; and the eye of Science strives to penetrate into the events of the
+future with the same clearness as it has been able to descry the events
+of the past. And what does it see? It will take too long to go into full
+detail: but I will shortly summarize the results. It sees this
+first&mdash;the day and the month both again equal, but both now about 1,400
+hours long. Neither of these bodies rotating with respect to each
+other&mdash;the two as if joined by a bar&mdash;and total cessation of
+tide-generating action between them.</p>
+
+<p>The date of this period is one hundred and fifty millions of years
+hence, but unless some unforeseen catastrophe intervenes, it must
+assuredly come. Yet neither will even this be the final stage; for the
+system is disturbed by the tide-generating force of the sun. It is a
+small effect, but it is cumulative; and gradually, by much slower
+degrees than anything we have yet contemplated, we are presented with a
+picture of the month getting gradually shorter than the day, the moon
+gradually approaching instead of receding, and so, incalculable myriads
+of ages hence, precipitating itself upon the surface of the earth whence
+it arose.</p>
+
+<p>Such a catastrophe is already imminent in a neighbouring planet&mdash;Mars.
+Mars' principal moon circulates round him at an absurd pace, completing
+a revolution in 7&frac12; hours, and it is now only 4,000 miles from his
+surface. The planet rotates in twenty-four hours as we do; but its tides
+are following its moon more quickly than it rotates after them; they are
+therefore tending to increase its rate<span class='pagenum'><a name="Page_394" id="Page_394">[Pg 394]</a></span> of spin, and to retard the
+revolution of the moon. Mars is therefore slowly but surely pulling its
+moon down on to itself, by a reverse action to that which separated our
+moon. The day shorter than the month forces a moon further away; the
+month shorter than the day tends to draw a satellite nearer.</p>
+
+<p>This moon of Mars is not a large body: it is only twenty or thirty miles
+in diameter, but it weighs some forty billion tons, and will ultimately
+crash along the surface with a velocity of 8,000 miles an hour. Such a
+blow must produce the most astounding effects when it occurs, but I am
+unable to tell you its probable date.</p>
+
+<p>So far we have dealt mainly with the earth and its moon; but is the
+existence of tides limited to these bodies? By no means. No body in the
+solar system is rigid, no body in the stellar universe is rigid. All
+must be susceptible of some tidal deformation, and hence, in all of
+them, agents like those we have traced in the history of the earth and
+moon must be at work: the motion of all must be complicated by the
+phenomena of tides. It is Prof. George Darwin who has worked out the
+astronomical influence of the tides, on the principles of Sir William
+Thomson: it is Sir Robert Ball who has extended Mr. Darwin's results to
+the past history of our own and other worlds.<a name="FNanchor_32_32" id="FNanchor_32_32"></a><a href="#Footnote_32_32" class="fnanchor">[32]</a></p>
+
+<div class="blockquot"><p>Tides are of course produced in the sun by the action of the
+planets, for the sun rotates in twenty-five days or thereabouts,
+while the planets revolve in much longer periods than that. The
+principal tide-generating bodies will be Venus and Jupiter; the
+greater nearness of one rather more than compensating for the
+greater mass of the other.</p>
+
+<p>It may be interesting to tabulate the relative tide-producing
+powers of the planets on the sun. They are as follows, calling that
+of the earth 1,000:&mdash;</p>
+
+<p>
+<span class='pagenum'><a name="Page_395" id="Page_395">[Pg 395]</a></span></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="Relative Tide-Producing Powers of the Planets">
+<tr>
+ <td align='center' colspan='2'><span class="smcap">Relative Tide-producing Powers of the Planets<br />on the Sun.</span></td></tr>
+<tr>
+ <td align='left'>Mercury</td>
+ <td align='right'>1,121</td>
+</tr>
+<tr>
+ <td align='left'>Venus</td>
+ <td align='right'>2,339</td>
+</tr>
+<tr>
+ <td align='left'>Earth</td>
+ <td align='right'>1,000</td>
+</tr>
+<tr>
+ <td align='left'>Mars</td>
+ <td align='right'>304</td>
+</tr>
+<tr>
+ <td align='left'>Jupiter</td>
+ <td align='right'>2,136</td>
+</tr>
+<tr>
+ <td align='left'>Saturn</td>
+ <td align='right'>1,033</td>
+</tr>
+<tr>
+ <td align='left'>Uranus</td>
+ <td align='right'>21</td>
+</tr>
+<tr>
+ <td align='left'>Neptune</td>
+ <td align='right'>9</td>
+</tr>
+</table></div>
+
+<p>The power of all of them is very feeble, and by acting on different
+sides they usually partly neutralize each other's action; but
+occasionally they get all on one side, and in that case some
+perceptible effect may be produced; the probable effect seems
+likely to be a gentle heaving tide in the solar surface, with
+breaking up of any incipient crust; and such an effect may be
+considered as evidenced periodically by the great increase in the
+number of solar spots which then break out.</p>
+
+<p>The solar tides are, however, much too small to appreciably push
+any planet away, hence we are not to suppose that the planets
+originated by budding from the sun, in contradiction of the nebular
+hypothesis. Nor is it necessary to assume that the satellites, as a
+class, originated in the way ours did; though they may have done
+so. They were more probably secondary rings. Our moon differs from
+other satellites in being exceptionally large compared with the
+size of its primary; it is as big as some of the moons of Jupiter
+and Saturn. The earth is the only one of the small planets that has
+an appreciable moon, and hence there is nothing forced or unnatural
+in supposing that it may have had an exceptional history.</p>
+
+<p>Evidently, however, tidal phenomena must be taken into
+consideration in any treatment of the solar system through enormous
+length of time, and it will probably play a large part in
+determining its future. </p></div>
+
+<p>When Laplace and Lagrange investigated the question of the stability or
+instability of the solar system, they did so on the hypothesis that the
+bodies composing it were rigid. They reached a grand conclusion&mdash;that
+all the mutual perturbations of the solar system were periodic&mdash;that
+whatever changes were going on would reach a maximum and then<span class='pagenum'><a name="Page_396" id="Page_396">[Pg 396]</a></span> begin to
+diminish; then increase again, then diminish, and so on. The system was
+stable, and its changes were merely like those of a swinging pendulum.</p>
+
+<p>But this conclusion is not final. The hypothesis that the bodies are
+rigid is not strictly true: and directly tidal deformation is taken into
+consideration it is perceived to be a potent factor, able in the long
+run to upset all their calculations. But it is so utterly and
+inconceivably minute&mdash;it only produces an appreciable effect after
+millions of years&mdash;whereas the ordinary perturbations go through their
+swings in some hundred thousand years or so at the most. Granted it is
+small, but it is terribly persistent; and it always acts in one
+direction. Never does it cease: never does it begin to act oppositely
+and undo what it has done. It is like the perpetual dropping of water.
+There may be only one drop in a twelvemonth, but leave it long enough,
+and the hardest stone must be worn away at last.</p>
+
+<hr style='width: 15%;' />
+
+<p>We have been speaking of millions of years somewhat familiarly; but
+what, after all, is a million years that we should not speak familiarly
+of it? It is longer than our lifetime, it is true. To the ephemeral
+insects whose lifetime is an hour, a year might seem an awful period,
+the mid-day sun might seem an almost stationary body, the changes of the
+seasons would be unknown, everything but the most fleeting and rapid
+changes would appear permanent and at rest. Conversely, if our
+life-period embraced myriads of &aelig;ons, things which now seem permanent
+would then appear as in a perpetual state of flux. A continent would be
+sometimes dry, sometimes covered with ocean; the stars we now call fixed
+would be moving visibly before our eyes; the earth would be humming on
+its axis like a top, and the whole of human history might seem as
+fleeting as a cloud of breath on a mirror.</p>
+
+<p><span class='pagenum'><a name="Page_397" id="Page_397">[Pg 397]</a></span></p><p>Evolution is always a slow process. To evolve such an animal as a
+greyhound from its remote ancestors, according to Mr. Darwin, needs
+immense tracts of time; and if the evolution of some feeble animal
+crawling on the surface of this planet is slow, shall the stately
+evolution of the planetary orbs themselves be hurried? It may be that we
+are able to trace the history of the solar system for some thousand
+million years or so; but for how much longer time must it not have a
+history&mdash;a history, and also a future&mdash;entirely beyond our ken?</p>
+
+<p>Those who study the stars have impressed upon them the existence of the
+most immeasurable distances, which yet are swallowed up as nothing in
+the infinitude of space. No less are we compelled to recognize the
+existence of incalculable &aelig;ons of time, and yet to perceive that these
+are but as drops in the ocean of eternity.</p>
+
+
+<p><span class='pagenum'><a name="Page_398" id="Page_398">[Pg 398]</a></span></p>
+
+<div class="footnotes"><h3>FOOTNOTES:</h3>
+
+<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The following account of Mars's motion is from the
+excellent small manual of astronomy by Dr. Haughton of Trinity College,
+Dublin:&mdash;(P. 151) "Mars's motion is very unequal; when he first appears
+in the morning emerging from the rays of the sun, his motion is direct
+and rapid; it afterwards becomes slower, and he becomes stationary when
+at an elongation of 137&deg; from the sun; then his motion becomes
+retrograde, and its velocity increases until he is in opposition to the
+sun at 180&deg;; at this time the retrograde motion is most rapid, and
+afterwards diminishes until he is 137&deg; distant from the sun on the other
+side, when Mars again becomes stationary; his motion then becomes
+direct, and increases in velocity until it reaches a maximum, when the
+planet is again in conjunction with the sun. The retrograde motion of
+this planet lasts for 73 days: and its arc of retrogradation is 16&deg;."</p></div>
+
+<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> It is not so easy to plot the path of the sun among the
+stars by direct observation, as it is to plot the path of a planet;
+because sun and stars are not visible together. Hipparchus used the moon
+as an intermediary; since sun and moon are visible together, and also
+moon and stars.</p></div>
+
+<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> This is, however, by no means the whole of the matter. The
+motion is not a simple circle nor has it a readily specifiable period.
+There are several disturbing causes. All that is given here is a first
+rough approximation.</p></div>
+
+<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> The proof is easy, and ought to occur in books on solid
+geometry. By a "regular" solid is meant one with all its faces, edges,
+angles, &amp;c., absolutely alike: it is of these perfectly symmetrical
+bodies that there are only five. Crystalline forms are practically
+infinite in number.</p></div>
+
+<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> Best known to us by his Christian name, as so many others
+of that time are known, <i>e.g.</i> Raphael Sanzio, Dante Alighieri, Michael
+Angelo Buonarotti. The rule is not universal. Tasso and Ariosto are
+surnames.</p></div>
+
+<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> It would seem that the fact that all bodies of every
+material tend to fall at the same rate is still not clearly known.
+Confusion is introduced by the resistance of the air. But a little
+thought should make it clear that the effect of the air is a mere
+disturbance, to be eliminated as far as possible, since the atmosphere
+has nothing to do with gravitation. The old fashioned "guinea and
+feather experiment" illustrates that in a vacuum things entirely
+different in specific gravity or surface drop at the same pace.</p></div>
+
+<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> Karl von Gebler (Galileo), p. 13.</p></div>
+
+<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> It is of course the "silver lining" of clouds that outside
+observers see.</p></div>
+
+<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> L.U.K., <i>Life of Galileo</i>, p. 26.</p></div>
+
+<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> <i>Note added September, 1892.</i> News from the Lick
+Observatory makes a very small fifth satellite not improbable.</p></div>
+
+<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> They remained there till this century. In 1835 they were
+quietly dropped.</p></div>
+
+<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> It was invented by van Helmont, a Belgian chemist, who
+died in 1644. He suggested two names <i>gas</i> and <i>blas</i>, and the first has
+survived. Blas was, I suppose, from <i>blasen</i>, to blow, and gas seems to
+be an attempt to get at the Sanskrit root underlying all such words as
+<i>geist</i>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> Such as this, among many others:&mdash;The duration of a flame
+under different conditions is well worth determining. A spoonful of warm
+spirits of wine burnt 116 pulsations. The same spoonful of spirits of
+wine with addition of one-sixth saltpetre burnt 94 pulsations. With
+one-sixth common salt, 83; with one-sixth gunpowder, 110; a piece of wax
+in the middle of the spirit, 87; a piece of <i>Kieselstein</i>, 94; one-sixth
+water, 86; and with equal parts water, only 4 pulse-beats. This, says
+Liebig, is given as an example of a "<i>licht-bringende Versuch</i>."</p></div>
+
+<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> Draper, <i>History of Civilization in Europe</i>, vol. ii. p.
+259.</p></div>
+
+<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> Professor Knight's series of Philosophical Classics.</p></div>
+
+<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> To explain why the entire system, horse and cart together,
+move forward, the forces acting on the ground must be attended to.</p></div>
+
+<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> The distance being proportional to the <i>square</i> of the
+time, see <a href="#Page_82">p. 82</a>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> The following letter, recently unearthed and published in
+<i>Nature</i>, May 12, 1881, seems to me well worth preserving. The feeling
+of a respiratory interval which it describes is familiar to students
+during the too few periods of really satisfactory occupation. The early
+guess concerning atmospheric electricity is typical of his extraordinary
+instinct for guessing right.
+</p>
+<p class="right">
+"<span class="smcap">London</span>, <i>Dec. 15, 1716</i>.<br />
+</p>
+<p>
+"<span class="smcap">Dear Doctor</span>,&mdash;He that in ye mine of knowledge deepest diggeth, hath,
+like every other miner, ye least breathing time, and must sometimes at
+least come to terr. alt. for air.
+</p><p>
+"In one of these respiratory intervals I now sit down to write to you,
+my friend.
+</p><p>
+"You ask me how, with so much study, I manage to retene my health. Ah,
+my dear doctor, you have a better opinion of your lazy friend than he
+hath of himself. Morpheous is my last companion; without 8 or 9 hours of
+him yr correspondent is not worth one scavenger's peruke. My practices
+did at ye first hurt my stomach, but now I eat heartily enou' as y' will
+see when I come down beside you.
+</p><p>
+"I have been much amused at ye singular <span class="greek" title="Greek: phenomena">&#966;&#949;&#957;&#8057;&#956;&#949;&#957;&#945;</span> resulting
+from bringing of a needle into contact with a piece of amber or resin
+fricated on silke clothe. Ye flame putteth me in mind of sheet lightning
+on a small&mdash;how very small&mdash;scale. But I shall in my epistles abjure
+Philosophy whereof when I come down to Sakly I'll give you enou'. I
+began to scrawl at 5 mins. from 9 of ye clk. and have in writing consmd.
+10 mins. My Ld. Somerset is announced.
+</p><p>
+"Farewell, Gd. bless you and help yr sincere friend.
+</p>
+<p class="right">
+"<span class="smcap">Isaac Newton.</span><br />
+</p>
+<p>
+"<i>To</i> <span class="smcap">Dr. Law</span>, Suffolk."<br />
+</p>
+</div>
+
+<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> Kepler's laws may be called respectively, the law of path,
+the law of speed, and the relationship law. By the "mass" of a body is
+meant the number of pounds or tons in it: the amount of matter it
+contains. The idea is involved in the popular word "massive."</p></div>
+
+<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> The equation we have to verify is
+</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Substituting gravity for weight equation">
+<tr class='tr2'>
+ <td align='center' rowspan='2'><i>gR<sup>2</sup></i>&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>4&#960;<sup>2</sup>r<sup>3</sup></i></td>
+ <td class='center' rowspan='2'>&nbsp;,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'>T<sup>2</sup></td>
+</tr>
+</table></div>
+
+<p class="noin">
+with the data that <i>r</i>, the moon's distance, is 60 times R, the earth's
+radius, which is 3,963 miles; while T, the time taken to complete the
+moon's orbit, is 27 days, 13 hours, 18 minutes, 37 seconds. Hence,
+suppose we calculate out <i>g</i>, the intensity of terrestrial gravity, from
+the above equation, we get</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="2" summary="Intensity of Terrestial Gravity Equation">
+<tr class='tr2'>
+ <td align='center' rowspan='2'><i>g</i>&nbsp;=&nbsp;</td>
+ <td class='tdcbb'><i>4&#960;<sup>2</sup></i></td>
+ <td class='center' rowspan='2'>&nbsp;&times;&nbsp;(60)<sup>3</sup>&nbsp;=&nbsp;</td>
+ <td class='tdcbb'>39&middot;92 &times; 216000 &times; 3963 miles</td>
+ <td class='center' rowspan='2'>&nbsp;=&nbsp;32&middot;92 feet-per-second per second,</td>
+</tr>
+<tr class='tr2'>
+ <td align='center'>T<sup>2</sup></td>
+ <td align='center'>(27 days, 13 hours, &amp;c.)<sup>2</sup></td>
+</tr>
+</table></div>
+
+<p class="noin">
+which is not far wrong.</p></div>
+
+<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> The two motions may be roughly compounded into a single
+motion, which for a few centuries may without much error be regarded as
+a conical revolution about a different axis with a different period; and
+Lieutenant-Colonel Drayson writes books emphasizing this simple fact,
+under the impression that it is a discovery.</p></div>
+
+<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> Members of the Accademia dei Lyncei, the famous old
+scientific Society established in the time of Cosmo de Medici&mdash;older
+than our own Royal Society.</p></div>
+
+<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> Newton suspected that the moon really did so oscillate,
+and so it may have done once; but any real or physical libration, if
+existing at all, is now extremely minute.</p></div>
+
+<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> An interesting picture in the New Gallery this year
+(1891), attempting to depict "Earth-rise in Moon-land," unfortunately
+errs in several particulars. First of all, the earth does not "rise,"
+but is fixed relatively to each place on the moon; and two-fifths of the
+moon never sees it. Next, the earth would not look like a map of the
+world with a haze on its edge. Lastly, whatever animal remains the moon
+may contain would probably be rather in the form of fossils than of
+skeletons. The skeleton is of course intended as an image of death and
+desolation. It is a matter of taste: but a skeleton, it seems to me,
+speaks too recently of life to be as appallingly weird and desolate as a
+blank stone or ice landscape, unshaded by atmosphere or by any trace of
+animal or plant life, could be made.</p></div>
+
+<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> Five of Jupiter's revolutions occupy 21,663 days; two of
+Saturn's revolutions occupy 21,526 days.</p></div>
+
+<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> <i>Excircularity</i> is what is meant by this term. It is
+called "excentricity" because the foci (not the centre) of an ellipse
+are regarded as the representatives of the centre of a circle. Their
+distance from the centre, compared with the radius of the unflattened
+circle, is called the excentricity.</p></div>
+
+<div class="footnote"><p><a name="Footnote_27_27" id="Footnote_27_27"></a><a href="#FNanchor_27_27"><span class="label">[27]</span></a> A curve of the <i>n</i>th degree has &frac12;<i>n</i>(<i>n</i>+3) arbitrary
+constants in its equation, hence this number of points specifically
+determine it. But special points, like focus or vertex, count as two
+ordinary ones. Hence three points plus the focus act as five points, and
+determine a conic or curve of the second degree. Three observations
+therefore fix an orbit round the sun.</p></div>
+
+<div class="footnote"><p><a name="Footnote_28_28" id="Footnote_28_28"></a><a href="#FNanchor_28_28"><span class="label">[28]</span></a> Its name suggests a measure of the diameter of the sun's
+disk, and this is one of its functions; but it can likewise measure
+planetary and other disks; and in general behaves as the most elaborate
+and expensive form of micrometer. The K&ouml;nigsberg instrument is shewn in
+fig. 92.</p></div>
+
+<div class="footnote"><p><a name="Footnote_29_29" id="Footnote_29_29"></a><a href="#FNanchor_29_29"><span class="label">[29]</span></a> It may be supposed that the terms "minute" and "second"
+have some necessary connection with time, but they are mere
+abbreviations for <i>partes minut&aelig;</i> and <i>partes minut&aelig; secund&aelig;</i>, and
+consequently may be applied to the subdivision of degrees just as
+properly as to the subdivision of hours. A "second" of arc means the
+3600th part of a degree, just as a second of time means the 3600th part
+of an hour.</p></div>
+
+<div class="footnote"><p><a name="Footnote_30_30" id="Footnote_30_30"></a><a href="#FNanchor_30_30"><span class="label">[30]</span></a> A group of flying particles, each one invisible, obstructs
+light singularly little, even when they are close together, as one can
+tell by the transparency of showers and snowstorms. The opacity of haze
+may be due not merely to dust particles, but to little eddies set up by
+radiation above each particle, so that the air becomes turbulent and of
+varying density. (See a similar suggestion by Mr. Poynting in <i>Nature</i>,
+vol. 39, p. 323.)</p></div>
+
+<div class="footnote"><p><a name="Footnote_31_31" id="Footnote_31_31"></a><a href="#FNanchor_31_31"><span class="label">[31]</span></a> The moon ought to be watched during the next great shower,
+if the line of fire happens to take effect on a visible part of the dark
+portion.</p></div>
+
+<div class="footnote"><p><a name="Footnote_32_32" id="Footnote_32_32"></a><a href="#FNanchor_32_32"><span class="label">[32]</span></a> Address to Birmingham Midland Institute, "A Glimpse
+through the Corridors of Time."</p></div>
+</div>
+
+
+<hr /><p><span class='pagenum'><a name="Page_399" id="Page_399">[Pg 399]</a></span></p><p><span class='pagenum'><a name="Page_400" id="Page_400">[Pg 400]</a></span></p>
+<h3><br /><br /><a name="INDEX" id="INDEX"></a>INDEX<br /><br /></h3>
+<hr />
+<p><span class='pagenum'><a name="Page_401" id="Page_401">[Pg 401]</a></span></p>
+<h3>INDEX</h3>
+
+
+<p class="index">
+<span style="margin-left: 4em;">A</span><br />
+<br />
+<span class="smcap">Abbott, T.K.</span>, on tides, <a href="#Page_369">369</a><br />
+<br />
+Adams, John Couch, <a href="#Page_193">193</a>, <a href="#Page_217">217</a>, <a href="#Page_302">302</a>, <a href="#Page_323">323</a>, <a href="#Page_324">324</a>, <a href="#Page_325">325</a>, <a href="#Page_327">327</a>, <a href="#Page_329">329</a>, <a href="#Page_330">330</a>, <a href="#Page_352">352</a>, <a href="#Page_385">385</a><br />
+<br />
+Airy, Sir George, <a href="#Page_193">193</a>, <a href="#Page_244">244</a>, <a href="#Page_302">302</a>, <a href="#Page_323">323</a>, <a href="#Page_324">324</a>, <a href="#Page_327">327</a>, <a href="#Page_367">367</a><br />
+<br />
+Anaxagoras, <a href="#Page_15">15</a><br />
+<br />
+Appian, <a href="#Page_218">218</a><br />
+<br />
+Arabs, the, form a link between the old and new science, <a href="#Page_9">9</a><br />
+<br />
+Archimedes, <a href="#Page_7">7</a>, <a href="#Page_8">8</a>, <a href="#Page_84">84</a>, <a href="#Page_87">87</a>, <a href="#Page_144">144</a>, <a href="#Page_177">177</a><br />
+<br />
+Aristarchus, <a href="#Page_34">34</a><br />
+<br />
+Aristotle, <a href="#Page_66">66</a>, <a href="#Page_69">69</a>, <a href="#Page_88">88</a>, <a href="#Page_94">94</a>, <a href="#Page_99">99</a>, <a href="#Page_167">167</a>.<br />
+<span style="margin-left: 1em;">He taught that the earth was a sphere, <a href="#Page_16">16</a>;</span><br />
+<span style="margin-left: 1em;">his theories did not allow of the earth's motion, <a href="#Page_34">34</a>;</span><br />
+<span style="margin-left: 1em;">he was regarded as inspired, <a href="#Page_89">89</a></span><br />
+<br />
+<br />
+<span style="margin-left: 4em;">B</span><br />
+<br />
+<span class="smcap">Bacon</span>, Francis, <a href="#Page_142">142</a>, <a href="#Page_143">143</a>, <a href="#Page_144">144</a>, <a href="#Page_145">145</a>.<br />
+<span style="margin-left: 1em;">His <i>Novum Organum</i>, <a href="#Page_141">141</a></span><br />
+<br />
+Bacon, Roger, <a href="#Page_96">96</a>, <a href="#Page_139">139</a>, <a href="#Page_140">140</a>.<br />
+<span style="margin-left: 1em;">The herald of the dawn of science, <a href="#Page_9">9</a></span><br />
+<br />
+Brah&eacute;, George, uncle of Tycho Brah&eacute;, <a href="#Page_39">39</a><br />
+<br />
+Brah&eacute;, Steno, brother of Tycho Brah&eacute;, <a href="#Page_39">39</a><br />
+<br />
+Brah&eacute;, Tycho, <a href="#Page_37">37</a>, <a href="#Page_39">39</a>, <a href="#Page_40">40</a>, <a href="#Page_44">44</a>, <a href="#Page_45">45</a>, <a href="#Page_49">49</a>, <a href="#Page_51">51</a>, <a href="#Page_53">53</a>,
+<a href="#Page_54">54</a>, <a href="#Page_55">55</a>, <a href="#Page_58">58</a>, <a href="#Page_63">63</a>, <a href="#Page_64">64</a>, <a href="#Page_65">65</a>, <a href="#Page_66">66</a>, <a href="#Page_68">68</a>, <a href="#Page_71">71</a>,
+<a href="#Page_72">72</a>, <a href="#Page_74">74</a>, <a href="#Page_75">75</a>, <a href="#Page_77">77</a>, <a href="#Page_78">78</a>, <a href="#Page_86">86</a>, <a href="#Page_94">94</a>, <a href="#Page_117">117</a>, <a href="#Page_137">137</a>,
+<a href="#Page_155">155</a>, <a href="#Page_165">165</a>, <a href="#Page_166">166</a>, <a href="#Page_200">200</a>, <a href="#Page_244">244</a>, <a href="#Page_281">281</a>, <a href="#Page_288">288</a>.<br />
+<span style="margin-left: 1em;">He tried to adopt the main features of the Copernican theory without admitting the motion of the earth, <a href="#Page_37">37</a>;</span><br />
+<span style="margin-left: 1em;">he was a poor theorist but a great observer, <a href="#Page_38">38</a>;</span><br />
+<span style="margin-left: 1em;">his medicine, <a href="#Page_44">44</a>;</span><br />
+<span style="margin-left: 1em;">his personal history, <a href="#Page_39">39</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his observatory, Uraniburg, <a href="#Page_47">47</a>;</span><br />
+<span style="margin-left: 1em;">his greatest invention, <a href="#Page_50">50</a>, note;</span><br />
+<span style="margin-left: 1em;">his maniac Lep, <a href="#Page_52">52</a>;</span><br />
+<span style="margin-left: 1em;">his kindness to Kepler, <a href="#Page_63">63</a></span><br />
+<br />
+Ball, Sir R., <a href="#Page_391">391</a>, <a href="#Page_394">394</a>;<br />
+<span style="margin-left: 1em;">his <i>Story of the Heavens</i>, <a href="#Page_377">377</a></span><br />
+<br />
+Barrow, Dr., <a href="#Page_165">165</a>, <a href="#Page_187">187</a><br />
+<br />
+Bessel, <a href="#Page_288">288</a>, <a href="#Page_310">310</a>, <a href="#Page_311">311</a>, <a href="#Page_313">313</a>, <a href="#Page_315">315</a>, <a href="#Page_316">316</a>, <a href="#Page_318">318</a>, <a href="#Page_323">323</a><br />
+<br />
+Biela, <a href="#Page_345">345</a>, <a href="#Page_346">346</a>, <a href="#Page_347">347</a><br />
+<br />
+Bode's Law, <a href="#Page_60">60</a>, <a href="#Page_296">296</a>, <a href="#Page_298">298</a>, <a href="#Page_299">299</a>, <a href="#Page_326">326</a><br />
+<br />
+Boyle, <a href="#Page_139">139</a>, <a href="#Page_188">188</a><br />
+<br />
+Bradley, Prof. James, <a href="#Page_233">233</a>, <a href="#Page_246">246</a>, <a href="#Page_247">247</a>, <a href="#Page_249">249</a>, <a href="#Page_252">252</a>, <a href="#Page_253">253</a>, <a href="#Page_308">308</a>, <a href="#Page_319">319</a><br />
+<br />
+Bremiker, <a href="#Page_328">328</a>, <a href="#Page_329">329</a><br />
+<br />
+Brewster, on Kepler, <a href="#Page_78">78</a><br />
+<br />
+Brinkley, <a href="#Page_308">308</a><br />
+<br />
+Bruno, Giordano, <a href="#Page_108">108</a>, <a href="#Page_127">127</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">C</span><br />
+<br />
+<span class="smcap">Castelli</span>, <a href="#Page_112">112</a>, <a href="#Page_133">133</a><br />
+<br />
+Cayley, Prof., <a href="#Page_385">385</a><br />
+<br />
+Challis, Prof., <a href="#Page_328">328</a>, <a href="#Page_329">329</a><br />
+<br />
+Clairut, <a href="#Page_193">193</a>, <a href="#Page_216">216</a>, <a href="#Page_217">217</a>, <a href="#Page_219">219</a>, <a href="#Page_234">234</a>, <a href="#Page_341">341</a><br />
+<br />
+Clark, Alvan and Sons, <a href="#Page_316">316</a><br />
+<br />
+Columbus, <a href="#Page_9">9</a>, <a href="#Page_144">144</a><br />
+<br />
+Copernicus, <a href="#Page_7">7</a>, <a href="#Page_10">10</a>, <i>seq.</i>, <a href="#Page_14">14</a>, <a href="#Page_26">26</a>, <a href="#Page_27">27</a>, <a href="#Page_29">29</a>, <a href="#Page_30">30</a>, <a href="#Page_31">31</a>, <a href="#Page_33">33</a>, <a href="#Page_34">34</a>,
+<a href="#Page_35">35</a>, <a href="#Page_37">37</a>, <a href="#Page_38">38</a>, <a href="#Page_62">62</a>, <a href="#Page_66">66</a>, <a href="#Page_68">68</a>, <a href="#Page_70">70</a>, <a href="#Page_78">78</a>, <a href="#Page_93">93</a>, <a href="#Page_95">95</a>,
+<a href="#Page_100">100</a>, <a href="#Page_108">108</a>, <a href="#Page_111">111</a>, <a href="#Page_121">121</a>, <a href="#Page_122">122</a>, <a href="#Page_137">137</a>, <a href="#Page_155">155</a>, <a href="#Page_166">166</a>, <a href="#Page_223">223</a>, <a href="#Page_234">234</a>,
+<a href="#Page_247">247</a>, <a href="#Page_307">307</a>;<br />
+<span style="margin-left: 1em;">his <i>De Revolutionibus Orbium C&#339;lestium</i>, <a href="#Page_11">11</a>, <a href="#Page_75">75</a>, <a href="#Page_138">138</a>;</span><br />
+<span style="margin-left: 1em;">he <i>proved</i> that the earth went round the sun, <a href="#Page_13">13</a>;</span><br />
+<span class='pagenum'><a name="Page_402" id="Page_402">[Pg 402]</a></span><span style="margin-left: 1em;">the influence of his theory on the Church, <a href="#Page_13">13</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his life-work summarised, <a href="#Page_30">30</a>;</span><br />
+<span style="margin-left: 1em;">his Life by Mr. E.J.C. Morton, <a href="#Page_31">31</a></span><br />
+<br />
+Copernican tables, <a href="#Page_40">40</a>;<br />
+<span style="margin-left: 1em;">Copernican theory, <a href="#Page_59">59</a>, <a href="#Page_60">60</a>, <a href="#Page_125">125</a>, <a href="#Page_144">144</a>, <a href="#Page_167">167</a></span><br />
+<br />
+Copernik, Nicolas; see Copernicus<br />
+<br />
+Cornu, <a href="#Page_238">238</a><br />
+<br />
+Croll, Dr., his <i>Climate and Time</i>, <a href="#Page_264">264</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">D</span><br />
+<br />
+<span class="smcap">D'Alembert</span>, <a href="#Page_193">193</a>, <a href="#Page_234">234</a><br />
+<br />
+Darwin, Charles, <a href="#Page_134">134</a>, <a href="#Page_138">138</a>, <a href="#Page_397">397</a><br />
+<br />
+Darwin, Prof. George, <a href="#Page_367">367</a>, <a href="#Page_394">394</a><br />
+<br />
+Delambre, <a href="#Page_253">253</a><br />
+<br />
+Descartes, <a href="#Page_145">145</a>, <a href="#Page_146">146</a>, <a href="#Page_148">148</a>, <a href="#Page_151">151</a>, <a href="#Page_153">153</a>,
+<a href="#Page_156">156</a>, <a href="#Page_158">158</a>, <a href="#Page_164">164</a>, <a href="#Page_165">165</a>, <a href="#Page_167">167</a>, <a href="#Page_178">178</a>,
+<a href="#Page_181">181</a>, <a href="#Page_224">224</a>, <a href="#Page_227">227</a>;<br />
+<span style="margin-left: 1em;">his <i>Discourse on Method</i>, <a href="#Page_142">142</a>;</span><br />
+<span style="margin-left: 1em;">his dream, <a href="#Page_147">147</a>;</span><br />
+<span style="margin-left: 1em;">his system of algebraic geometry, <a href="#Page_149">149</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his doctrine of vortices, <a href="#Page_151">151</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his <i>Principia Mathematica</i>, <a href="#Page_154">154</a>;</span><br />
+<span style="margin-left: 1em;">his Life by Mr. Mahaffy, <a href="#Page_154">154</a></span><br />
+<br />
+<br />
+<span style="margin-left: 4em;">E</span><br />
+<br />
+<span class="smcap">Earth</span>, the difficulties in the way of believing that it moved, <a href="#Page_34">34</a>, <i>seq.</i><br />
+<br />
+"Earth-rise in Moon-land," <a href="#Page_258">258</a>, note<br />
+<br />
+Encke, <a href="#Page_345">345</a>, <a href="#Page_346">346</a><br />
+<br />
+Epicyclic orbits explained, <a href="#Page_23">23</a>, <i>seq.</i><br />
+<br />
+Equinoxes, their precession discovered by Hipparchus, <a href="#Page_27">27</a><br />
+<br />
+Eudoxus, <a href="#Page_19">19</a><br />
+<br />
+Euler, <a href="#Page_193">193</a>, <a href="#Page_234">234</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">F</span><br />
+<br />
+<span class="smcap">Faraday</span>, <a href="#Page_84">84</a><br />
+<br />
+Fizeau, <a href="#Page_238">238</a>, <a href="#Page_239">239</a><br />
+<br />
+Flamsteed, <a href="#Page_215">215</a>, <a href="#Page_246">246</a>, <a href="#Page_284">284</a>, <a href="#Page_308">308</a>, <a href="#Page_319">319</a><br />
+<br />
+Fraunhofer, <a href="#Page_311">311</a><br />
+<br />
+Froude, Prof.; his <i>Oceania</i>, <a href="#Page_387">387</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">G</span><br />
+<br />
+<span class="smcap">Galen</span>, <a href="#Page_87">87</a><br />
+<br />
+Galileo, Galilei, <a href="#Page_63">63</a>, <a href="#Page_75">75</a>, <a href="#Page_84">84</a>, <a href="#Page_88">88</a>, <a href="#Page_90">90</a>, <a href="#Page_92">92</a>, <a href="#Page_93">93</a>, <a href="#Page_97">97</a>, <a href="#Page_98">98</a>, <a href="#Page_101">101</a>, <a href="#Page_104">104</a>, <a href="#Page_106">106</a>,
+<a href="#Page_107">107</a>, <a href="#Page_108">108</a>, <a href="#Page_109">109</a>, <a href="#Page_110">110</a>, <a href="#Page_112">112</a>, <a href="#Page_114">114</a>, <a href="#Page_116">116</a>, <a href="#Page_117">117</a>, <a href="#Page_118">118</a>, <a href="#Page_120">120</a>, <a href="#Page_121">121</a>, <a href="#Page_122">122</a>,
+<a href="#Page_123">123</a>, <a href="#Page_125">125</a>, <a href="#Page_127">127</a>, <a href="#Page_133">133</a>, <a href="#Page_134">134</a>, <a href="#Page_137">137</a>, <a href="#Page_144">144</a>, <a href="#Page_145">145</a>, <a href="#Page_153">153</a>, <a href="#Page_154">154</a>, <a href="#Page_157">157</a>, <a href="#Page_165">165</a>,
+<a href="#Page_166">166</a>, <a href="#Page_167">167</a>, <a href="#Page_168">168</a>, <a href="#Page_177">177</a>, <a href="#Page_188">188</a>, <a href="#Page_200">200</a>, <a href="#Page_224">224</a>, <a href="#Page_227">227</a>, <a href="#Page_256">256</a>, <a href="#Page_281">281</a>, <a href="#Page_288">288</a>, <a href="#Page_309">309</a>,
+<a href="#Page_361">361</a>;<br />
+<span style="margin-left: 1em;">his youth, <a href="#Page_85">85</a>;</span><br />
+<span style="margin-left: 1em;">his discovery of the pendulum, <a href="#Page_86">86</a>;</span><br />
+<span style="margin-left: 1em;">his first observations about falling bodies, <a href="#Page_88">88</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">he invents a telescope, <a href="#Page_95">95</a>;</span><br />
+<span style="margin-left: 1em;">he adopts the Copernican theory, <a href="#Page_94">94</a>;</span><br />
+<span style="margin-left: 1em;">he conceives "earth-shine," <a href="#Page_100">100</a>;</span><br />
+<span style="margin-left: 1em;">he discovers Jupiter's moons, <a href="#Page_103">103</a>;</span><br />
+<span style="margin-left: 1em;">he studies Saturn, <a href="#Page_114">114</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his <i>Dialogues on the Ptolemaic and Copernican Systems</i>, <a href="#Page_124">124</a>;</span><br />
+<span style="margin-left: 1em;">his abjuration, <a href="#Page_130">130</a>;</span><br />
+<span style="margin-left: 1em;">he becomes blind, <a href="#Page_132">132</a>;</span><br />
+<span style="margin-left: 1em;">he discovered the Laws of Motion, <a href="#Page_167">167</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">he guessed that sight was not instantaneous, <a href="#Page_236">236</a>, <a href="#Page_237">237</a></span><br />
+<br />
+Galle, Dr., <a href="#Page_245">245</a>, <a href="#Page_329">329</a><br />
+<br />
+Gauss, <a href="#Page_299">299</a>, <a href="#Page_300">300</a><br />
+<br />
+Gilbert, Dr., <a href="#Page_139">139</a>, <a href="#Page_140">140</a>, <a href="#Page_157">157</a>, <a href="#Page_188">188</a>;<br />
+<span style="margin-left: 1em;">his <i>De Magnete</i>, <a href="#Page_140">140</a>, <a href="#Page_144">144</a></span><br />
+<br />
+Greeks, their scientific methods, <a href="#Page_7">7</a><br />
+<br />
+Groombridge's Catalogue, <a href="#Page_315">315</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">H</span><br />
+<br />
+<span class="smcap">Hadley</span>, <a href="#Page_185">185</a><br />
+<br />
+Halley, <a href="#Page_192">192</a>, <a href="#Page_193">193</a>, <a href="#Page_194">194</a>, <a href="#Page_195">195</a>, <a href="#Page_197">197</a>, <a href="#Page_215">215</a>, <a href="#Page_218">218</a>, <a href="#Page_219">219</a>, <a href="#Page_246">246</a>, <a href="#Page_258">258</a>, <a href="#Page_260">260</a>, <a href="#Page_261">261</a>, <a href="#Page_340">340</a>, <a href="#Page_341">341</a>;<br />
+<span style="margin-left: 1em;">he discovered the <i>Principia</i>, <a href="#Page_194">194</a></span><br />
+<br />
+Harvey, <a href="#Page_144">144</a>, <a href="#Page_149">149</a><br />
+<br />
+Haughton, Dr., <a href="#Page_321">321</a>;<br />
+<span style="margin-left: 1em;">his manual on Astronomy, <a href="#Page_21">21</a>, note</span><br />
+<br />
+Heliometer, described, <a href="#Page_311">311</a><br />
+<br />
+Helmholtz, <a href="#Page_378">378</a><br />
+<br />
+Helmont, Van, invented the word "gas," <a href="#Page_141">141</a><br />
+<br />
+Henderson, <a href="#Page_310">310</a>, <a href="#Page_314">314</a><br />
+<br />
+Herschel, Alexander, <a href="#Page_275">275</a>, <a href="#Page_277">277</a>, <a href="#Page_278">278</a>, <a href="#Page_279">279</a><br />
+<br />
+Herschel, Caroline, <a href="#Page_275">275</a>, <a href="#Page_276">276</a>, <a href="#Page_279">279</a>, <a href="#Page_286">286</a>, <a href="#Page_345">345</a>;<br />
+<span style="margin-left: 1em;">her journal quoted, <a href="#Page_277">277</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">her work with William H. described, <a href="#Page_284">284</a></span><br />
+<br />
+Herschel, Sir John, <a href="#Page_283">283</a>, <a href="#Page_285">285</a>, <a href="#Page_327">327</a>, <a href="#Page_329">329</a><br />
+<br />
+<span class='pagenum'><a name="Page_403" id="Page_403">[Pg 403]</a></span>Herschel, William, <a href="#Page_185">185</a>, <a href="#Page_234">234</a>, <a href="#Page_235">235</a>, <a href="#Page_244">244</a>, <a href="#Page_249">249</a>, <a href="#Page_274">274</a>, <a href="#Page_275">275</a>, <a href="#Page_280">280</a>, <a href="#Page_281">281</a>, <a href="#Page_282">282</a>, <a href="#Page_284">284</a>, <a href="#Page_288">288</a>, <a href="#Page_289">289</a>, <a href="#Page_290">290</a>, <a href="#Page_293">293</a>, <a href="#Page_295">295</a>, <a href="#Page_305">305</a>, <a href="#Page_309">309</a>, <a href="#Page_310">310</a>, <a href="#Page_318">318</a>, <a href="#Page_319">319</a>, <a href="#Page_327">327</a>;<br />
+<span style="margin-left: 1em;">he "sweeps" the heavens, <a href="#Page_280">280</a>;</span><br />
+<span style="margin-left: 1em;">his discovery of Uranus, <a href="#Page_281">281</a>, <a href="#Page_287">287</a>;</span><br />
+<span style="margin-left: 1em;">his artificial Saturn, <a href="#Page_281">281</a>, <a href="#Page_282">282</a>;</span><br />
+<span style="margin-left: 1em;">his methods of work with his sister, described, <a href="#Page_284">284</a>;</span><br />
+<span style="margin-left: 1em;">he founded the science of Astronomy, <a href="#Page_287">287</a></span><br />
+<br />
+Hind, <a href="#Page_300">300</a><br />
+<br />
+Hipparchus, <a href="#Page_7">7</a>, <a href="#Page_18">18</a>, <a href="#Page_20">20</a>, <a href="#Page_27">27</a>, <a href="#Page_28">28</a>, and note, <a href="#Page_30">30</a>, <a href="#Page_40">40</a>, <a href="#Page_66">66</a>, <a href="#Page_223">223</a>, <a href="#Page_253">253</a>;<br />
+<span style="margin-left: 1em;">an explanation of his discovery of the precession of the equinoxes, <a href="#Page_27">27</a>, seq.</span><br />
+<br />
+Hippocrates, <a href="#Page_87">87</a><br />
+<br />
+Homeric Cosmogony, <a href="#Page_15">15</a>, <i>seq.</i><br />
+<br />
+Hooke, <a href="#Page_139">139</a>, <a href="#Page_188">188</a>, <a href="#Page_192">192</a>, <a href="#Page_193">193</a>, <a href="#Page_196">196</a>, <a href="#Page_197">197</a>, <a href="#Page_308">308</a><br />
+<br />
+H&ocirc;pital, Marquis de l', <a href="#Page_228">228</a><br />
+<br />
+Horkey, Martin, <a href="#Page_106">106</a><br />
+<br />
+Horrebow, <a href="#Page_244">244</a><br />
+<br />
+Huxley, Prof., <a href="#Page_149">149</a><br />
+<br />
+Huyghens, <a href="#Page_86">86</a>, <a href="#Page_166">166</a>, <a href="#Page_185">185</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">K</span><br />
+<br />
+<span class="smcap">Kant</span>, <a href="#Page_267">267</a>, <a href="#Page_270">270</a><br />
+<br />
+Kelvin, Lord, see Thomson, Sir W.<br />
+<br />
+Kepler, John, <a href="#Page_59">59</a>, <a href="#Page_60">60</a>, <a href="#Page_63">63</a>, <a href="#Page_64">64</a>, <a href="#Page_65">65</a>, <a href="#Page_66">66</a>, <a href="#Page_70">70</a>, <a href="#Page_72">72</a>, <a href="#Page_73">73</a>, <a href="#Page_75">75</a>, <a href="#Page_77">77</a>, <a href="#Page_79">79</a>, <a href="#Page_84">84</a>,
+<a href="#Page_93">93</a>, <a href="#Page_94">94</a>, <a href="#Page_95">95</a>, <a href="#Page_104">104</a>, <a href="#Page_106">106</a>, <a href="#Page_107">107</a>, <a href="#Page_110">110</a>, <a href="#Page_122">122</a>, <a href="#Page_137">137</a>, <a href="#Page_145">145</a>, <a href="#Page_153">153</a>, <a href="#Page_158">158</a>,
+<a href="#Page_164">164</a>, <a href="#Page_165">165</a>, <a href="#Page_166">166</a>, <a href="#Page_167">167</a>, <a href="#Page_192">192</a>, <a href="#Page_200">200</a>, <a href="#Page_208">208</a>, <a href="#Page_209">209</a>, <a href="#Page_210">210</a>, <a href="#Page_211">211</a>, <a href="#Page_212">212</a>, <a href="#Page_214">214</a>,
+<a href="#Page_218">218</a>, <a href="#Page_224">224</a>, <a href="#Page_227">227</a>, <a href="#Page_253">253</a>, <a href="#Page_256">256</a>, <a href="#Page_259">259</a>, <a href="#Page_260">260</a>, <a href="#Page_262">262</a>, <a href="#Page_288">288</a>, <a href="#Page_295">295</a>, <a href="#Page_296">296</a>, <a href="#Page_332">332</a>,
+<a href="#Page_338">338</a>, <a href="#Page_361">361</a>, <a href="#Page_389">389</a>;<br />
+<span style="margin-left: 1em;">he replaced epicycles by an ellipse, <a href="#Page_27">27</a>;</span><br />
+<span style="margin-left: 1em;">he was a pupil of Tycho Brah&eacute;, <a href="#Page_54">54</a>;</span><br />
+<span style="margin-left: 1em;">he was a speculator more than an observer, <a href="#Page_58">58</a>;</span><br />
+<span style="margin-left: 1em;">his personal life, <a href="#Page_58">58</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his theories about the numbers and distances of the planets, <a href="#Page_60">60</a>, <a href="#Page_62">62</a>;</span><br />
+<span style="margin-left: 1em;">he was helped by Tycho, <a href="#Page_63">63</a>;</span><br />
+<span style="margin-left: 1em;">his main work, <a href="#Page_65">65</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">he gave up circular motion, <a href="#Page_69">69</a>;</span><br />
+<span style="margin-left: 1em;">his <i>Mysterium Cosmographicon</i>, <a href="#Page_105">105</a>;</span><br />
+<span style="margin-left: 1em;">his Laws, <a href="#Page_71">71</a>, <a href="#Page_74">74</a>, <a href="#Page_173">173</a>, <a href="#Page_174">174</a>, <a href="#Page_176">176</a>, <a href="#Page_179">179</a>, <a href="#Page_180">180</a>, <a href="#Page_206">206</a>, <i>seq.</i></span><br />
+<br />
+<br />
+<span style="margin-left: 4em;">L</span><br />
+<br />
+<span class="smcap">Lagrange</span>, <a href="#Page_193">193</a>, <a href="#Page_234">234</a>, <a href="#Page_255">255</a>, <a href="#Page_256">256</a>, <a href="#Page_257">257</a>, <a href="#Page_258">258</a>, <a href="#Page_263">263</a><br />
+<br />
+Lagrange and Laplace, <a href="#Page_258">258</a>, <a href="#Page_266">266</a>, <a href="#Page_395">395</a>;<br />
+<span style="margin-left: 1em;">they laid the foundations of the planetary theory, <a href="#Page_259">259</a></span><br />
+<br />
+Laplace, <a href="#Page_68">68</a>, <a href="#Page_193">193</a>, <a href="#Page_218">218</a>, <a href="#Page_234">234</a>, <a href="#Page_255">255</a>, <a href="#Page_261">261</a>, <a href="#Page_262">262</a>, <a href="#Page_267">267</a>, <a href="#Page_268">268</a>, <a href="#Page_269">269</a>, <a href="#Page_270">270</a>, <a href="#Page_272">272</a>, <a href="#Page_288">288</a>, <a href="#Page_301">301</a>, <a href="#Page_317">317</a>, <a href="#Page_384">384</a>, <a href="#Page_385">385</a>, <a href="#Page_390">390</a>;<br />
+<span style="margin-left: 1em;">his nebular hypothesis, <a href="#Page_267">267</a>, <a href="#Page_292">292</a>;</span><br />
+<span style="margin-left: 1em;">his <i>M&eacute;canique C&eacute;leste</i>, <a href="#Page_323">323</a></span><br />
+<br />
+Lassell, Mr., <a href="#Page_283">283</a>, <a href="#Page_284">284</a><br />
+<br />
+Leibnitz, <a href="#Page_192">192</a>, <a href="#Page_197">197</a>, <a href="#Page_233">233</a><br />
+<br />
+Le Monnier, <a href="#Page_319">319</a><br />
+<br />
+Leonardo, see Vinci, Leonardo da<br />
+<br />
+Leverrier, <a href="#Page_193">193</a>, <a href="#Page_327">327</a>, <a href="#Page_328">328</a>, <a href="#Page_329">329</a>, <a href="#Page_330">330</a>, <a href="#Page_352">352</a><br />
+<br />
+Lippershey, Hans, <a href="#Page_95">95</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">M</span><br />
+<br />
+<span class="smcap">Maskelyne</span>, <a href="#Page_281">281</a><br />
+<br />
+Maxwell, Clerk, <a href="#Page_302">302</a>, <a href="#Page_303">303</a><br />
+<br />
+Molyneux, <a href="#Page_248">248</a>, <a href="#Page_249">249</a><br />
+<br />
+Morton, Mr. E.J. C, his Life of Copernicus, <a href="#Page_31">31</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">N</span><br />
+<br />
+<span class="smcap">Newton</span>, Prof. H.A., <a href="#Page_347">347</a><br />
+<br />
+Newton, Sir Isaac, <a href="#Page_7">7</a>, <a href="#Page_30">30</a>, <a href="#Page_79">79</a>, <a href="#Page_138">138</a>, <a href="#Page_139">139</a>, <a href="#Page_144">144</a>, <a href="#Page_145">145</a>, <a href="#Page_149">149</a>,
+<a href="#Page_153">153</a>, <a href="#Page_157">157</a>, <a href="#Page_158">158</a>, <a href="#Page_165">165</a>, <a href="#Page_166">166</a>, <a href="#Page_167">167</a>, <a href="#Page_174">174</a>, <a href="#Page_176">176</a>, <a href="#Page_184">184</a>,
+<a href="#Page_187">187</a>, <a href="#Page_188">188</a>, <a href="#Page_189">189</a>, <a href="#Page_191">191</a>, <a href="#Page_192">192</a>, <a href="#Page_194">194</a>, <a href="#Page_196">196</a>, <a href="#Page_198">198</a>, <a href="#Page_199">199</a>,
+<a href="#Page_201">201</a>, <a href="#Page_213">213</a>, <a href="#Page_216">216</a>, <a href="#Page_219">219</a>, <a href="#Page_220">220</a>, <a href="#Page_221">221</a>, <a href="#Page_224">224</a>, <a href="#Page_226">226</a>, <a href="#Page_227">227</a>,
+<a href="#Page_228">228</a>, <a href="#Page_233">233</a>, <a href="#Page_242">242</a>, <a href="#Page_253">253</a>, <a href="#Page_255">255</a>, <a href="#Page_256">256</a>, <a href="#Page_274">274</a>, <a href="#Page_288">288</a>, <a href="#Page_317">317</a>,
+<a href="#Page_340">340</a>, <a href="#Page_378">378</a>;<br />
+<span style="margin-left: 1em;">his <i>Principia</i>, <a href="#Page_191">191</a>, <a href="#Page_192">192</a>, <a href="#Page_193">193</a>, <a href="#Page_194">194</a>, <a href="#Page_195">195</a>, <a href="#Page_196">196</a>, <a href="#Page_197">197</a>, <a href="#Page_207">207</a>, <a href="#Page_214">214</a>, <a href="#Page_216">216</a>, <a href="#Page_218">218</a>, <a href="#Page_228">228</a>, <a href="#Page_233">233</a>, <a href="#Page_242">242</a>, <a href="#Page_253">253</a>;</span><br />
+<span style="margin-left: 1em;">his early life, <a href="#Page_161">161</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his first experiments, <a href="#Page_163">163</a>;</span><br />
+<span style="margin-left: 1em;">his work at Cambridge, <a href="#Page_164">164</a>;</span><br />
+<span style="margin-left: 1em;">his Laws, <a href="#Page_168">168</a>;</span><br />
+<span style="margin-left: 1em;">his application of the Laws of Gravity to Astronomy, <a href="#Page_177">177</a>, <a href="#Page_178">178</a>, <a href="#Page_179">179</a>, <a href="#Page_185">185</a>, <a href="#Page_190">190</a>;</span><br />
+<span style="margin-left: 1em;">his reticence, <a href="#Page_178">178</a>;</span><br />
+<span style="margin-left: 1em;">his discoveries in Optics, <a href="#Page_181">181</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his work summarised, <a href="#Page_186">186</a>;</span><br />
+<span style="margin-left: 1em;">his <i>Optics</i>, <a href="#Page_189">189</a>;</span><br />
+<span style="margin-left: 1em;">anecdotes of him, <a href="#Page_191">191</a>;</span><br />
+<span style="margin-left: 1em;">his appearance in a Court of Justice, <a href="#Page_195">195</a>;</span><br />
+<span style="margin-left: 1em;">some of his manuscripts very recently discovered, <a href="#Page_217">217</a>;</span><br />
+<span style="margin-left: 1em;">his theories of the Equinoxes and tides, <a href="#Page_223">223</a>, <i>seq.</i>, <a href="#Page_225">225</a>, <a href="#Page_363">363</a>, <i>seq.</i></span><br />
+<br />
+<br />
+<span class='pagenum'><a name="Page_404" id="Page_404">[Pg 404]</a></span>
+<span style="margin-left: 4em;">O</span><br />
+<br />
+<span class="smcap">Olbers</span>, <a href="#Page_299">299</a>, <a href="#Page_300">300</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">P</span><br />
+<br />
+<span class="smcap">Peters</span>, Prof., <a href="#Page_300">300</a>, <a href="#Page_316">316</a><br />
+<br />
+Piazzi, <a href="#Page_298">298</a>, <a href="#Page_299">299</a>, <a href="#Page_308">308</a>, <a href="#Page_313">313</a><br />
+<br />
+Picard, <a href="#Page_190">190</a>, <a href="#Page_242">242</a>, <a href="#Page_244">244</a>, <a href="#Page_247">247</a><br />
+<br />
+Pioneers, genuine, <a href="#Page_7">7</a><br />
+<br />
+Planets and days of the week, <a href="#Page_18">18</a><br />
+<br />
+Poynting, <a href="#Page_332">332</a><br />
+<br />
+Printing, <a href="#Page_9">9</a><br />
+<br />
+Ptolemy, <a href="#Page_18">18</a>, <a href="#Page_20">20</a>, <a href="#Page_27">27</a>, <a href="#Page_38">38</a>, <a href="#Page_153">153</a>, <a href="#Page_155">155</a>, <a href="#Page_166">166</a>, <a href="#Page_214">214</a>;<br />
+<span style="margin-left: 1em;">his system of the Heavens simplified by Copernicus, <a href="#Page_11">11</a>, <a href="#Page_30">30</a>;</span><br />
+<span style="margin-left: 1em;">his system described, <a href="#Page_19">19</a>, <i>seq.</i>;</span><br />
+<span style="margin-left: 1em;">his system taught, <a href="#Page_34">34</a>;</span><br />
+<span style="margin-left: 1em;">his harmonies, <a href="#Page_74">74</a></span><br />
+<br />
+Pythagoras, <a href="#Page_19">19</a>, <a href="#Page_20">20</a>, <a href="#Page_34">34</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">Q</span><br />
+<br />
+<span class="smcap">Quadrant</span>, an early, <a href="#Page_42">42</a>, <a href="#Page_43">43</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">R</span><br />
+<br />
+<span class="smcap">Rheiter</span>, <a href="#Page_107">107</a><br />
+<br />
+Ricci, Ostillio, <a href="#Page_86">86</a>, <a href="#Page_87">87</a><br />
+<br />
+Roberts, Isaac, <a href="#Page_268">268</a><br />
+<br />
+Roemer, <a href="#Page_239">239</a>, <a href="#Page_240">240</a>, <a href="#Page_242">242</a>, <a href="#Page_244">244</a>, <a href="#Page_249">249</a>, <a href="#Page_251">251</a>, <a href="#Page_308">308</a><br />
+<br />
+Rosse, Lord, his telescope, <a href="#Page_186">186</a>, <a href="#Page_268">268</a><br />
+<br />
+Rudolphine tables, <a href="#Page_65">65</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">S</span><br />
+<br />
+<span class="smcap">Scheiner</span>, <a href="#Page_107">107</a><br />
+<br />
+Sizzi, Francesca, an orthodox astronomer, <a href="#Page_106">106</a><br />
+<br />
+Snell, Willebrod, and the law of refraction, <a href="#Page_65">65</a><br />
+<br />
+Solar system, its fate, <a href="#Page_265">265</a><br />
+<br />
+Stars, a list of, <a href="#Page_307">307</a><br />
+<br />
+Struve, <a href="#Page_308">308</a>, <a href="#Page_310">310</a>, <a href="#Page_311">311</a>, <a href="#Page_313">313</a><br />
+<br />
+Stuart, Prof., quoted, <a href="#Page_52">52</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">T</span><br />
+<br />
+<span class="smcap">Tatius</span>, <a href="#Page_296">296</a><br />
+<br />
+Telescopes, early, <a href="#Page_96">96</a><br />
+<br />
+Thales, <a href="#Page_7">7</a>, <a href="#Page_140">140</a>, <a href="#Page_317">317</a><br />
+<br />
+Thomson, Sir William, <a href="#Page_367">367</a>, <a href="#Page_372">372</a>, <a href="#Page_373">373</a>, <a href="#Page_378">378</a>, <a href="#Page_394">394</a><br />
+<br />
+Tide-gauge, described, <a href="#Page_373">373</a>, <i>seq.</i><br />
+<br />
+Tides, <a href="#Page_354">354</a>, <i>seq.</i><br />
+<br />
+Time, is not exactly uniform, <a href="#Page_384">384</a><br />
+<br />
+Torricelli, <a href="#Page_133">133</a>, <a href="#Page_168">168</a><br />
+<br />
+Tycho, see Brah&eacute;, Tycho<br />
+<br />
+<br />
+<span style="margin-left: 4em;">V</span><br />
+<br />
+<span class="smcap">Vinci</span>, Leonardo da, <a href="#Page_9">9</a>, <a href="#Page_100">100</a>, <a href="#Page_144">144</a>, <a href="#Page_184">184</a><br />
+<br />
+Viviani, <a href="#Page_133">133</a>, <a href="#Page_168">168</a><br />
+<br />
+Voltaire, <a href="#Page_181">181</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">W</span><br />
+<br />
+<span class="smcap">Watson</span>, Prof., <a href="#Page_300">300</a><br />
+<br />
+Whewell, <a href="#Page_227">227</a><br />
+<br />
+Wren, Sir Christopher, <a href="#Page_188">188</a>, <a href="#Page_192">192</a>, <a href="#Page_193">193</a>, <a href="#Page_197">197</a><br />
+<br />
+<br />
+<span style="margin-left: 4em;">Z</span><br />
+<br />
+<span class="smcap">Zach</span>, Von, <a href="#Page_296">296</a>, <a href="#Page_299">299</a><br />
+<br />
+Zone of Asteroids, <a href="#Page_300">300</a>, <i>seq.</i><br />
+</p>
+
+<p class="center"><br />
+THE END.<br />
+<br />
+<small>RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.</small><br />
+</p>
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of the Project Gutenberg EBook of Pioneers of Science, by Oliver Lodge
+
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+</pre>
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+</body>
+</html>
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+The Project Gutenberg EBook of Pioneers of Science, by Oliver Lodge
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Pioneers of Science
+
+Author: Oliver Lodge
+
+Release Date: April 26, 2009 [EBook #28613]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK PIONEERS OF SCIENCE ***
+
+
+
+
+Produced by Audrey Longhurst, Greg Bergquist and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by The Internet Archive/American Libraries.)
+
+
+
+
+
+
+Transcriber's Note
+
+The punctuation and spelling from the original text have been faithfully
+preserved. Only obvious typographical errors have been corrected.
+
+There are several mathematical formulas within the text. They are
+represented as follows:
+ Superscripts: x^3
+ Subscripts: x_3
+ Square Root: [square root] Greek Letters: [pi], [theta].
+
+Greek star names are represented as [alpha], [gamma], for example.
+
+
+
+
+PIONEERS OF SCIENCE
+
+[Illustration]
+
+[Illustration: NEWTON
+
+_From the picture by Kneller, 1689, now at Cambridge_]
+
+
+
+
+  PIONEERS OF SCIENCE
+
+  BY
+  OLIVER LODGE, F.R.S.
+
+
+  PROFESSOR OF PHYSICS IN VICTORIA UNIVERSITY COLLEGE, LIVERPOOL
+
+  _WITH PORTRAITS AND OTHER ILLUSTRATIONS_
+
+
+  London
+  MACMILLAN AND CO.
+  AND NEW YORK
+  1893
+
+  RICHARD CLAY AND SONS, LIMITED,
+  LONDON AND BUNGAY.
+
+
+
+
+PREFACE
+
+
+This book takes its origin in a course of lectures on the history and
+progress of Astronomy arranged for me in the year 1887 by three of my
+colleagues (A.C.B., J.M., G.H.R.), one of whom gave the course its name.
+
+The lectures having been found interesting, it was natural to write them
+out in full and publish.
+
+If I may claim for them any merit, I should say it consists in their
+simple statement and explanation of scientific facts and laws. The
+biographical details are compiled from all readily available sources,
+there is no novelty or originality about them; though it is hoped that
+there may be some vividness. I have simply tried to present a living
+figure of each Pioneer in turn, and to trace his influence on the
+progress of thought.
+
+I am indebted to many biographers and writers, among others to Mr.
+E.J.C. Morton, whose excellent set of lives published by the S.P.C.K.
+saved me much trouble in the early part of the course.
+
+As we approach recent times the subject grows more complex, and the men
+more nearly contemporaries; hence the biographical aspect diminishes and
+the scientific treatment becomes fuller, but in no case has it been
+allowed to become technical and generally unreadable.
+
+To the friends (C.C.C., F.W.H.M., E.F.R.) who with great kindness have
+revised the proofs, and have indicated places where the facts could be
+made more readily intelligible by a clearer statement, I express my
+genuine gratitude.
+
+  UNIVERSITY COLLEGE, LIVERPOOL,
+  _November, 1892_.
+
+
+
+
+CONTENTS
+
+
+  _PART I_
+
+  LECTURE I
+
+                                                   PAGE
+
+  COPERNICUS AND THE MOTION OF THE EARTH              2
+
+
+  LECTURE II
+
+  TYCHO BRAHE AND THE EARLIEST OBSERVATORY           32
+
+
+  LECTURE III
+
+  KEPLER AND THE LAWS OF PLANETARY MOTION            56
+
+
+  LECTURE IV
+
+  GALILEO AND THE INVENTION OF THE TELESCOPE         80
+
+
+  LECTURE V
+
+  GALILEO AND THE INQUISITION                       108
+
+
+  LECTURE VI
+
+  DESCARTES AND HIS THEORY OF VORTICES              136
+
+
+  LECTURE VII
+
+  SIR ISAAC NEWTON                                  159
+
+
+  LECTURE VIII
+
+  NEWTON AND THE LAW OF GRAVITATION                 180
+
+
+  LECTURE IX
+
+  NEWTON'S "PRINCIPIA"                              203
+
+
+  _PART II_
+
+  LECTURE X
+
+  ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT      232
+
+
+  LECTURE XI
+
+  LAGRANGE AND LAPLACE--THE STABILITY OF THE SOLAR SYSTEM,
+  AND THE NEBULAR HYPOTHESIS                        254
+
+
+  LECTURE XII
+
+  HERSCHEL AND THE MOTION OF THE FIXED STARS        273
+
+
+  LECTURE XIII
+
+  THE DISCOVERY OF THE ASTEROIDS                    294
+
+
+  LECTURE XIV
+
+  BESSEL--THE DISTANCES OF THE STARS, AND THE DISCOVERY OF
+  STELLAR PLANETS                                   304
+
+
+  LECTURE XV
+
+  THE DISCOVERY OF NEPTUNE                          317
+
+
+  LECTURE XVI
+
+  COMETS AND METEORS                                331
+
+
+  LECTURE XVII
+
+  THE TIDES                                         353
+
+
+  LECTURE XVIII
+
+  THE TIDES, AND PLANETARY EVOLUTION                379
+
+
+
+
+ILLUSTRATIONS
+
+
+  FIG.                                                                PAGE
+
+  1. ARCHIMEDES                                                          8
+
+  2. LEONARDO DA VINCI                                                  10
+
+  3. COPERNICUS                                                         12
+
+  4. HOMERIC COSMOGONY                                                  15
+
+  5. EGYPTIAN SYMBOL OF THE UNIVERSE                                    16
+
+  6. HINDOO EARTH                                                       17
+
+  7. ORDER OF ANCIENT PLANETS CORRESPONDING TO THE DAYS OF
+     THE WEEK                                                           19
+
+  8. PTOLEMAIC SYSTEM                                                   20
+
+  9. SPECIMENS OF APPARENT PATHS OF VENUS AND OF MARS
+     AMONG THE STARS                                                    21
+
+  10. APPARENT EPICYCLIC ORBITS OF JUPITER AND SATURN                   22
+
+  11. EGYPTIAN SYSTEM                                                   24
+
+  12. TRUE ORBITS OF EARTH AND JUPITER                                  25
+
+  13. ORBITS OF MERCURY AND EARTH                                       25
+
+  14. COPERNICAN SYSTEM AS FREQUENTLY REPRESENTED                       26
+
+  15. SLOW MOVEMENT OF THE NORTH POLE IN A CIRCLE AMONG
+      THE STARS                                                         29
+
+  16. TYCHONIC SYSTEM, SHOWING THE SUN WITH ALL THE PLANETS
+      REVOLVING ROUND THE EARTH                                         38
+
+  17. PORTRAIT OF TYCHO                                                 41
+
+  18. EARLY OUT-DOOR QUADRANT OF TYCHO                                  43
+
+  19. MAP OF DENMARK, SHOWING THE ISLAND OF HUEN                        45
+
+  20. URANIBURG                                                         46
+
+  21. ASTROLABE                                                         47
+
+  22. TYCHO'S LARGE SEXTANT                                             48
+
+  23. THE QUADRANT IN URANIBURG                                         49
+
+  24. TYCHO'S FORM OF TRANSIT CIRCLE                                    50
+
+  25. A MODERN TRANSIT CIRCLE                                           51
+
+  26. ORBITS OF SOME OF THE PLANETS DRAWN TO SCALE                      60
+
+  27. MANY-SIDED POLYGON OR APPROXIMATE CIRCLE ENVELOPED
+      BY STRAIGHT LINES                                                 61
+
+  28. KEPLER'S IDEA OF THE REGULAR SOLIDS                               62
+
+  29. DIAGRAM OF EQUANT                                                 67
+
+  30. EXCENTRIC CIRCLE SUPPOSED TO BE DIVIDED INTO EQUAL AREAS          68
+
+  31. MODE OF DRAWING AN ELLIPSE                                        70
+
+  32. KEPLER'S DIAGRAM PROVING EQUABLE DESCRIPTION OF AREAS
+      FOR AN ELLIPSE                                                    71
+
+  33. DIAGRAM OF A PLANET'S VELOCITY IN DIFFERENT PARTS OF ITS ORBIT    72
+
+  34. PORTRAIT OF KEPLER                                                76
+
+  35. CURVE DESCRIBED BY A PROJECTILE                                   82
+
+  36. TWO FORMS OF PULSILOGY                                            87
+
+  37. TOWER OF PISA                                                     91
+
+  38. VIEW OF THE HALF-MOON IN SMALL TELESCOPE                          97
+
+  39. PORTION OF THE LUNAR SURFACE MORE HIGHLY MAGNIFIED                98
+
+  40. ANOTHER PORTION OF THE LUNAR SURFACE                              99
+
+  41. LUNAR LANDSCAPE SHOWING EARTH                                    100
+
+  42. GALILEO'S METHOD OF ESTIMATING THE HEIGHT OF LUNAR MOUNTAIN      101
+
+  43. SOME CLUSTERS AND NEBULAE                                         102
+
+  44. STAGES OF THE DISCOVERY OF JUPITER'S SATELLITES                  103
+
+  45. ECLIPSES OF JUPITER'S SATELLITES                                 105
+
+  46. OLD DRAWINGS OF SATURN BY DIFFERENT OBSERVERS, WITH
+      THE IMPERFECT INSTRUMENTS OF THAT DAY                            111
+
+  47. PHASES OF VENUS                                                  112
+
+  48. SUNSPOTS AS SEEN WITH LOW POWER                                  113
+
+  49. A PORTION OF THE SUN'S DISK AS SEEN IN A POWERFUL MODERN
+      TELESCOPE                                                        114
+
+  50. SATURN AND HIS RINGS                                             115
+
+  51. MAP OF ITALY                                                     118
+
+  52. PORTRAIT OF GALILEO                                              126
+
+  53. PORTRAIT OF DESCARTES                                            148
+
+  54. DESCARTES'S EYE DIAGRAM                                          151
+
+  55. DESCARTES'S DIAGRAM OF VORTICES FROM HIS "PRINCIPIA"             152
+
+  56. MANOR-HOUSE OF WOOLSTHORPE                                       162
+
+  57. PROJECTILE DIAGRAM                                               170
+
+  58.   }                                                            { 171
+  59.   } DIAGRAMS ILLUSTRATIVE OF THOSE NEAR THE BEGINNING          { 174
+  60.   } OF NEWTON'S "PRINCIPIA"                                    { 175
+  61-2. }                                                            { 175
+
+  63. PRISMATIC DISPERSION                                             182
+
+  64. A SINGLE CONSTITUENT OF WHITE LIGHT IS CAPABLE OF NO
+      MORE DISPERSION                                                  183
+
+  65. PARALLEL BEAM PASSING THROUGH A LENS                             184
+
+  66. NEWTON'S TELESCOPE                                               186
+
+  67. THE SEXTANT, AS NOW MADE                                         187
+
+  68. NEWTON WHEN YOUNG                                                196
+
+  69. SIR ISAAC NEWTON                                                 200
+
+  70. ANOTHER "PRINCIPIA" DIAGRAM                                      207
+
+  71. WELL-KNOWN MODEL EXHIBITING THE OBLATE SPHEROIDAL
+      FORM AS A CONSEQUENCE OF SPINNING ABOUT A CENTRAL
+      AXIS                                                             219
+
+  72. JUPITER                                                          221
+
+  73. DIAGRAM OF EYE LOOKING AT A LIGHT REFLECTED IN A DISTANT
+      MIRROR THROUGH THE TEETH OF A REVOLVING WHEEL                    238
+
+  74. FIZEAU'S WHEEL, SHOWING THE APPEARANCE OF DISTANT
+      IMAGE SEEN THROUGH ITS TEETH                                     239
+
+  75. ECLIPSES OF ONE OF JUPITER'S SATELLITES                          241
+
+  76. A TRANSIT INSTRUMENT FOR THE BRITISH ASTRONOMICAL EXPEDITION,
+      1874                                                             243
+
+  77. DIAGRAM OF EQUATORIALLY MOUNTED TELESCOPE                        245
+
+  78. ABERRATION DIAGRAM                                               250
+
+  79. SHOWING THE THREE CONJUNCTION PLACES IN THE ORBITS OF
+      JUPITER AND SATURN                                               259
+
+  80. LORD ROSSE'S DRAWING OF THE SPIRAL NEBULA IN CANES
+      VENATICI                                                         269
+
+  81. SATURN                                                           271
+
+  82. PRINCIPLE OF NEWTONIAN REFLECTOR                                 278
+
+  83. HERSCHEL'S 40-FOOT TELESCOPE                                     283
+
+  84. WILLIAM HERSCHEL                                                 285
+
+  85. CAROLINE HERSCHEL                                                287
+
+  86. DOUBLE STARS                                                     288
+
+  87. OLD DRAWING OF THE CLUSTER IN HERCULES                           290
+
+  88. OLD DRAWING OF THE ANDROMEDA NEBULA                              291
+
+  89. THE GREAT NEBULA IN ORION                                        292
+
+  90. PLANETARY ORBITS TO SCALE                                        297
+
+  91. DIAGRAM ILLUSTRATING PARALLAX                                    307
+
+  92. THE KOeNIGSBERG HELIOMETER                                        312
+
+  93. PERTURBATIONS OF URANUS                                          320
+
+  94. URANUS' AND NEPTUNE'S RELATIVE POSITIONS                         325
+
+  95. METEORITE                                                        333
+
+  96. METEOR STREAM CROSSING FIELD OF TELESCOPE                        334
+
+  97. DIAGRAM OF DIRECTION OF EARTH'S ORBITAL MOTION                   335
+
+  98. PARABOLIC AND ELLIPTIC ORBITS                                    340
+
+  99. ORBIT OF HALLEY'S COMET                                          341
+
+  100. VARIOUS APPEARANCES OF HALLEY'S COMET WHEN LAST SEEN            342
+
+  101. HEAD OF DONATI'S COMET OF 1858                                  343
+
+  102. COMET                                                           344
+
+  103. ENCKE'S COMET                                                   345
+
+  104. BIELA'S COMET AS LAST SEEN IN TWO PORTIONS                      346
+
+  105. RADIANT POINT PERSPECTIVE                                       348
+
+  106. PRESENT ORBIT OF NOVEMBER METEORS                               349
+
+  107. ORBIT OF NOVEMBER METEORS BEFORE AND AFTER ENCOUNTER
+       WITH URANUS                                                     351
+
+  108. THE MERSEY                                                      355
+
+  109. CO-TIDAL LINES, SHOWING THE WAY THE TIDAL WAVE
+       REACHES THE BRITISH ISLES FROM THE ATLANTIC                     359
+
+  110. WHIRLING EARTH MODEL                                            364
+
+  111. EARTH AND MOON MODEL                                            365
+
+  112. EARTH AND MOON (EARTH'S ROTATION NEGLECTED)                     366
+
+  113. MAPS SHOWING HOW COMPARATIVELY FREE FROM LAND OBSTRUCTION
+       THE OCEAN IN THE SOUTHERN HEMISPHERE IS                         369
+
+  114. SPRING AND NEAP TIDES                                           370
+
+  115. TIDAL CLOCK                                                     371
+
+  116. SIR WILLIAM THOMSON (LORD KELVIN)                               373
+
+  117. TIDE-GAUGE FOR RECORDING LOCAL TIDES                            375
+
+  118. HARMONIC ANALYZER                                               375
+
+  119. TIDE-PREDICTER                                                  376
+
+  120. WEEKLY SHEET OF CURVES                                          377
+
+
+
+
+PIONEERS OF SCIENCE
+
+
+
+
+PART I
+
+_FROM DUSK TO DAYLIGHT_
+
+
+
+
+DATES AND SUMMARY OF FACTS FOR LECTURE I
+
+
+_Physical Science of the Ancients._ Thales 640 B.C., Anaximander 610
+B.C., PYTHAGORAS 600 B.C., Anaxagoras 500 B.C., Eudoxus 400 B.C.,
+ARISTOTLE 384 B.C., Aristarchus 300 B.C., ARCHIMEDES 287 B.C.,
+Eratosthenes 276 B.C., HIPPARCHUS 160 B.C., Ptolemy 100 A.D.
+
+_Science of the Middle Ages._ Cultivated only among the Arabs; largely
+in the forms of astrology, alchemy, and algebra.
+
+_Return of Science to Europe._ Roger Bacon 1240, Leonardo da Vinci 1480,
+(Printing 1455), Columbus 1492, Copernicus 1543.
+
+_A sketch of Copernik's life and work._ Born 1473 at Thorn in Poland.
+Studied mathematics at Bologna. Became an ecclesiastic. Lived at
+Frauenburg near mouth of Vistula. Substituted for the apparent motion of
+the heavens the real motion of the earth. Published tables of planetary
+motions. Motion still supposed to be in epicycles. Worked out his ideas
+for 36 years, and finally dedicated his work to the Pope. Died just as
+his book was printed, aged 72, a century before the birth of Newton. A
+colossal statue by Thorwaldsen erected at Warsaw in 1830.
+
+
+
+
+PIONEERS OF SCIENCE
+
+
+
+
+LECTURE I
+
+COPERNICUS AND THE MOTION OF THE EARTH
+
+
+The ordinary run of men live among phenomena of which they know nothing
+and care less. They see bodies fall to the earth, they hear sounds, they
+kindle fires, they see the heavens roll above them, but of the causes
+and inner working of the whole they are ignorant, and with their
+ignorance they are content.
+
+"Understand the structure of a soap-bubble?" said a cultivated literary
+man whom I know; "I wouldn't cross the street to know it!"
+
+And if this is a prevalent attitude now, what must have been the
+attitude in ancient times, when mankind was emerging from savagery, and
+when history seems composed of harassments by wars abroad and
+revolutions at home? In the most violently disturbed times indeed, those
+with which ordinary history is mainly occupied, science is quite
+impossible. It needs as its condition, in order to flourish, a fairly
+quiet, untroubled state, or else a cloister or university removed from
+the din and bustle of the political and commercial world. In such places
+it has taken its rise, and in such peaceful places and quiet times true
+science will continue to be cultivated.
+
+The great bulk of mankind must always remain, I suppose, more or less
+careless of scientific research and scientific result, except in so far
+as it affects their modes of locomotion, their health and pleasure, or
+their purse.
+
+But among a people hurried and busy and preoccupied, some in the pursuit
+of riches, some in the pursuit of pleasure, and some, the majority, in
+the struggle for existence, there arise in every generation, here and
+there, one or two great souls--men who seem of another age and country,
+who look upon the bustle and feverish activity and are not infected by
+it, who watch others achieving prizes of riches and pleasure and are not
+disturbed, who look on the world and the universe they are born in with
+quite other eyes. To them it appears not as a bazaar to buy and to sell
+in; not as a ladder to scramble up (or down) helter-skelter without
+knowing whither or why; but as a fact--a great and mysterious fact--to
+be pondered over, studied, and perchance in some small measure
+understood. By the multitude these men were sneered at as eccentric or
+feared as supernatural. Their calm, clear, contemplative attitude seemed
+either insane or diabolic; and accordingly they have been pitied as
+enthusiasts or killed as blasphemers. One of these great souls may have
+been a prophet or preacher, and have called to his generation to bethink
+them of why and what they were, to struggle less and meditate more, to
+search for things of true value and not for dross. Another has been a
+poet or musician, and has uttered in words or in song thoughts dimly
+possible to many men, but by them unutterable and left inarticulate.
+Another has been influenced still more _directly_ by the universe around
+him, has felt at times overpowered by the mystery and solemnity of it
+all, and has been impelled by a force stronger than himself to study it,
+patiently, slowly, diligently; content if he could gather a few crumbs
+of the great harvest of knowledge, happy if he could grasp some great
+generalization or wide-embracing law, and so in some small measure enter
+into the mind and thought of the Designer of all this wondrous frame of
+things.
+
+These last have been the men of science, the great and heaven-born men
+of science; and they are few. In our own day, amid the throng of
+inventions, there are a multitude of small men using the name of science
+but working for their own ends, jostling and scrambling just as they
+would jostle and scramble in any other trade or profession. These may be
+workers, they may and do advance knowledge, but they are never pioneers.
+Not to them is it given to open out great tracts of unexplored
+territory, or to view the promised land as from a mountain-top. Of them
+we shall not speak; we will concern ourselves only with the greatest,
+the epoch-making men, to whose life and work we and all who come after
+them owe so much. Such a man was Thales. Such was Archimedes,
+Hipparchus, Copernicus. Such pre-eminently was Newton.
+
+Now I am not going to attempt a history of science. Such a work in ten
+lectures would be absurd. I intend to pick out a few salient names here
+and there, and to study these in some detail, rather than by attempting
+to deal with too many to lose individuality and distinctness.
+
+We know so little of the great names of antiquity, that they are for
+this purpose scarcely suitable. In some departments the science of the
+Greeks was remarkable, though it is completely overshadowed by their
+philosophy; yet it was largely based on what has proved to be a wrong
+method of procedure, viz the introspective and conjectural, rather than
+the inductive and experimental methods. They investigated Nature by
+studying their own minds, by considering the meanings of words, rather
+than by studying things and recording phenomena. This wrong (though by
+no means, on the face of it, absurd) method was not pursued exclusively,
+else would their science have been valueless, but the influence it had
+was such as materially to detract from the value of their speculations
+and discoveries. For when truth and falsehood are inextricably woven
+into a statement, the truth is as hopelessly hidden as if it had never
+been stated, for we have no criterion to distinguish the false from the
+true.
+
+[Illustration: FIG. 1.--Archimedes.]
+
+Besides this, however, many of their discoveries were ultimately lost to
+the world, some, as at Alexandria, by fire--the bigoted work of a
+Mohammedan conqueror--some by irruption of barbarians; and all were
+buried so long and so completely by the night of the dark ages, that
+they had to be rediscovered almost as absolutely and completely as
+though they had never been. Some of the names of antiquity we shall have
+occasion to refer to; so I have arranged some of them in chronological
+order on page 4, and as a representative one I may specially emphasize
+Archimedes, one of the greatest men of science there has ever been, and
+the father of physics.
+
+The only effective link between the old and the new science is afforded
+by the Arabs. The dark ages come as an utter gap in the scientific
+history of Europe, and for more than a thousand years there was not a
+scientific man of note except in Arabia; and with the Arabs knowledge
+was so mixed up with magic and enchantment that one cannot contemplate
+it with any degree of satisfaction, and little real progress was made.
+In some of the _Waverley Novels_ you can realize the state of matters in
+these times; and you know how the only approach to science is through
+some Arab sorcerer or astrologer, maintained usually by a monarch, and
+consulted upon all great occasions, as the oracles were of old.
+
+In the thirteenth century, however, a really great scientific man
+appeared, who may be said to herald the dawn of modern science in
+Europe. This man was Roger Bacon. He cannot be said to do more than
+herald it, however, for we must wait two hundred years for the next name
+of great magnitude; moreover he was isolated, and so far in advance of
+his time that he left no followers. His own work suffered from the
+prevailing ignorance, for he was persecuted and imprisoned, not for the
+commonplace and natural reason that he frightened the Church, but merely
+because he was eccentric in his habits and knew too much.
+
+The man I spoke of as coming two hundred years later is Leonardo da
+Vinci. True he is best known as an artist, but if you read his works you
+will come to the conclusion that he was the most scientific artist who
+ever lived. He teaches the laws of perspective (then new), of light and
+shade, of colour, of the equilibrium of bodies, and of a multitude of
+other matters where science touches on art--not always quite correctly
+according to modern ideas, but in beautiful and precise language. For
+clear and conscious power, for wide-embracing knowledge and skill,
+Leonardo is one of the most remarkable men that ever lived.
+
+About this time the tremendous invention of printing was achieved, and
+Columbus unwittingly discovered the New World. The middle of the next
+century must be taken as the real dawn of modern science; for the year
+1543 marks the publication of the life-work of Copernicus.
+
+[Illustration: FIG. 2.--Leonardo da Vinci.]
+
+Nicolas Copernik was his proper name. Copernicus is merely the Latinized
+form of it, according to the then prevailing fashion. He was born at
+Thorn, in Polish Prussia, in 1473. His father is believed to have been a
+German. He graduated at Cracow as doctor in arts and medicine, and was
+destined for the ecclesiastical profession. The details of his life are
+few; it seems to have been quiet and uneventful, and we know very little
+about it. He was instructed in astronomy at Cracow, and learnt
+mathematics at Bologna. Thence he went to Rome, where he was made
+Professor of Mathematics; and soon afterwards he went into orders. On
+his return home, he took charge of the principal church in his native
+place, and became a canon. At Frauenburg, near the mouth of the Vistula,
+he lived the remainder of his life. We find him reporting on coinage for
+the Government, but otherwise he does not appear as having entered into
+the life of the times.
+
+He was a quiet, scholarly monk of studious habits, and with a reputation
+which drew to him several earnest students, who received _viva voce_
+instruction from him; so, in study and meditation, his life passed.
+
+He compiled tables of the planetary motions which were far more correct
+than any which had hitherto appeared, and which remained serviceable for
+long afterwards. The Ptolemaic system of the heavens, which had been the
+orthodox system all through the Christian era, he endeavoured to improve
+and simplify by the hypothesis that the sun was the centre of the system
+instead of the earth; and the first consequences of this change he
+worked out for many years, producing in the end a great book: his one
+life-work. This famous work, "De Revolutionibus Orbium Coelestium,"
+embodied all his painstaking calculations, applied his new system to
+each of the bodies in the solar system in succession, and treated
+besides of much other recondite matter. Towards the close of his life it
+was put into type. He can scarcely be said to have lived to see it
+appear, for he was stricken with paralysis before its completion; but a
+printed copy was brought to his bedside and put into his hands, so that
+he might just feel it before he died.
+
+[Illustration: FIG. 3.--Copernicus.]
+
+That Copernicus was a giant in intellect or power--such as had lived in
+the past, and were destined to live in the near future--I see no reason
+whatever to believe. He was just a quiet, earnest, patient, and
+God-fearing man, a deep student, an unbiassed thinker, although with no
+specially brilliant or striking gifts; yet to him it was given to effect
+such a revolution in the whole course of man's thoughts as is difficult
+to parallel.
+
+You know what the outcome of his work was. It proved--he did not merely
+speculate, he proved--that the earth is a planet like the others, and
+that it revolves round the sun.
+
+Yes, it can be summed up in a sentence, but what a revelation it
+contains. If you have never made an effort to grasp the full
+significance of this discovery you will not appreciate it. The doctrine
+is very familiar to us now, we have heard it, I suppose, since we were
+four years old, but can you realize it? I know it was a long time before
+I could. Think of the solid earth, with trees and houses, cities and
+countries, mountains and seas--think of the vast tracts of land in Asia,
+Africa, and America--and then picture the whole mass spinning like a
+top, and rushing along its annual course round the sun at the rate of
+nineteen miles every second.
+
+Were we not accustomed to it, the idea would be staggering. No wonder it
+was received with incredulity. But the difficulties of the conception
+are not only physical, they are still more felt from the speculative and
+theological points of view. With this last, indeed, the reconcilement
+cannot be considered complete even yet. Theologians do not, indeed, now
+_deny_ the fact of the earth's subordination in the scheme of the
+universe, but many of them ignore it and pass it by. So soon as the
+Church awoke to a perception of the tremendous and revolutionary import
+of the new doctrines, it was bound to resist them or be false to its
+traditions. For the whole tenor of men's thought must have been changed
+had they accepted it. If the earth were not the central and
+all-important body in the universe, if the sun and planets and stars
+were not attendant and subsidiary lights, but were other worlds larger
+and perhaps superior to ours, where was man's place in the universe?
+and where were the doctrines they had maintained as irrefragable? I by
+no means assert that the new doctrines were really utterly
+irreconcilable with the more essential parts of the old dogmas, if only
+theologians had had patience and genius enough to consider the matter
+calmly. I suppose that in that case they might have reached the amount
+of reconciliation at present attained, and not only have left scientific
+truth in peace to spread as it could, but might perhaps themselves have
+joined the band of earnest students and workers, as so many of the
+higher Catholic clergy do at the present day.
+
+But this was too much to expect. Such a revelation was not to be
+accepted in a day or in a century--the easiest plan was to treat it as a
+heresy, and try to crush it out.
+
+Not in Copernik's life, however, did they perceive the dangerous
+tendency of the doctrine--partly because it was buried in a ponderous
+and learned treatise not likely to be easily understood; partly,
+perhaps, because its propounder was himself an ecclesiastic; mainly
+because he was a patient and judicious man, not given to loud or
+intolerant assertion, but content to state his views in quiet
+conversation, and to let them gently spread for thirty years before he
+published them. And, when he did publish them, he used the happy device
+of dedicating his great book to the Pope, and a cardinal bore the
+expense of printing it. Thus did the Roman Church stand sponsor to a
+system of truth against which it was destined in the next century to
+hurl its anathemas, and to inflict on its conspicuous adherents torture,
+imprisonment, and death.
+
+To realize the change of thought, the utterly new view of the universe,
+which the Copernican theory introduced, we must go back to preceding
+ages, and try to recall the views which had been held as probable
+concerning the form of the earth and the motion of the heavenly bodies.
+
+[Illustration: FIG. 4.--Homeric Cosmogony.]
+
+The earliest recorded notion of the earth is the very natural one that
+it is a flat area floating in an illimitable ocean. The sun was a god
+who drove his chariot across the heavens once a day; and Anaxagoras was
+threatened with death and punished with banishment for teaching that the
+sun was only a ball of fire, and that it might perhaps be as big as the
+country of Greece. The obvious difficulty as to how the sun got back to
+the east again every morning was got over--not by the conjecture that he
+went back in the dark, nor by the idea that there was a fresh sun every
+day; though, indeed, it was once believed that the moon was created once
+a month, and periodically cut up into stars--but by the doctrine that in
+the northern part of the earth was a high range of mountains, and that
+the sun travelled round on the surface of the sea behind these.
+Sometimes, indeed, you find a representation of the sun being rowed
+round in a boat. Later on it was perceived to be necessary that the sun
+should be able to travel beneath the earth, and so the earth was
+supposed to be supported on pillars or on roots, or to be a dome-shaped
+body floating in air--much like Dean Swift's island of Laputa. The
+elephant and tortoise of the Hindu earth are, no doubt, emblematic or
+typical, not literal.
+
+[Illustration: FIG. 5.--Egyptian Symbol of the Universe.
+
+The earth a figure with leaves, the heaven a figure with stars, the
+principle of equilibrium and support, the boats of the rising and
+setting sun.]
+
+Aristotle, however, taught that the earth must be a sphere, and used all
+the orthodox arguments of the present children's geography-books about
+the way you see ships at sea, and about lunar eclipses.
+
+To imagine a possible antipodes must, however, have been a tremendous
+difficulty in the way of this conception of a sphere, and I scarcely
+suppose that any one can at that time have contemplated the possibility
+of such upside-down regions being inhabited. I find that intelligent
+children invariably feel the greatest difficulty in realizing the
+existence of inhabitants on the opposite side of the earth. Stupid
+children, like stupid persons in general, will of course believe
+anything they are told, and much good may the belief do them; but the
+kind of difficulties felt by intelligent and thoughtful children are
+most instructive, since it is quite certain that the early philosophers
+must have encountered and overcome those very same difficulties by their
+own genius.
+
+[Illustration: FIG. 6.--Hindoo Earth.]
+
+However, somehow or other the conception of a spherical earth was
+gradually grasped, and the heavenly bodies were perceived all to revolve
+round it: some moving regularly, as the stars, all fixed together into
+one spherical shell or firmament; some moving irregularly and apparently
+anomalously--these irregular bodies were therefore called planets [or
+wanderers]. Seven of them were known, viz. Moon, Mercury, Venus, Sun,
+Mars, Jupiter, Saturn, and there is little doubt that this number seven,
+so suggested, is the origin of the seven days of the week.
+
+     The above order of the ancient planets is that of their supposed
+     distance from the earth. Not always, however, are they thus quoted
+     by the ancients: sometimes the sun is supposed nearer than Mercury
+     or Venus. It has always been known that the moon was the nearest of
+     the heavenly bodies; and some rough notion of its distance was
+     current. Mars, Jupiter, and Saturn were placed in that order
+     because that is the order of their apparent motions, and it was
+     natural to suppose that the slowest moving bodies were the furthest
+     off.
+
+     The order of the days of the week shows what astrologers considered
+     to be the order of the planets; on their system of each successive
+     hour of the day being ruled over by the successive planets taken in
+     order. The diagram (fig. 7) shows that if the Sun rule the first
+     hour of a certain day (thereby giving its name to the day) Venus
+     will rule the second hour, Mercury the third, and so on; the Sun
+     will thus be found to rule the eighth, fifteenth, and twenty-second
+     hour of that day, Venus the twenty-third, and Mercury the
+     twenty-fourth hour; so the Moon will rule the first hour of the
+     next day, which will therefore be Monday. On the same principle
+     (numbering round the hours successively, with the arrows) the first
+     hour of the next day will be found to be ruled by Mars, or by the
+     Saxon deity corresponding thereto; the first hour of the day after,
+     by Mercury (_Mercredi_), and so on (following the straight lines of
+     the pattern).
+
+     The order of the planets round the circle counter-clockwise, _i.e._
+     the direction of their proper motions, is that quoted above in the
+     text.
+
+To explain the motion of the planets and reduce them to any sort of law
+was a work of tremendous difficulty. The greatest astronomer of ancient
+times was Hipparchus, and to him the system known as the Ptolemaic
+system is no doubt largely due. But it was delivered to the world mainly
+by Ptolemy, and goes by his name. This was a fine piece of work, and a
+great advance on anything that had gone before; for although it is of
+course saturated with error, still it is based on a large substratum of
+truth. Its superiority to all the previously mentioned systems is
+obvious. And it really did in its more developed form describe the
+observed motions of the planets.
+
+Each planet was, in the early stages of this system, as taught, say, by
+Eudoxus, supposed to be set in a crystal sphere, which revolved so as to
+carry the planet with it. The sphere had to be of crystal to account for
+the visibility of other planets and the stars through it. Outside the
+seven planetary spheres, arranged one inside the other, was a still
+larger one in which were set the stars. This was believed to turn all
+the others, and was called the _primum mobile_. The whole system was
+supposed to produce, in its revolution, for the few privileged to hear
+the music of the spheres, a sound as of some magnificent harmony.
+
+[Illustration: FIG. 7.--Order of ancient planets corresponding to the
+days of the week.]
+
+The enthusiastic disciples of Pythagoras believed that their master was
+privileged to hear this noble chant; and far be it from us to doubt
+that the rapt and absorbing pleasure of contemplating the harmony of
+nature, to a man so eminently great as Pythagoras, must be truly and
+adequately represented by some such poetic conception.
+
+[Illustration: FIG. 8.--Ptolemaic system.]
+
+The precise kind of motion supposed to be communicated from the _primum
+mobile_ to the other spheres so as to produce the observed motions of
+the planets was modified and improved by various philosophers until it
+developed into the epicyclic train of Hipparchus and of Ptolemy.
+
+It is very instructive to observe a planet (say Mars or Jupiter) night
+after night and plot down its place with reference to the fixed stars
+on a celestial globe or star-map. Or, instead of direct observation by
+alignment with known stars, it is easier to look out its right ascension
+and declination in _Whitaker's Almanac_, and plot those down. If this be
+done for a year or two, it will be found that the motion of the planet
+is by no means regular, but that though on the whole it advances it
+sometimes is stationary and sometimes goes back.[1]
+
+[Illustration: FIG. 9.--Specimens of Apparent paths of Venus and of Mars
+among the stars.]
+
+[Illustration: FIG. 10.--Apparent epicyclic orbits of Jupiter and
+Saturn; the Earth being supposed fixed at the centre, with the Sun
+revolving in a small circle. A loop is made by each planet every year.]
+
+These "stations" and "retrogressions" of the planets were well known to
+the ancients. It was not to be supposed for a moment that the crystal
+spheres were subject to any irregularity, neither was uniform circular
+motion to be readily abandoned; so it was surmised that the main sphere
+carried, not the planet itself, but the centre or axis of a subordinate
+sphere, and that the planet was carried by this. The minor sphere could
+be allowed to revolve at a different uniform pace from the main sphere,
+and so a curve of some complexity could be obtained.
+
+A curve described in space by a point of a circle or sphere, which
+itself is carried along at the same time, is some kind of cycloid; if
+the centre of the tracing circle travels along a straight line, we get
+the ordinary cycloid, the curve traced in air by a nail on a
+coach-wheel; but if the centre of the tracing circle be carried round
+another circle the curve described is called an epicycloid. By such
+curves the planetary stations and retrogressions could be explained. A
+large sphere would have to revolve once for a "year" of the particular
+planet, carrying with it a subsidiary sphere in which the planet was
+fixed; this latter sphere revolving once for a "year" of the earth. The
+actual looped curve thus described is depicted for Jupiter and Saturn in
+the annexed diagram (fig. 10.)
+
+     It was long ago perceived that real material spheres were
+     unnecessary; such spheres indeed, though possibly transparent to
+     light, would be impermeable to comets: any other epicyclic gearing
+     would serve, and as a mere description of the motion it is simpler
+     to think of a system of jointed bars, one long arm carrying a
+     shorter arm, the two revolving at different rates, and the end of
+     the short one carrying the planet. This does all that is needful
+     for the first approximation to a planet's motion. In so far as the
+     motion cannot be thus truly stated, the short arm may be supposed
+     to carry another, and that another, and so on, so that the
+     resultant motion of the planet is compounded of a large number of
+     circular motions of different periods; by this device any required
+     amount of complexity could be attained. We shall return to this at
+     greater length in Lecture III.
+
+     The main features of the motion, as shown in the diagram, required
+     only two arms for their expression; one arm revolving with the
+     average motion of the planet, and the other revolving with the
+     apparent motion of the sun, and always pointing in the same
+     direction as the single arm supposed to carry the sun. This last
+     fact is of course because the motion to be represented does not
+     really belong to the planet at all, but to the earth, and so all
+     the main epicyclic motions for the superior planets were the same.
+     As for the inferior planets (Mercury and Venus) they only appear
+     to oscillate like the bob of a pendulum about the sun, and so it is
+     very obvious that they must be really revolving round it. An
+     ancient Egyptian system perceived this truth; but the Ptolemaic
+     system imagined them to revolve round the earth like the rest, with
+     an artificial system of epicycles to prevent their ever getting far
+     away from the neighbourhood of the sun.
+
+     It is easy now to see how the Copernican system explains the main
+     features of planetary motion, the stations and retrogressions,
+     quite naturally and without any complexity.
+
+     [Illustration: FIG. 11.--Egyptian system.]
+
+     Let the outer circle represent the orbit of Jupiter, and the inner
+     circle the orbit of the earth, which is moving faster than Jupiter
+     (since Jupiter takes 4332 days to make one revolution); then
+     remember that the apparent position of Jupiter is referred to the
+     infinitely distant fixed stars and refer to fig. 12.
+
+     Let E_1, E_2, &c., be successive positions of the earth; J_1,
+     J_2, &c., corresponding positions of Jupiter. Produce the lines
+     E_1 J_1, E_2 J_2, &c., to an enormously greater circle
+     outside, and it will be seen that the termination of these lines,
+     representing apparent positions of Jupiter among the stars,
+     advances while the earth goes from E_1 to E_3; is almost
+     stationary from somewhere about E_3 to E_4; and recedes from
+     E_4 to E_5; so that evidently the recessions of Jupiter are
+     only apparent, and are due to the orbital motion of the earth. The
+     apparent complications in the path of Jupiter, shown in Fig. 10,
+     are seen to be caused simply by the motion of the earth, and to be
+     thus completely and easily explained.
+
+     [Illustration: FIG. 12.--True orbits of Earth and Jupiter.]
+
+     The same thing for an inferior planet, say Mercury, is even still
+     more easily seen (_vide_ figure 13).
+
+     The motion of Mercury is direct from M'' to M''', retrograde from
+     M''' to M'', and stationary at M'' and M'''. It appears to
+     oscillate, taking 72.5 days for its direct swing, and 43.5 for its
+     return swing.
+
+     [Illustration: FIG. 13.--Orbit of Mercury and Earth.]
+
+     On this system no artificiality is required to prevent Mercury's
+     ever getting far from the sun: the radius of its orbit limits its
+     real and apparent excursions. Even if the earth were stationary,
+     the motions of Mercury and Venus would not be _essentially_
+     modified, but the stations and retrogressions of the superior
+     planets, Mars, Jupiter, &c., would wholly cease.
+
+     The complexity of the old mode of regarding apparent motion may be
+     illustrated by the case of a traveller in a railway train unaware
+     of his own motion. It is as though trees, hedges, distant objects,
+     were all flying past him and contorting themselves as you may see
+     the furrows of a ploughed field do when travelling, while you
+     yourself seem stationary amidst it all. How great a simplicity
+     would be introduced by the hypothesis that, after all, these things
+     might be stationary and one's self moving.
+
+[Illustration: FIG. 14.--Copernican system as frequently represented.
+But the cometary orbit is a much later addition, and no attempt is made
+to show the relative distances of the planets.]
+
+Now you are not to suppose that the system of Copernicus swept away the
+entire doctrine of epicycles; that doctrine can hardly be said to be
+swept away even now. As a description of a planet's motion it is not
+incorrect, though it is geometrically cumbrous. If you describe the
+motion of a railway train by stating that every point on the rim of each
+wheel describes a cycloid with reference to the earth, and a circle with
+reference to the train, and that the motion of the train is compounded
+of these cycloidal and circular motions, you will not be saying what is
+false, only what is cumbrous.
+
+The Ptolemaic system demanded large epicycles, depending on the motion
+of the earth, these are what Copernicus overthrew; but to express the
+minuter details of the motion smaller epicycles remained, and grew more
+and more complex as observations increased in accuracy, until a greater
+man than either Copernicus or Ptolemy, viz. Kepler, replaced them all by
+a simple ellipse.
+
+One point I must not omit from this brief notice of the work of
+Copernicus. Hipparchus had, by most sagacious interpretation of certain
+observations of his, discovered a remarkable phenomenon called the
+precession of the equinoxes. It was a discovery of the first magnitude,
+and such as would raise to great fame the man who should have made it in
+any period of the world's history, even the present. It is scarcely
+expressible in popular language, and without some technical terms; but I
+can try.
+
+The plane of the earth's orbit produced into the sky gives the apparent
+path of the sun throughout a year. This path is known as the ecliptic,
+because eclipses only happen when the moon is in it. The sun keeps to it
+accurately, but the planets wander somewhat above and below it (fig. 9),
+and the moon wanders a good deal. It is manifest, however, in order that
+there may be an eclipse of any kind, that a straight line must be able
+to be drawn through earth and moon and sun (not necessarily through
+their centres of course), and this is impossible unless some parts of
+the three bodies are in one plane, viz. the ecliptic, or something very
+near it. The ecliptic is a great circle of the sphere, and is usually
+drawn on both celestial and terrestrial globes.
+
+The earth's equator also produced into the sky, where it may still be
+called the equator (sometimes it is awkwardly called "the equinoctial"),
+gives another great circle inclined to the ecliptic and cutting it at
+two opposite points, labelled respectively [Aries symbol] and [Libra
+symbol], and together called "the equinoxes." The reason for the name is
+that when the sun is in that part of the ecliptic it is temporarily also
+on the equator, and hence is symmetrically situated with respect to the
+earth's axis of rotation, and consequently day and night are equal all
+over the earth.
+
+Well, Hipparchus found, by plotting the position of the sun for a long
+time,[2] that these points of intersection, or equinoxes, were not
+stationary from century to century, but slowly moved among the stars,
+moving as it were to meet the sun, so that he gets back to one of these
+points again 20 minutes 23-1/4 seconds before it has really completed a
+revolution, _i.e._ before the true year is fairly over. This slow
+movement forward of the goal-post is called precession--the precession
+of the equinoxes. (One result of it is to shorten our years by about 20
+minutes each; for the shortened period has to be called a year, because
+it is on the position of the sun with respect to the earth's axis that
+our seasons depend.) Copernicus perceived that, assuming the motion of
+the earth, a clearer account of this motion could be given. The ordinary
+approximate statement concerning the earth's axis is that it remains
+parallel to itself, _i.e._ has a fixed direction as the earth moves
+round the sun. But if, instead of being thus fixed, it be supposed to
+have a slow movement of revolution, so that it traces out a cone in the
+course of about 26,000 years, then, since the equator of course goes
+with it, the motion of its intersection with the fixed ecliptic is so
+far accounted for. That is to say, the precession of the equinoxes is
+seen to be dependent on, and caused by, a slow conical movement of the
+earth's axis.
+
+The prolongation of each end of the earth's axis into the sky, or the
+celestial north and south poles, will thus slowly trace out an
+approximate circle among the stars; and the course of the north pole
+during historic time is exhibited in the annexed diagram.
+
+It is now situated near one of the stars of the Lesser Bear, which we
+therefore call the Pole star; but not always was it so, nor will it be
+so in the future. The position of the north pole 4000 years ago is shown
+in the figure; and a revolution will be completed in something like
+26,000 years.[3]
+
+[Illustration: FIG. 15.--Slow movement of the north pole in a circle
+among the stars. (Copied from Sir R. Ball.)]
+
+This perception of the conical motion of the earth's axis was a
+beautiful generalization of Copernik's, whereby a multitude of facts
+were grouped into a single phenomenon. Of course he did not explain the
+motion of the axis itself. He stated the fact that it so moved, and I do
+not suppose it ever struck him to seek for an explanation.
+
+An explanation was given later, and that a most complete one; but the
+idea even of seeking for it is a brilliant and striking one: the
+achievement of the explanation by a single individual in the way it
+actually was accomplished is one of the most astounding things in the
+history of science; and were it not that the same individual
+accomplished a dozen other things, equally and some still more
+extraordinary, we should rank that man as one of the greatest
+astronomers that ever lived.
+
+As it is, he is Sir Isaac Newton.
+
+We are to remember, then, as the life-work of Copernicus, that he placed
+the sun in its true place as the centre of the solar system, instead of
+the earth; that he greatly simplified the theory of planetary motion by
+this step, and also by the simpler epicyclic chain which now sufficed,
+and which he worked out mathematically; that he exhibited the precession
+of the equinoxes (discovered by Hipparchus) as due to a conical motion
+of the earth's axis; and that, by means of his simpler theory and more
+exact planetary tables, he reduced to some sort of order the confused
+chaos of the Ptolemaic system, whose accumulation of complexity and of
+outstanding errors threatened to render astronomy impossible by the mere
+burden of its detail.
+
+There are many imperfections in his system, it is true; but his great
+merit is that he dared to look at the facts of Nature with his own eyes,
+unhampered by the prejudice of centuries. A system venerable with age,
+and supported by great names, was universally believed, and had been
+believed for centuries. To doubt this system, and to seek after another
+and better one, at a time when all men's minds were governed by
+tradition and authority, and when to doubt was sin--this required a
+great mind and a high character. Such a mind and such a character had
+this monk of Frauenburg. And it is interesting to notice that the
+so-called religious scruples of smaller and less truly religious men did
+not affect Copernicus; it was no dread of consequences to one form of
+truth that led him to delay the publication of the other form of truth
+specially revealed to him. In his dedication he says:--
+
+"If there be some babblers who, though ignorant of all mathematics, take
+upon them to judge of these things, and dare to blame and cavil at my
+work, because of some passage of Scripture which they have wrested to
+their own purpose, I regard them not, and will not scruple to hold their
+judgment in contempt."
+
+I will conclude with the words of one of his biographers (Mr. E.J.C.
+Morton):--
+
+"Copernicus cannot be said to have flooded with light the dark places of
+nature--in the way that one stupendous mind subsequently did--but still,
+as we look back through the long vista of the history of science, the
+dim Titanic figure of the old monk seems to rear itself out of the dull
+flats around it, pierces with its head the mists that overshadow them,
+and catches the first gleam of the rising sun,
+
+  "'... like some iron peak, by the Creator
+    Fired with the red glow of the rushing morn.'"
+
+
+
+
+DATES AND SUMMARY OF FACTS FOR LECTURE II
+
+
+Copernicus lived from 1473 to 1543, and was contemporary with Paracelsus
+and Raphael.
+
+  Tycho Brahe     from 1546 to 1601.
+  Kepler          from 1571 to 1630.
+  Galileo         from 1564 to 1642.
+  Gilbert         from 1540 to 1603.
+  Francis Bacon   from 1561 to 1626.
+  Descartes       from 1596 to 1650.
+
+_A sketch of Tycho Brahe's life and work._ Tycho was a Danish noble,
+born on his ancestral estate at Knudstorp, near Helsinborg, in 1546.
+Adopted by his uncle, and sent to the University of Copenhagen to study
+law. Attracted to astronomy by the occurrence of an eclipse on its
+predicted day, August 21st, 1560. Began to construct astronomical
+instruments, especially a quadrant and a sextant. Observed at Augsburg
+and Wittenberg. Studied alchemy, but was recalled to astronomy by the
+appearance of a new star. Overcame his aristocratic prejudices, and
+delivered a course of lectures at Copenhagen, at the request of the
+king. After this he married a peasant girl. Again travelled and observed
+in Germany. In 1576 was sent for to Denmark by Frederick II., and
+established in the island of Huen, with an endowment enabling him to
+devote his life to astronomy. Built Uraniburg, furnished it with
+splendid instruments, and became the founder of accurate instrumental
+astronomy. His theories were poor, but his observations were admirable.
+In 1592 Frederick died, and five years later, Tycho was impoverished and
+practically banished. After wandering till 1599, he was invited to
+Prague by the Emperor Rudolf, and there received John Kepler among other
+pupils. But the sentence of exile was too severe, and he died in 1601,
+aged 54 years.
+
+A man of strong character, untiring energy, and devotion to accuracy,
+his influence on astronomy has been immense.
+
+
+
+
+LECTURE II
+
+TYCHO BRAHE AND THE EARLIEST OBSERVATORY
+
+
+We have seen how Copernicus placed the earth in its true position in the
+solar system, making it merely one of a number of other worlds revolving
+about a central luminary. And observe that there are two phenomena to be
+thus accounted for and explained: first, the diurnal revolution of the
+heavens; second, the annual motion of the sun among the stars.
+
+The effect of the diurnal motion is conspicuous to every one, and
+explains the rising, southing, and setting of the whole visible
+firmament. The effect of the annual motion, _i.e._ of the apparent
+annual motion, of the sun among the stars, is less obvious, but it may
+be followed easily enough by observing the stars visible at any given
+time of evening at different seasons of the year. At midnight, for
+instance, the position of the sun is definite, viz. due north always,
+but the constellation which at that time is due south or is rising or
+setting varies with the time of year; an interval of one month producing
+just the same effect on the appearance of the constellations as an
+interval of two hours does (because the day contains twice as many hours
+as the year contains months), _e.g._ the sky looks the same at midnight
+on the 1st of October as it does at 10 p.m. on the 1st of November.
+
+All these simple consequences of the geocentric as opposed to the
+heliocentric point of view were pointed out by Copernicus, in addition
+to his greater work of constructing improved planetary tables on the
+basis of his theory. But it must be admitted that he himself felt the
+hypothesis of the motion of the earth to be a difficulty. Its acceptance
+is by no means such an easy and childish matter as we are apt now to
+regard it, and the hostility to it is not at all surprising. The human
+race, after having ridiculed and resisted the truth for a long time, is
+apt to end in accepting it so blindly and unimaginatively as to fail to
+recognize the real achievement of its first propounders, or the
+difficulties which they had to overcome. The majority of men at the
+present day have grown accustomed to hear the motion of the earth spoken
+of: their acceptance of it means nothing: the attitude of the paradoxer
+who denies it is more intelligent.
+
+It is not to be supposed that the idea of thus explaining some of the
+phenomena of the heavens, especially the daily motion of the entire
+firmament, by a diurnal rotation of the earth had not struck any one. It
+was often at this time referred to as the Pythagorean theory, and it had
+been taught, I believe, by Aristarchus. But it was new to the modern
+world, and it had the great weight of Aristotle against it.
+Consequently, for long after Copernicus, only a few leading spirits
+could be found to support it, and the long-established venerable
+Ptolemaic system continued to be taught in all Universities.
+
+The main objections to the motion of the earth were such as the
+following:--
+
+1. The motion is unfelt and difficult to imagine.
+
+     That it is unfelt is due to its uniformity, and can be explained
+     mechanically. That it is difficult to imagine is and remains true,
+     but a most important lesson we have to learn is that difficulty of
+     conception is no valid argument against reality.
+
+2. That the stars do not alter their relative positions according to
+the season of the year, but the constellations preserve always the same
+aspect precisely, even to careful measurement.
+
+     This is indeed a difficulty, and a great one. In June the earth is
+     184 million miles away from where it was in December: how can we
+     see precisely the same fixed stars? It is not possible, unless they
+     are at a practically infinite distance. That is the only answer
+     that can be given. It was the tentative answer given by Copernicus.
+     It is the correct answer. Not only from every position of the
+     earth, but from every planet of the solar system, the same
+     constellations are visible, and the stars have the same aspect. The
+     whole immensity of the solar system shrinks to practically a point
+     when confronted with the distance of the stars.
+
+     Not, however, so entirely a speck as to resist the terrific
+     accuracy of the present century, and their microscopic relative
+     displacement with the season of the year has now at length been
+     detected, and the distance of many thereby measured.
+
+3. That, if the earth revolved round the sun, Mercury and Venus ought to
+show phases like the moon.
+
+     So they ought. Any globe must show phases if it live nearer the sun
+     than we do and if we go round it, for we shall see varying amounts
+     of its illuminated half. The only answer that Copernicus could give
+     to this was that they might be difficult to see without extra
+     powers of sight, but he ventured to predict that the phases would
+     be seen if ever our powers of vision should be enhanced.
+
+4. That if the earth moved, or even revolved on its own axis, a stone or
+other dropped body ought to be left far behind.
+
+     This difficulty is not a real one, like the two last, and it is
+     based on an ignorance of the laws of mechanics, which had not at
+     that time been formulated. We know now that a ball dropped from a
+     high tower, so far from lagging, drops a minute trifle _in front_
+     of the foot of a perpendicular, because the top of the tower is
+     moving a trace faster than the bottom, by reason of the diurnal
+     rotation. But, ignoring this, a stone dropped from the lamp of a
+     railway carriage drops in the centre of the floor, whether the
+     carriage be moving steadily or standing still; a slant direction of
+     fall could only be detected if the carriage were being accelerated
+     or if the brake were applied. A body dropped from a moving carriage
+     shares the motion of the carriage, and starts with that as its
+     initial velocity. A ball dropped from a moving balloon does not
+     simply drop, but starts off in whatever direction the car was
+     moving, its motion being immediately modified by gravity, precisely
+     in the same way as that of a thrown ball is modified. This is,
+     indeed, the whole philosophy of throwing--to drop a ball from a
+     moving carriage. The carriage is the hand, and, to throw far, a run
+     is taken and the body is jerked forward; the arm is also moved as
+     rapidly as possible on the shoulder as pivot. The fore-arm can be
+     moved still faster, and the wrist-joint gives yet another motion:
+     the art of throwing is to bring all these to bear at the same
+     instant, and then just as they have all attained their maximum
+     velocity to let the ball go. It starts off with the initial
+     velocity thus imparted, and is abandoned to gravity. If the vehicle
+     were able to continue its motion steadily, as a balloon does, the
+     ball when let go from it would appear to the occupant simply to
+     drop; and it would strike the ground at a spot vertically under the
+     moving vehicle, though by no means vertically below the place where
+     it started. The resistance of the air makes observations of this
+     kind inaccurate, except when performed inside a carriage so that
+     the air shares in the motion. Otherwise a person could toss and
+     catch a ball out of a train window just as well as if he were
+     stationary; though to a spectator outside he would seem to be using
+     great skill to throw the ball in the parabola adapted to bring it
+     back to his hand.
+
+     The same circumstance enhances the apparent difficulty of the
+     circus rider's jumping feats. All he has to do is to jump up and
+     down on the horse; the forward motion which carries him through
+     hoops belongs to him by virtue of the motion of the horse, without
+     effort on his part.
+
+     Thus, then, it happens that a stone dropped sixteen feet on the
+     earth appears to fall straight down, although its real path in
+     space is a very flat trajectory of nineteen miles base and sixteen
+     feet height; nineteen miles being the distance traversed by the
+     earth every second in the course of its annual journey round the
+     sun.
+
+     No wonder that it was thought that bodies must be left behind if
+     the earth was subject to such terrific speed as this. All that
+     Copernicus could suggest on this head was that perhaps the
+     atmosphere might help to carry things forward, and enable them to
+     keep pace with the earth.
+
+There were thus several outstanding physical difficulties in the way of
+the acceptance of the Copernican theory, besides the Biblical
+difficulty.
+
+It was quite natural that the idea of the earth's motion should be
+repugnant, and take a long time to sink into the minds of men; and as
+scientific progress was vastly slower then than it is now, we find not
+only all priests but even some astronomers one hundred years afterwards
+still imagining the earth to be at rest. And among them was a very
+eminent one, Tycho Brahe.
+
+It is interesting to note, moreover, that the argument about the motion
+of the earth being contrary to Scripture appealed not only to
+ecclesiastics in those days, but to scientific men also; and Tycho
+Brahe, being a man of great piety, and highly superstitious also, was so
+much influenced by it, that he endeavoured to devise some scheme by
+which the chief practical advantages of the Copernican system could be
+retained, and yet the earth be kept still at the centre of the whole.
+This was done by making all the celestial sphere, with stars and
+everything, rotate round the earth once a day, as in the Ptolemaic
+scheme; and then besides this making all the planets revolve round the
+sun, and this to revolve round the earth. Such is the Tychonic system.
+
+So far as _relative_ motion is concerned it comes to the same thing;
+just as when you drop a book you may say either that the earth rises to
+meet the book, or that the book falls to meet the earth. Or when a fly
+buzzes round your head, you may say that you are revolving round the
+fly. But the absurdity of making the whole gigantic system of sun and
+planets and stars revolve round our insignificant earth was too great to
+be swallowed by other astronomers after they had once had a taste of the
+Copernican theory; and accordingly the Tychonic system died a speedy and
+an easy death at the same time as its inventor.
+
+Wherein then lay the magnitude of the man?--not in his theories, which
+were puerile, but in his observations, which were magnificent. He was
+the first observational astronomer, the founder of the splendid system
+of practical astronomy which has culminated in the present Greenwich
+Observatory.
+
+[Illustration: FIG. 16.--Tychonic system showing the sun with all the
+planets revolving round the earth.]
+
+Up to Tycho the only astronomical measurements had been of the rudest
+kind. Copernicus even improved upon what had gone before, with measuring
+rules made with his own hands. Ptolemy's observations could never be
+trusted to half a degree. Tycho introduced accuracy before undreamed of,
+and though his measurements, reckoned by modern ideas, are of course
+almost ludicrously rough (remember no such thing as a telescope or
+microscope was then dreamed of), yet, estimated by the era in which they
+were made, they are marvels of accuracy, and not a single mistake due
+to carelessness has ever been detected in them. In fact they may be
+depended on almost to minutes of arc, _i.e._ to sixtieths of a degree.
+
+For certain purposes connected with the proper motion of stars they are
+still appealed to, and they served as the certain and trustworthy data
+for succeeding generations of theorists to work upon. It was long,
+indeed, after Tycho's death before observations approaching in accuracy
+to his were again made.
+
+In every sense, therefore, he was a pioneer: let us proceed to trace his
+history.
+
+Born the eldest son of a noble family--"as noble and ignorant as sixteen
+undisputed quarterings could make them," as one of his biographers
+says--in a period when, even more than at present, killing and hunting
+were the only natural aristocratic pursuits, when all study was regarded
+as something only fit for monks, and when science was looked at askance
+as something unsavoury, useless, and semi-diabolic, there was little in
+his introduction to the world urging him in the direction where his
+genius lay. Of course he was destined for a soldier; but fortunately his
+uncle, George Brahe, a more educated man than his father, having no son
+of his own, was anxious to adopt him, and though not permitted to do so
+for a time, succeeded in getting his way on the birth of a second son,
+Steno--who, by the way, ultimately became Privy Councillor to the King
+of Denmark.
+
+Tycho's uncle gave him what he would never have got at home--a good
+education; and ultimately put him to study law. At the age of thirteen
+he entered the University of Copenhagen, and while there occurred the
+determining influence of his life.
+
+An eclipse of the sun in those days was not regarded with the
+cold-blooded inquisitiveness or matter-of-fact apathy, according as
+there is or is not anything to be learnt from it, with which such an
+event is now regarded. Every occurrence in the heavens was then
+believed to carry with it the destiny of nations and the fate of
+individuals, and accordingly was of surpassing interest. Ever since the
+time of Hipparchus it had been possible for some capable man here and
+there to predict the occurrence of eclipses pretty closely. The thing is
+not difficult. The prediction was not, indeed, to the minute and second,
+as it is now; but the day could usually be hit upon pretty accurately
+some time ahead, much as we now manage to hit upon the return of a
+comet--barring accidents; and the hour could be predicted as the event
+approached.
+
+Well, the boy Tycho, among others, watched for this eclipse on August
+21st, 1560; and when it appeared at its appointed time, every instinct
+for the marvellous, dormant in his strong nature, awoke to strenuous
+life, and he determined to understand for himself a science permitting
+such wonderful possibilities of prediction. He was sent to Leipzig with
+a tutor to go on with his study of law, but he seems to have done as
+little law as possible: he spent all his money on books and instruments,
+and sat up half the night studying and watching the stars.
+
+In 1563 he observed a conjunction of Jupiter and Saturn, the precursor,
+and _cause_ as he thought it, of the great plague. He found that the old
+planetary tables were as much as a month in error in fixing this event,
+and even the Copernican tables were several days out; so he formed the
+resolve to devote his life to improving astronomical tables. This
+resolve he executed with a vengeance. His first instrument was a jointed
+ruler with sights for fixing the position of planets with respect to the
+stars, and observing their stations and retrogressions. By thus
+measuring the angles between a planet and two fixed stars, its position
+can be plotted down on a celestial map or globe.
+
+[Illustration: FIG. 17.--Portrait of Tycho.]
+
+In 1565 his uncle George died, and made Tycho his heir. He returned to
+Denmark, but met with nothing but ridicule and contempt for his absurd
+drivelling away of time over useless pursuits. So he went back to
+Germany--first to Wittenberg, thence, driven by the plague, to Rostock.
+
+Here his fiery nature led him into an absurd though somewhat dangerous
+adventure. A quarrel at some feast, on a mathematical point, with a
+countryman, Manderupius, led to the fixing of a duel, and it was fought
+with swords at 7 p.m. at the end of December, when, if there was any
+light at all, it must have been of a flickering and unsatisfactory
+nature. The result of this insane performance was that Tycho got his
+nose cut clean off.
+
+He managed however to construct an artificial one, some say of gold and
+silver, some say of putty and brass; but whatever it was made of there
+is no doubt that he wore it for the rest of his life, and it is a most
+famous feature. It excited generally far more interest than his
+astronomical researches. It is said, moreover, to have very fairly
+resembled the original, but whether this remark was made by a friend or
+by an enemy I cannot say. One account says that he used to carry about
+with him a box of cement to apply whenever his nose came off, which it
+periodically did.
+
+About this time he visited Augsburg, met with some kindred and
+enlightened spirits in that town, and with much enthusiasm and spirit
+constructed a great quadrant. These early instruments were tremendous
+affairs. A great number of workmen were employed upon this quadrant, and
+it took twenty men to carry it to its place and erect it. It stood in
+the open air for five years, and then was destroyed by a storm. With it
+he made many observations.
+
+[Illustration: FIG. 18.--Early out-door quadrant of Tycho; for
+observing altitudes by help of the sights _D_, _L_ and the plumb line.]
+
+On his return to Denmark in 1571, his fame preceded him, and he was
+much better received; and in order to increase his power of constructing
+instruments he took up the study of alchemy, and like the rest of the
+persuasion tried to make gold. The precious metals were by many old
+philosophers considered to be related in some way to the heavenly
+bodies: silver to the moon, for instance--as we still see by the name
+lunar caustic applied to nitrate of silver; gold to the sun, copper to
+Mars, lead to Saturn. Hence astronomy and alchemy often went together.
+Tycho all his life combined a little alchemy with his astronomical
+labours, and he constructed a wonderful patent medicine to cure all
+disorders, which had as wide a circulation in Europe in its time as
+Holloway's pills; he gives a tremendous receipt for it, with liquid gold
+and all manner of ingredients in it; among them, however, occurs a
+little antimony--a well-known sudorific--and to this, no doubt, whatever
+efficacy the medicine possessed was due.
+
+So he might have gone on wasting his time, were it not that in November,
+1572, a new star made its appearance, as they have done occasionally
+before and since. On the average one may say that about every fifty
+years a new star of fair magnitude makes its temporary appearance. They
+are now known to be the result of some catastrophe or collision, whereby
+immense masses of incandescent gas are produced. This one seen by Tycho
+became as bright as Jupiter, and then died away in about a year and a
+half. Tycho observed all its changes, and endeavoured to measure its
+distance from the earth, with the result that it was proved to belong to
+the region of the fixed stars, at an immeasurable distance, and was not
+some nearer and more trivial phenomenon.
+
+He was asked by the University of Copenhagen to give a course of
+lectures on astronomy; but this was a step he felt some aristocratic
+aversion to, until a little friendly pressure was brought to bear upon
+him by a request from the king, and delivered they were.
+
+He now seems to have finally thrown off his aristocratic prejudices, and
+to have indulged himself in treading on the corns of nearly all the high
+and mighty people he came into contact with. In short, he became what we
+might now call a violent Radical; but he was a good-hearted man,
+nevertheless, and many are the tales told of his visits to sick
+peasants, of his consulting the stars as to their fate--all in perfect
+good faith--and of the medicines which he concocted and prescribed for
+them.
+
+The daughter of one of these peasants he married, and very happy the
+marriage seems to have been.
+
+[Illustration: FIG. 19.--Map of Denmark, showing the island of Huen.
+
+_Walker & Boutallse._]
+
+Now comes the crowning episode in Tycho's life. Frederick II., realizing
+how eminent a man they had among them, and how much he could do if only
+he had the means--for we must understand that Tycho, though of good
+family and well off, was by no means what we would call a wealthy
+man--Frederick II. made him a splendid and enlightened offer. The offer
+was this: that if Tycho would agree to settle down and make his
+astronomical observations in Denmark, he should have an estate in Norway
+settled upon him, a pension of L400 a year for life, a site for a large
+observatory, and L20,000 to build it with.
+
+[Illustration: FIG. 20.--Uraniburg.]
+
+[Illustration: FIG. 21.--Astrolabe. An old instrument with sights for
+marking the positions of the celestial bodies roughly. A sort of
+skeleton celestial globe.]
+
+[Illustration:
+
+  SEXTANS ASTRONOMICVS
+  TRIGONICVS PRO DISTANTIIS
+  rimandis.
+
+FIG. 22.--Tycho's large sextant; for measuring the angular distance
+between two bodies by direct sighting.]
+
+Well, if ever money was well spent, this was. By its means Denmark
+before long headed the nations of Europe in the matter of science--a
+thing it has not done before or since. The site granted was the island
+of Huen, between Copenhagen and Elsinore; and here the most magnificent
+observatory ever built was raised, and called Uraniburg--the castle of
+the heavens. It was built on a hill in the centre of the island, and
+included gardens, printing shops, laboratory, dwelling-houses, and four
+observatories--all furnished with the most splendid instruments that
+Tycho could devise, and that could then be constructed. It was decorated
+with pictures and sculptures of eminent men, and altogether was a most
+gorgeous place. L20,000 no doubt went far in those days, but the
+original grant was supplemented by Tycho himself, who is said to have
+spent another equal sum out of his own pocket on the place.
+
+[Illustration: QVADRANS MAXIMVS CHALIBEUS QUADRATO INCLUSUS, ET
+Horizonti Azimuthali chalybeo insistens.
+
+FIG. 23.--The Quadrant in Uraniburg; or altitude and azimuth
+instrument.]
+
+For twenty years this great temple of science was continually worked in
+by him, and he soon became the foremost scientific man in Europe.
+Philosophers, statesmen, and occasionally kings, came to visit the great
+astronomer, and to inspect his curiosities.
+
+[Illustration:
+
+  QVADRANS MVRALIS
+  SIVE TICHONICUS.
+
+FIG. 24.--Tycho's form of transit circle.
+
+The method of utilising the extremely uniform rotation of the earth by
+watching the planets and stars as they cross the meridian, and recording
+their times of transit; observing also at the same time their meridian
+altitudes (see observer _F_), was the invention of Tycho, and
+constitutes his greatest achievement. His method is followed to this day
+in all observatories.]
+
+[Illustration: FIG. 25.--A modern transit circle, showing essentially
+the same parts as in Tycho's instrument, viz. the observer watching the
+transit, the clock, the recorder of the observation, and the graduated
+circle; the latter to be read by a second observer.]
+
+And very wholesome for some of these great personages must have been the
+treatment they met with. For Tycho was no respecter of persons. His
+humbly-born wife sat at the head of the table, whoever was there; and he
+would snub and contradict a chancellor just as soon as he would a serf.
+Whatever form his pride may have taken when a youth, in his maturity it
+impelled him to ignore differences of rank not substantially justified,
+and he seemed to take a delight in exposing the ignorance of shallow
+titled persons, to whom contradiction and exposure were most unusual
+experiences.
+
+For sick peasants he would take no end of trouble, and went about
+doctoring them for nothing, till he set all the professional doctors
+against him; so that when his day of misfortune came, as come it did,
+their influence was not wanting to help to ruin one who spoilt their
+practice, and whom they derided as a quack.
+
+But some of the great ignorant folk who came to visit his temple of
+science, and to inspect its curiosities, felt themselves insulted--not
+always without reason. He kept a tame maniac in the house, named Lep,
+and he used to regard the sayings of this personage as oracular,
+presaging future events, and far better worth listening to than ordinary
+conversation. Consequently he used to have him at his banquets and feed
+him himself; and whenever Lep opened his mouth to speak, every one else
+was peremptorily ordered to hold his tongue, so that Lep's words might
+be written down. In fact it was something like an exaggerated edition of
+Betsy Trotwood and Mr. Dick.
+
+"It must have been an odd dinner party" (says Prof. Stuart), "with this
+strange, wild, terribly clever man, with his red hair and brazen nose,
+sometimes flashing with wit and knowledge, sometimes making the whole
+company, princes and servants alike, hold their peace and listen humbly
+to the ravings of a poor imbecile."
+
+To people he despised he did not show his serious instruments. He had
+other attractions, in the shape of a lot of toy machinery, little
+windmills, and queer doors, and golden globes, and all manner of
+ingenious tricks and automata, many of which he had made himself, and
+these he used to show them instead; and no doubt they were well enough
+pleased with them. Those of the visitors, however, who really cared to
+see and understand his instruments, went away enchanted with his genius
+and hospitality.
+
+I may, perhaps, be producing an unfair impression of imperiousness and
+insolence. Tycho was fiery, no doubt, but I think we should wrong him
+if we considered him insolent. Most of the nobles of his day were
+haughty persons, accustomed to deal with serfs, and very likely to sneer
+at and trample on any meek man of science whom they could easily
+despise. So Tycho was not meek; he stood up for the honour of his
+science, and paid them back in their own coin, with perhaps a little
+interest. That this behaviour was not worldly-wise is true enough, but I
+know of no commandment enjoining us to be worldly-wise.
+
+If we knew more about his so-called imbecile _protege_ we should
+probably find some reason for the interest which Tycho took in him.
+Whether he was what is now called a "clairvoyant" or not, Tycho
+evidently regarded his utterances as oracular, and of course when one is
+receiving what may be a revelation from heaven it is natural to suppress
+ordinary conversation.
+
+Among the noble visitors whom he received and entertained, it is
+interesting to notice James I. of England, who spent eight days at
+Uraniburg on the occasion of his marriage with Anne of Denmark in 1590,
+and seems to have been deeply impressed by his visit.
+
+Among other gifts, James presented Tycho with a dog (depicted in Fig.
+24), and this same animal was subsequently the cause of trouble. For it
+seems that one day the Chancellor of Denmark, Walchendorf, brutally
+kicked the poor beast; and Tycho, who was very fond of animals, gave him
+a piece of his mind in no measured language. Walchendorf went home
+determined to ruin him. King Frederick, however, remained his true
+friend, doubtless partly influenced thereto by his Queen Sophia, an
+enlightened woman who paid many visits to Uraniburg, and knew Tycho
+well. But unfortunately Frederick died; and his son, a mere boy, came to
+the throne.
+
+Now was the time for the people whom Tycho had offended, for those who
+were jealous of his great fame and importance, as well as for those who
+cast longing eyes on his estate and endowments. The boy-king, too,
+unfortunately paid a visit to Tycho, and, venturing upon a decided
+opinion on some recondite subject, received a quiet setting down which
+he ill relished.
+
+Letters written by Tycho about this time are full of foreboding. He
+greatly dreads having to leave Uraniburg, with which his whole life has
+for twenty years been bound up. He tries to comfort himself with the
+thought that, wherever he is sent, he will have the same heavens and the
+same stars over his head.
+
+Gradually his Norwegian estate and his pension were taken away, and in
+five years poverty compelled him to abandon his magnificent temple, and
+to take a small house in Copenhagen.
+
+Not content with this, Walchendorf got a Royal Commission appointed to
+inquire into the value of his astronomical labours. This sapient body
+reported that his work was not only useless, but noxious; and soon after
+he was attacked by the populace in the public street.
+
+Nothing was left for him now but to leave the country, and he went into
+Germany, leaving his wife and instruments to follow him whenever he
+could find a home for them.
+
+His wanderings in this dark time--some two years--are not quite clear;
+but at last the enlightened Emperor of Bohemia, Rudolph II., invited him
+to settle in Prague. Thither he repaired, a castle was given him as an
+observatory, a house in the city, and 3000 crowns a year for life. So
+his instruments were set up once more, students flocked to hear him and
+to receive work at his hands--among them a poor youth, John Kepler, to
+whom he was very kind, and who became, as you know, a still greater man
+than his master.
+
+But the spirit of Tycho was broken, and though some good work was done
+at Prague--more observations made, and the Rudolphine tables begun--yet
+the hand of death was upon him. A painful disease seized him, attended
+with sleeplessness and temporary delirium, during the paroxysms of
+which he frequently exclaimed, _Ne frustra vixisse videar_. ("Oh that it
+may not appear that I have lived in vain!")
+
+Quietly, however, at last, and surrounded by his friends and relatives,
+this fierce, passionate soul passed away, on the 24th of October, 1601.
+
+His beloved instruments, which were almost a part of himself, were
+stored by Rudolph in a museum with scrupulous care, until the taking of
+Prague by the Elector Palatine's troops. In this disturbed time they got
+smashed, dispersed, and converted to other purposes. One thing only was
+saved--the great brass globe, which some thirty years after was
+recognized by a later king of Denmark as having belonged to Tycho, and
+deposited in the Library of the Academy of Sciences at Copenhagen, where
+I believe it is to this day.
+
+The island of Huen was overrun by the Danish nobility, and nothing now
+remains of Uraniburg but a mound of earth and two pits.
+
+As to the real work of Tycho, that has become immortal enough,--chiefly
+through the labours of his friend and scholar whose life we shall
+consider in the next lecture.
+
+
+
+
+SUMMARY OF FACTS FOR LECTURE III
+
+
+_Life and work of Kepler._ Kepler was born in December, 1571, at Weil in
+Wuertemberg. Father an officer in the duke's army, mother something of a
+virago, both very poor. Kepler was utilized as a tavern pot-boy, but
+ultimately sent to a charity school, and thence to the University of
+Tuebingen. Health extremely delicate; he was liable to violent attacks
+all his life. Studied mathematics, and accepted an astronomical
+lectureship at Graz as the first post which offered. Endeavoured to
+discover some connection between the number of the planets, their times
+of revolution, and their distances from the sun. Ultimately hit upon his
+fanciful regular-solid hypothesis, and published his first book in 1597.
+In 1599 was invited by Tycho to Prague, and there appointed Imperial
+mathematician, at a handsome but seldom paid salary. Observed the new
+star of 1604. Endeavoured to find the law of refraction of light from
+Vitellio's measurements, but failed. Analyzed Tycho's observations to
+find the true law of motion of Mars. After incredible labour, through
+innumerable wrong guesses, and six years of almost incessant
+calculation, he at length emerged in his two "laws"--discoveries which
+swept away all epicycles, deferents, equants, and other remnants of the
+Greek system, and ushered in the dawn of modern astronomy.
+
+LAW I. _Planets move in ellipses, with the Sun in one focus._
+
+LAW II. _The radius vector (or line joining sun and planet) sweeps out
+equal areas in equal times._
+
+Published his second book containing these laws in 1609. Death of
+Rudolph in 1612, and subsequent increased misery and misfortune of
+Kepler. Ultimately discovered the connection between the times and
+distances of the planets for which he had been groping all his mature
+life, and announced it in 1618:--
+
+LAW III. _The square of the time of revolution (or year) of each planet
+is proportional to the cube of its mean distance from the sun._
+
+The book in which this law was published ("On Celestial Harmonies") was
+dedicated to James of England. In 1620 had to intervene to protect his
+mother from being tortured for witchcraft. Accepted a professorship at
+Linz. Published the Rudolphine tables in 1627, embodying Tycho's
+observations and his own theory. Made a last effort to overcome his
+poverty by getting the arrears of his salary paid at Prague, but was
+unsuccessful, and, contracting brain fever on the journey, died in
+November, 1630, aged 59.
+
+A man of keen imagination, indomitable perseverance, and uncompromising
+love of truth, Kepler overcame ill-health, poverty, and misfortune, and
+placed himself in the very highest rank of scientific men. His laws, so
+extraordinarily discovered, introduced order and simplicity into what
+else would have been a chaos of detailed observations; and they served
+as a secure basis for the splendid erection made on them by Newton.
+
+  _Seven planets of the Ptolemaic system--_
+      Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.
+
+  _Six planets of the Copernican system--_
+      Mercury, Venus, Earth, Mars, Jupiter, Saturn.
+
+  _The five regular solids, in appropriate order--_
+      Octahedron, Icosahedron, Dodecahedron, Tetrahedron, Cube.
+
+_Table illustrating Kepler's third law._
+
+ +---------+---------------+-----------+---------------+----------------+
+ |         | Mean distance |   Length  |  Cube of the  |  Square of the |
+ | Planet.  |   from Sun.  |  of Year. |   Distance.   |     Time.      |
+ |         |       D       |     T     |      D^3      |      T^2       |
+ +---------+---------------+-----------+---------------+----------------+
+ | Mercury |     .3871     |   .24084  |     .05801    |     .05801     |
+ | Venus   |     .7233     |   .61519  |     .37845    |     .37846     |
+ | Earth   |    1.0000     |  1.0000   |    1.0000     |    1.0000      |
+ | Mars    |    1.5237     |  1.8808   |    3.5375     |    3.5375      |
+ | Jupiter |    5.2028     | 11.862    |  140.83       |  140.70        |
+ | Saturn  |    9.5388     | 29.457    |  867.92       |  867.70        |
+ +---------+---------------+-----------+---------------+----------------+
+
+The length of the earth's year is 365.256 days; its mean distance from
+the sun, taken above as unity, is 92,000,000 miles.
+
+
+
+
+LECTURE III
+
+KEPLER AND THE LAWS OF PLANETARY MOTION
+
+
+It is difficult to imagine a stronger contrast between two men engaged
+in the same branch of science than exists between Tycho Brahe, the
+subject of last lecture, and Kepler, our subject on the present
+occasion.
+
+The one, rich, noble, vigorous, passionate, strong in mechanical
+ingenuity and experimental skill, but not above the average in
+theoretical and mathematical power.
+
+The other, poor, sickly, devoid of experimental gifts, and unfitted by
+nature for accurate observation, but strong almost beyond competition in
+speculative subtlety and innate mathematical perception.
+
+The one is the complement of the other; and from the fact of their
+following each other so closely arose the most surprising benefits to
+science.
+
+The outward life of Kepler is to a large extent a mere record of poverty
+and misfortune. I shall only sketch in its broad features, so that we
+may have more time to attend to his work.
+
+He was born (so his biographer assures us) in longitude 29 deg. 7', latitude
+48 deg. 54', on the 21st of December, 1571. His parents seem to have been of
+fair condition, but by reason, it is said, of his becoming surety for a
+friend, the father lost all his slender income, and was reduced to
+keeping a tavern. Young John Kepler was thereupon taken from school,
+and employed as pot-boy between the ages of nine and twelve. He was a
+sickly lad, subject to violent illnesses from the cradle, so that his
+life was frequently despaired of. Ultimately he was sent to a monastic
+school and thence to the University of Tuebingen, where he graduated
+second on the list. Meanwhile home affairs had gone to rack and ruin.
+His father abandoned the home, and later died abroad. The mother
+quarrelled with all her relations, including her son John; who was
+therefore glad to get away as soon as possible.
+
+All his connection with astronomy up to this time had been the hearing
+the Copernican theory expounded in University lectures, and defending it
+in a college debating society.
+
+An astronomical lectureship at Graz happening to offer itself, he was
+urged to take it, and agreed to do so, though stipulating that it should
+not debar him from some more brilliant profession when there was a
+chance.
+
+For astronomy in those days seems to have ranked as a minor science,
+like mineralogy or meteorology now. It had little of the special dignity
+with which the labours of Kepler himself were destined so greatly to aid
+in endowing it.
+
+Well, he speedily became a thorough Copernican, and as he had a most
+singularly restless and inquisitive mind, full of appreciation of
+everything relating to number and magnitude--was a born speculator and
+thinker just as Mozart was a born musician, or Bidder a born
+calculator--he was agitated by questions such as these: Why are there
+exactly six planets? Is there any connection between their orbital
+distances, or between their orbits and the times of describing them?
+These things tormented him, and he thought about them day and night. It
+is characteristic of the spirit of the times--this questioning why there
+should be six planets. Nowadays, we should simply record the fact and
+look out for a seventh. Then, some occult property of the number six was
+groped for, such as that it was equal to 1 + 2 + 3 and likewise equal to
+1 x 2 x 3, and so on. Many fine reasons had been given for the seven
+planets of the Ptolemaic system (see, for instance, p. 106), but for
+the six planets of the Copernican system the reasons were not so cogent.
+
+Again, with respect to their successive distances from the sun, some law
+would seem to regulate their distance, but it was not known.
+(Parenthetically I may remark that it is not known even now: a crude
+empirical statement known as Bode's law--see page 294--is all that has
+been discovered.)
+
+Once more, the further the planet the slower it moved; there seemed to
+be some law connecting speed and distance. This also Kepler made
+continual attempts to discover.
+
+[Illustration: FIG. 26.--Orbits of some of the planets drawn to scale:
+showing the gap between Mars and Jupiter.]
+
+One of his ideas concerning the law of the successive distances was
+based on the inscription of a triangle in a circle. If you inscribe in a
+circle a large number of equilateral triangles, they envelop another
+circle bearing a definite ratio to the first: these might do for the
+orbits of two planets (see Fig. 27). Then try inscribing and
+circumscribing squares, hexagons, and other figures, and see if the
+circles thus defined would correspond to the several planetary orbits.
+But they would not give any satisfactory result. Brooding over this
+disappointment, the idea of trying solid figures suddenly strikes him.
+"What have plane figures to do with the celestial orbits?" he cries out;
+"inscribe the regular solids." And then--brilliant idea--he remembers
+that there are but five. Euclid had shown that there could be only five
+regular solids.[4] The number evidently corresponds to the gaps between
+the six planets. The reason of there being only six seems to be
+attained. This coincidence assures him he is on the right track, and
+with great enthusiasm and hope he "represents the earth's orbit by a
+sphere as the norm and measure of all"; round it he circumscribes a
+dodecahedron, and puts another sphere round that, which is approximately
+the orbit of Mars; round that, again, a tetrahedron, the corners of
+which mark the sphere of the orbit of Jupiter; round that sphere, again,
+he places a cube, which roughly gives the orbit of Saturn.
+
+[Illustration: FIG. 27.--Many-sided polygon or approximate circle
+enveloped by straight lines, as for instance by a number of equilateral
+triangles.]
+
+On the other hand, he inscribes in the sphere of the earth's orbit an
+icosahedron; and inside the sphere determined by that, an octahedron;
+which figures he takes to inclose the spheres of Venus and of Mercury
+respectively.
+
+The imagined discovery is purely fictitious and accidental. First of
+all, eight planets are now known; and secondly, their real distances
+agree only very approximately with Kepler's hypothesis.
+
+[Illustration: FIG. 28.--Frameworks with inscribed and circumscribed
+spheres, representing the five regular solids distributed as Kepler
+supposed them to be among the planetary orbits. (See "Summary" at
+beginning of this lecture, p. 57.)]
+
+Nevertheless, the idea gave him great delight. He says:--"The intense
+pleasure I have received from this discovery can never be told in words.
+I regretted no more the time wasted; I tired of no labour; I shunned no
+toil of reckoning, days and nights spent in calculation, until I could
+see whether my hypothesis would agree with the orbits of Copernicus, or
+whether my joy was to vanish into air."
+
+He then went on to speculate as to the cause of the planets' motion.
+The old idea was that they were carried round by angels or celestial
+intelligences. Kepler tried to establish some propelling force emanating
+from the sun, like the spokes of a windmill.
+
+This first book of his brought him into notice, and served as an
+introduction to Tycho and to Galileo.
+
+Tycho Brahe was at this time at Prague under the patronage of the
+Emperor Rudolph; and as he was known to have by far the best planetary
+observations of any man living, Kepler wrote to him to know if he might
+come and examine them so as to perfect his theory.
+
+Tycho immediately replied, "Come, not as a stranger, but as a very
+welcome friend; come and share in my observations with such instruments
+as I have with me, and as a dearly beloved associate." After this visit,
+Tycho wrote again, offering him the post of mathematical assistant,
+which after hesitation was accepted. Part of the hesitation Kepler
+expresses by saying that "for observations his sight was dull, and for
+mechanical operations his hand was awkward. He suffered much from weak
+eyes, and dare not expose himself to night air." In all this he was, of
+course, the antipodes of Tycho, but in mathematical skill he was greatly
+his superior.
+
+On his way to Prague he was seized with one of his periodical illnesses,
+and all his means were exhausted by the time he could set forward again,
+so that he had to apply for help to Tycho.
+
+It is clear, indeed, that for some time now he subsisted entirely on the
+bounty of Tycho, and he expresses himself most deeply grateful for all
+the kindness he received from that noble and distinguished man, the head
+of the scientific world at that date.
+
+To illustrate Tycho's kindness and generosity, I must read you a letter
+written to him by Kepler. It seems that Kepler, on one of his absences
+from Prague, driven half mad with poverty and trouble, fell foul of
+Tycho, whom he thought to be behaving badly in money matters to him and
+his family, and wrote him a violent letter full of reproaches and
+insults. Tycho's secretary replied quietly enough, pointing out the
+groundlessness and ingratitude of the accusation.
+
+Kepler repents instantly, and replies:--
+
+     "MOST NOBLE TYCHO," (these are the words of his letter), "how shall
+     I enumerate or rightly estimate your benefits conferred on me? For
+     two months you have liberally and gratuitously maintained me, and
+     my whole family; you have provided for all my wishes; you have done
+     me every possible kindness; you have communicated to me everything
+     you hold most dear; no one, by word or deed, has intentionally
+     injured me in anything; in short, not to your children, your wife,
+     or yourself have you shown more indulgence than to me. This being
+     so, as I am anxious to put on record, I cannot reflect without
+     consternation that I should have been so given up by God to my own
+     intemperance as to shut my eyes on all these benefits; that,
+     instead of modest and respectful gratitude, I should indulge for
+     three weeks in continual moroseness towards all your family, in
+     headlong passion and the utmost insolence towards yourself, who
+     possess so many claims on my veneration, from your noble family,
+     your extraordinary learning, and distinguished reputation. Whatever
+     I have said or written against the person, the fame, the honour,
+     and the learning of your excellency; or whatever, in any other way,
+     I have injuriously spoken or written (if they admit no other more
+     favourable interpretation), as, to my grief, I have spoken and
+     written many things, and more than I can remember; all and
+     everything I recant, and freely and honestly declare and profess to
+     be groundless, false, and incapable of proof."
+
+Tycho accepted the apology thus heartily rendered, and the temporary
+breach was permanently healed.
+
+In 1601, Kepler was appointed "Imperial mathematician," to assist Tycho
+in his calculations.
+
+The Emperor Rudolph did a good piece of work in thus maintaining these
+two eminent men, but it is quite clear that it was as astrologers that
+he valued them; and all he cared for in the planetary motions was
+limited to their supposed effect on his own and his kingdom's destiny.
+He seems to have been politically a weak and superstitious prince, who
+was letting his kingdom get into hopeless confusion, and entangling
+himself in all manner of political complications. While Bohemia
+suffered, however, the world has benefited at his hands; and the tables
+upon which Tycho was now engaged are well called the Rudolphine tables.
+
+These tables of planetary motion Tycho had always regarded as the main
+work of his life; but he died before they were finished, and on his
+death-bed he intrusted the completion of them to Kepler, who loyally
+undertook their charge.
+
+The Imperial funds were by this time, however, so taxed by wars and
+other difficulties that the tables could only be proceeded with very
+slowly, a staff of calculators being out of the question. In fact,
+Kepler could not get even his own salary paid: he got orders, and
+promises, and drafts on estates for it; but when the time came for them
+to be honoured they were worthless, and he had no power to enforce his
+claims.
+
+So everything but brooding had to be abandoned as too expensive, and he
+proceeded to study optics. He gave a very accurate explanation of the
+action of the human eye, and made many hypotheses, some of them shrewd
+and close to the mark, concerning the law of refraction of light in
+dense media: but though several minor points of interest turned up,
+nothing of the first magnitude came out of this long research.
+
+The true law of refraction was discovered some years after by a Dutch
+professor, Willebrod Snell.
+
+We must now devote a little time to the main work of Kepler's life. All
+the time he had been at Prague he had been making a severe study of the
+motion of the planet Mars, analyzing minutely Tycho's books of
+observations, in order to find out, if possible, the true theory of his
+motion. Aristotle had taught that circular motion was the only perfect
+and natural motion, and that the heavenly bodies therefore necessarily
+moved in circles.
+
+So firmly had this idea become rooted in men's minds, that no one ever
+seems to have contemplated the possibility of its being false or
+meaningless.
+
+When Hipparchus and others found that, as a matter of fact, the planets
+did _not_ revolve in simple circles, they did not try other curves, as
+we should at once do now, but they tried combinations of circles, as we
+saw in Lecture I. The small circle carried by a bigger one was called an
+Epicycle. The carrying circle was called the Deferent. If for any reason
+the earth had to be placed out of the centre, the main planetary orbit
+was called an Excentric, and so on.
+
+But although the planetary paths might be roughly represented by a
+combination of circles, their speeds could not, on the hypothesis of
+uniform motion in each circle round the earth as a fixed body. Hence was
+introduced the idea of an Equant, _i.e._ an arbitrary point, not the
+earth, about which the speed might be uniform. Copernicus, by making the
+sun the centre, had been able to simplify a good deal of this, and to
+abolish the equant.
+
+But now that Kepler had the accurate observations of Tycho to refer to,
+he found immense difficulty in obtaining the true positions of the
+planets for long together on any such theory.
+
+He specially attacked the motion of the planet Mars, because that was
+sufficiently rapid in its changes for a considerable collection of data
+to have accumulated with respect to it. He tried all manner of circular
+orbits for the earth and for Mars, placing them in all sorts of aspects
+with respect to the sun. The problem to be solved was to choose such an
+orbit and such a law of speed, for both the earth and Mars, that a line
+joining them, produced out to the stars, should always mark correctly
+the apparent position of Mars as seen from the earth. He had to arrange
+the size of the orbits that suited best, then the positions of their
+centres, both being supposed excentric with respect to the sun; but he
+could not get any such arrangement to work with uniform motion about the
+sun. So he reintroduced the equant, and thus had another variable at his
+disposal--in fact, two, for he had an equant for the earth and another
+for Mars, getting a pattern of the kind suggested in Fig. 29.
+
+The equants might divide the line in any arbitrary ratio. All sorts of
+combinations had to be tried, the relative positions of the earth and
+Mars to be worked out for each, and compared with Tycho's recorded
+observations. It was easy to get them to agree for a short time, but
+sooner or later a discrepancy showed itself.
+
+[Illustration: FIG. 29.--_S_ represents the sun; _EC_, the centre of the
+earth's orbit, to be placed as best suited; _MC_, the same for Mars;
+_EE_, the earth's equant, or point about which the earth uniformly
+revolved (_i.e._ the point determining the law of speed about the sun),
+likewise to be placed anywhere, but supposed to be in the line joining
+_S_ to _EC_; _ME_, the same thing for Mars; with _?ME_ for an
+alternative hypothesis that perhaps Mars' equant was on line joining
+_EC_ with _MC_.]
+
+I need not say that all these attempts and gropings, thus briefly
+summarized, entailed enormous labour, and required not only great
+pertinacity, but a most singularly constituted mind, that could thus
+continue groping in the dark without a possible ray of theory to
+illuminate its search. Grope he did, however, with unexampled diligence.
+
+At length he hit upon a point that seemed nearly right. He thought he
+had found the truth; but no, before long the position of the planet, as
+calculated, and as recorded by Tycho, differed by eight minutes of arc,
+or about one-eighth of a degree. Could the observation be wrong by this
+small amount? No, he had known Tycho, and knew that he was never wrong
+eight minutes in an observation.
+
+So he set out the whole weary way again, and said that with those eight
+minutes he would yet find out the law of the universe. He proceeded to
+see if by making the planet librate, or the plane of its orbit tilt up
+and down, anything could be done. He was rewarded by finding that at any
+rate the plane of the orbit did not tilt up and down: it was fixed, and
+this was a simplification on Copernicus's theory. It is not an absolute
+fixture, but the changes are very small (see Laplace, page 266).
+
+[Illustration: FIG. 30.--Excentric circle supposed to be divided into
+equal areas. The sun, _S_, being placed at a selected point, it was
+possible to represent the varying speed of a planet by saying that it
+moved from _A_ to _B_, from _B_ to _C_, and so on, in equal times.]
+
+At last he thought of giving up the idea of _uniform_ circular motion,
+and of trying _varying_ circular motion, say inversely as its distance
+from the sun. To simplify calculation, he divided the orbit into
+triangles, and tried if making the triangles equal would do. A great
+piece of luck, they did beautifully: the rate of description of areas
+(not arcs) is uniform. Over this discovery he greatly rejoices. He feels
+as though he had been carrying on a war against the planet and had
+triumphed; but his gratulation was premature. Before long fresh little
+errors appeared, and grew in importance. Thus he announces it himself:--
+
+"While thus triumphing over Mars, and preparing for him, as for one
+already vanquished, tabular prisons and equated excentric fetters, it is
+buzzed here and there that the victory is vain, and that the war is
+raging anew as violently as before. For the enemy left at home a
+despised captive has burst all the chains of the equations, and broken
+forth from the prisons of the tables."
+
+Still, a part of the truth had been gained, and was not to be abandoned
+any more. The law of speed was fixed: that which is now known as his
+second law. But what about the shape of the orbit--Was it after all
+possible that Aristotle, and every philosopher since Aristotle, had been
+wrong? that circular motion was not the perfect and natural motion, but
+that planets might move in some other closed curve?
+
+Suppose he tried an oval. Well, there are a great variety of ovals, and
+several were tried: with the result that they could be made to answer
+better than a circle, but still were not right.
+
+Now, however, the geometrical and mathematical difficulties of
+calculation, which before had been tedious and oppressive, threatened to
+become overwhelming; and it is with a rising sense of despondency that
+Kepler sees his six years' unremitting labour leading deeper and deeper
+into complication.
+
+One most disheartening circumstance appeared, viz. that when he made the
+circuit oval his law of equable description of areas broke down. That
+seemed to require the circular orbit, and yet no circular orbit was
+quite accurate.
+
+While thinking and pondering for weeks and months over this new dilemma
+and complication of difficulties, till his brain reeled, an accidental
+ray of light broke upon him in a way not now intelligible, or barely
+intelligible. Half the extreme breadth intercepted between the circle
+and oval was 429/100,000 of the radius, and he remembered that the
+"optical inequality" of Mars was also about 429/100,000. This
+coincidence, in his own words, woke him out of sleep; and for some
+reason or other impelled him instantly to try making the planet
+oscillate in the diameter of its epicycle instead of revolve round it--a
+singular idea, but Copernicus had had a similar one to explain the
+motions of Mercury.
+
+[Illustration: FIG. 31.--Mode of drawing an ellipse. The two pins _F_
+are the foci.]
+
+Away he started through his calculations again. A long course of work
+night and day was rewarded by finding that he was now able to hit off
+the motions better than before; but what a singularly complicated motion
+it was. Could it be expressed no more simply? Yes, the curve so
+described by the planet is a comparatively simple one: it is a special
+kind of oval--the ellipse. Strange that he had not thought of it before.
+It was a famous curve, for the Greek geometers had studied it as one of
+the sections of a cone, but it was not so well known in Kepler's time.
+The fact that the planets move in it has raised it to the first
+importance, and it is familiar enough to us now. But did it satisfy the
+law of speed? Could the rate of description of areas be uniform with
+it? Well, he tried the ellipse, and to his inexpressible delight he
+found that it did satisfy the condition of equable description of areas,
+if the sun was in one focus. So, moving the planet in a selected
+ellipse, with the sun in one focus, at a speed given by the equable area
+description, its position agreed with Tycho's observations within the
+limits of the error of experiment. Mars was finally conquered, and
+remains in his prison-house to this day. The orbit was found.
+
+[Illustration: FIG. 32.]
+
+In a paroxysm of delight Kepler celebrates his victory by a triumphant
+figure, sketched actually on his geometrical diagram--the diagram which
+proves that the law of equable description of areas can hold good with
+an ellipse. The above is a tracing of it.
+
+Such is a crude and bald sketch of the steps by which Kepler rose to his
+great generalizations--the two laws which have immortalized his name.
+
+All the complications of epicycle, equant, deferent, excentric, and the
+like, were swept at once away, and an orbit of striking and beautiful
+properties substituted. Well might he be called, as he was, "the
+legislator," or law interpreter, "of the heavens."
+
+[Illustration: FIG. 33.--If _S_ is the sun, a planet or comet moves from
+_P_ to _P_1_, from _P_2_ to _P_3_, and from _P_4_ to _P_5_ in
+the same time; if the shaded areas are equal.]
+
+He concludes his book on the motions of Mars with a half comic appeal to
+the Emperor to provide him with the sinews of war for an attack on
+Mars's relations--father Jupiter, brother Mercury, and the rest--but the
+death of his unhappy patron in 1612 put an end to all these schemes, and
+reduced Kepler to the utmost misery. While at Prague his salary was in
+continual arrear, and it was with difficulty that he could provide
+sustenance for his family. He had been there eleven years, but they had
+been hard years of poverty, and he could leave without regret were it
+not that he should have to leave Tycho's instruments and observations
+behind him. While he was hesitating what best to do, and reduced to the
+verge of despair, his wife, who had long been suffering from low spirits
+and despondency, and his three children, were taken ill; one of the sons
+died of small-pox, and the wife eleven days after of low fever and
+epilepsy. No money could be got at Prague, so after a short time he
+accepted a professorship at Linz, and withdrew with his two quite young
+remaining children.
+
+He provided for himself now partly by publishing a prophesying almanack,
+a sort of Zadkiel arrangement--a thing which he despised, but the
+support of which he could not afford to do without. He is continually
+attacking and throwing sarcasm at astrology, but it was the only thing
+for which people would pay him, and on it after a fashion he lived. We
+do not find that his circumstances were ever prosperous, and though
+8,000 crowns were due to him from Bohemia he could not manage to get
+them paid.
+
+About this time occurred a singular interruption to his work. His old
+mother, of whose fierce temper something has already been indicated, had
+been engaged in a law-suit for some years near their old home in
+Wuertemberg. A change of judge having in process of time occurred, the
+defendant saw his way to turn the tables on the old lady by accusing her
+of sorcery. She was sent to prison, and condemned to the torture, with
+the usual intelligent idea of extracting a "voluntary" confession.
+Kepler had to hurry from Linz to interpose. He succeeded in saving her
+from the torture, but she remained in prison for a year or so. Her
+spirit, however, was unbroken, for no sooner was she released than she
+commenced a fresh action against her accuser. But fresh trouble was
+averted by the death of the poor old dame at the age of nearly eighty.
+
+This narration renders the unflagging energy shown by her son in his
+mathematical wrestlings less surprising.
+
+Interspersed with these domestic troubles, and with harassing and
+unsuccessful attempts to get his rights, he still brooded over his old
+problem of some possible connection between the distances of the planets
+from the sun and their times of revolution, _i.e._ the length of their
+years.
+
+It might well have been that there was no connection, that it was purely
+imaginary, like his old idea of the law of the successive distances of
+the planets, and like so many others of the guesses and fancies which
+he entertained and spent his energies in probing. But fortunately this
+time there was a connection, and he lived to have the joy of discovering
+it.
+
+The connection is this, that if one compares the distance of the
+different planets from the sun with the length of time they take to go
+round him, the cube of the respective distances is proportional to the
+square of the corresponding times. In other words, the ratio of r^3
+to T^2 for every planet is the same. Or, again, the length of a
+planet's year depends on the 3/2th power of its distance from the sun.
+Or, once more, the speed of each planet in its orbit is as the inverse
+square-root of its distance from the sun. The product of the distance
+into the square of the speed is the same for each planet.
+
+This (however stated) is called Kepler's third law. It welds the planets
+together, and shows them to be one system. His rapture on detecting the
+law was unbounded, and he breaks out into an exulting rhapsody:--
+
+"What I prophesied two-and-twenty years ago, as soon as I discovered the
+five solids among the heavenly orbits--what I firmly believed long
+before I had seen Ptolemy's _Harmonies_--what I had promised my friends
+in the title of this book, which I named before I was sure of my
+discovery--what sixteen years ago, I urged as a thing to be sought--that
+for which I joined Tycho Brahe, for which I settled in Prague, for which
+I have devoted the best part of my life to astronomical contemplations,
+at length I have brought to light, and recognized its truth beyond my
+most sanguine expectations. It is not eighteen months since I got the
+first glimpse of light, three months since the dawn, very few days since
+the unveiled sun, most admirable to gaze upon, burst upon me. Nothing
+holds me; I will indulge my sacred fury; I will triumph over mankind by
+the honest confession that I have stolen the golden vases of the
+Egyptians to build up a tabernacle for my God far away from the
+confines of Egypt. If you forgive me, I rejoice; if you are angry, I can
+bear it; the die is cast, the book is written, to be read either now or
+by posterity, I care not which; it may well wait a century for a reader,
+as God has waited six thousand years for an observer."
+
+Soon after this great work his third book appeared: it was an epitome of
+the Copernican theory, a clear and fairly popular exposition of it,
+which had the honour of being at once suppressed and placed on the list
+of books prohibited by the Church, side by side with the work of
+Copernicus himself, _De Revolutionibus Orbium Coelestium_.
+
+This honour, however, gave Kepler no satisfaction--it rather occasioned
+him dismay, especially as it deprived him of all pecuniary benefit, and
+made it almost impossible for him to get a publisher to undertake
+another book.
+
+Still he worked on at the Rudolphine tables of Tycho, and ultimately,
+with some small help from Vienna, completed them; but he could not get
+the means to print them. He applied to the Court till he was sick of
+applying: they lay idle four years. At last he determined to pay for the
+type himself. What he paid it with, God knows, but he did pay it, and he
+did bring out the tables, and so was faithful to the behest of his
+friend.
+
+This great publication marks an era in astronomy. They were the first
+really accurate tables which navigators ever possessed; they were the
+precursors of our present _Nautical Almanack_.
+
+After this, the Grand Duke of Tuscany sent Kepler a golden chain, which
+is interesting inasmuch as it must really have come from Galileo, who
+was in high favour at the Italian Court at this time.
+
+Once more Kepler made a determined attempt to get his arrears of salary
+paid, and rescue himself and family from their bitter poverty. He
+travelled to Prague on purpose, attended the imperial meeting, and
+pleaded his own cause, but it was all fruitless; and exhausted by the
+journey, weakened by over-study, and disheartened by the failure, he
+caught a fever, and died in his fifty-ninth year. His body was buried at
+Ratisbon, and a century ago a proposal was made to erect a marble
+monument to his memory, but nothing was done. It matters little one way
+or the other whether Germany, having almost refused him bread during his
+life, should, a century and a half after his death, offer him a stone.
+
+[Illustration: FIG. 34.--Portrait of Kepler, older.]
+
+The contiguity of the lives of Kepler and Tycho furnishes a moral too
+obvious to need pointing out. What Kepler might have achieved had he
+been relieved of those ghastly struggles for subsistence one cannot
+tell, but this much is clear, that had Tycho been subjected to the same
+misfortune, instead of being born rich and being assisted by generous
+and enlightened patrons, he could have accomplished very little. His
+instruments, his observatory--the tools by which he did his work--would
+have been impossible for him. Frederick and Sophia of Denmark, and
+Rudolph of Bohemia, are therefore to be remembered as co-workers with
+him.
+
+Kepler, with his ill-health and inferior physical energy, was unable to
+command the like advantages. Much, nevertheless, he did; more one cannot
+but feel he might have done had he been properly helped. Besides, the
+world would have been free from the reproach of accepting the fruits of
+his bright genius while condemning the worker to a life of misery,
+relieved only by the beauty of his own thoughts and the ecstasy awakened
+in him by the harmony and precision of Nature.
+
+Concerning the method of Kepler, the mode by which he made his
+discoveries, we must remember that he gives us an account of all the
+steps, unsuccessful as well as successful, by which he travelled. He
+maps out his route like a traveller. In fact he compares himself to
+Columbus or Magellan, voyaging into unknown lands, and recording his
+wandering route. This being remembered, it will be found that his
+methods do not differ so utterly from those used by other philosophers
+in like case. His imagination was perhaps more luxuriant and was allowed
+freer play than most men's, but it was nevertheless always controlled by
+rigid examination and comparison of hypotheses with fact.
+
+Brewster says of him:--"Ardent, restless, burning to distinguish
+himself by discovery, he attempted everything; and once having obtained
+a glimpse of a clue, no labour was too hard in following or verifying
+it. A few of his attempts succeeded--a multitude failed. Those which
+failed seem to us now fanciful, those which succeeded appear to us
+sublime. But his methods were the same. When in search of what really
+existed he sometimes found it; when in pursuit of a chimaera he could not
+but fail; but in either case he displayed the same great qualities, and
+that obstinate perseverance which must conquer all difficulties except
+those really insurmountable."
+
+To realize what he did for astronomy, it is necessary for us now to
+consider some science still in its infancy. Astronomy is so clear and so
+thoroughly explored now, that it is difficult to put oneself into a
+contemporary attitude. But take some other science still barely
+developed: meteorology, for instance. The science of the weather, the
+succession of winds and rain, sunshine and frost, clouds and fog, is now
+very much in the condition of astronomy before Kepler.
+
+We have passed through the stage of ascribing atmospheric
+disturbances--thunderstorms, cyclones, earthquakes, and the like--to
+supernatural agency; we have had our Copernican era: not perhaps brought
+about by a single individual, but still achieved. Something of the laws
+of cyclone and anticyclone are known, and rude weather predictions
+across the Atlantic are roughly possible. Barometers and thermometers
+and anemometers, and all their tribe, represent the astronomical
+instruments in the island of Huen; and our numerous meteorological
+observatories, with their continual record of events, represent the work
+of Tycho Brahe.
+
+Observation is heaped on observation; tables are compiled; volumes are
+filled with data; the hours of sunshine are recorded, the fall of rain,
+the moisture in the air, the kind of clouds, the temperature--millions
+of facts; but where is the Kepler to study and brood over them? Where
+is the man to spend his life in evolving the beginnings of law and order
+from the midst of all this chaos?
+
+Perhaps as a man he may not come, but his era will come. Through this
+stage the science must pass, ere it is ready for the commanding
+intellect of a Newton.
+
+But what a work it will be for the man, whoever he be that undertakes
+it--a fearful monotonous grind of calculation, hypothesis, hypothesis,
+calculation, a desperate and groping endeavour to reconcile theories
+with facts.
+
+A life of such labour, crowned by three brilliant discoveries, the world
+owes (and too late recognizes its obligation) to the harshly treated
+German genius, Kepler.
+
+
+
+
+SUMMARY OF FACTS FOR LECTURES IV AND V
+
+
+In 1564, Michael Angelo died and Galileo was born; in 1642, Galileo died
+and Newton was born. Milton lived from 1608 to 1674.
+
+For teaching the plurality of worlds, with other heterodox doctrines,
+and refusing to recant, Bruno, after six years' imprisonment in Rome,
+was burnt at the stake on the 16th of February, 1600 A.D. A "natural"
+death in the dungeons of the Inquisition saved Antonio de Dominis, the
+explainer of the rainbow, from the same fate, but his body and books
+were publicly burned at Rome in 1624.
+
+The persecution of Galileo began in 1615, became intense in 1632, and so
+lasted till his death and after.
+
+*      *      *      *      *
+
+Galileo Galilei, eldest son of Vincenzo de Bonajuti de Galilei, a noble
+Florentine, was born at Pisa, 18th of February, 1564. At the age of 17
+was sent to the University of Pisa to study medicine. Observed the swing
+of a pendulum and applied it to count pulse-beats. Read Euclid and
+Archimedes, and could be kept at medicine no more. At 26 was appointed
+Lecturer in Mathematics at Pisa. Read Bruno and became smitten with the
+Copernican theory. Controverted the Aristotelians concerning falling
+bodies, at Pisa. Hence became unpopular and accepted a chair at Padua,
+1592. Invented a thermometer. Wrote on astronomy, adopting the Ptolemaic
+system provisionally, and so opened up a correspondence with Kepler,
+with whom he formed a friendship. Lectured on the new star of 1604, and
+publicly renounced the old systems of astronomy. Invented a calculating
+compass or "Gunter's scale." In 1609 invented a telescope, after hearing
+of a Dutch optician's discovery. Invented the microscope soon after.
+Rapidly completed a better telescope and began a survey of the heavens.
+On the 8th of January, 1610, discovered Jupiter's satellites. Observed
+the mountains in the moon, and roughly measured their height. Explained
+the visibility of the new moon by _earth-shine_. Was invited to the
+Grand Ducal Court of Tuscany by Cosmo de Medici, and appointed
+philosopher to that personage. Discovered innumerable new stars, and the
+nebulae. Observed a triple appearance of Saturn. Discovered the phases
+of Venus predicted by Copernicus, and spots on the sun. Wrote on
+floating bodies. Tried to get his satellites utilized for determining
+longitude at sea.
+
+Went to Rome to defend the Copernican system, then under official
+discussion, and as a result was formally forbidden ever to teach it. On
+the accession of Pope Urban VIII. in 1623, Galileo again visited Rome to
+pay his respects, and was well received. In 1632 appeared his
+"Dialogues" on the Ptolemaic and Copernican systems. Summoned to Rome,
+practically imprisoned, and "rigorously questioned." Was made to recant
+22nd of June, 1633. Forbidden evermore to publish anything, or to teach,
+or receive friends. Retired to Arcetri in broken down health. Death of
+his favourite daughter, Sister Maria Celeste. Wrote and meditated on the
+laws of motion. Discovered the moon's libration. In 1637 he became
+blind. The rigour was then slightly relaxed and many visited him: among
+them John Milton. Died 8th of January, 1642, aged 78. As a prisoner of
+the Inquisition his right to make a will or to be buried in consecrated
+ground was disputed. Many of his manuscripts were destroyed.
+
+Galileo, besides being a singularly clear-headed thinker and
+experimental genius, was also something of a musician, a poet, and an
+artist. He was full of humour as well as of solid common-sense, and his
+literary style is brilliant. Of his scientific achievements those now
+reckoned most weighty, are the discovery of the Laws of Motion, and the
+laying of the foundations of Mechanics.
+
+_Particulars of Jupiter's Satellites,
+Illustrating their obedience to Kepler's third law._
+
+--------------------------------------------------------------------------
+          |       |            |  Distance|         |         |   T^2
+          |       |  Time of   |    from  |         |         |   ----
+Satellite.|Diameter revolution | Jupiter, |  T^2    |  d^3    |   d^3
+          | miles.|  in hours. |in Jovian |         |         | which is
+          | miles |   (T)      |   radii. |         |         |practically
+          |       |            |    (d)   |         |         | constant.
+----------|-------|------------|----------|---------|---------|-----------
+No. 1.    |  2437 |   42.47    |   6.049  |  1803.7 |  221.44 |  8.149
+No. 2.    |  2188 |   85.23    |   9.623  |  7264.1 |  891.11 |  8.152
+No. 3.    |  3575 |  177.72    |  15.350  | 29488.  | 3916.8  |  8.153
+No. 4.    |  3059 |  400.53    |  26.998  |160426.  |19679.   |  8.152
+--------------------------------------------------------------------------
+
+The diameter of Jupiter is 85,823 miles.
+
+
+_Falling Bodies._
+
+
+Since all bodies fall at the same rate, except for the disturbing effect
+of the resistance of the air, a statement of their rates of fall is of
+interest. In one second a freely falling body near the earth is found to
+drop 16 feet. In two seconds it drops 64 feet altogether, viz. 16 feet
+in the first, and 48 feet in the next second; because at the beginning
+of every second after the first it has the accumulated velocity of
+preceding seconds. The height fallen by a dropped body is not
+proportional to the time simply, but to what is rather absurdly called
+the square of the time, _i.e._ the time multiplied by itself.
+
+For instance, in 3 seconds it drops 9 x 16 = 144 feet; in 4 seconds 16 x
+16, or 256 feet, and so on. The distances travelled in 1, 2, 3, 4, &c.,
+seconds by a body dropped from rest and not appreciably resisted by the
+air, are 1, 4, 9, 16, 25, &c., respectively, each multiplied by the
+constant 16 feet.
+
+Another way of stating the law is to say that the heights travelled in
+successive seconds proceed in the proportion 1, 3, 5, 7, 9, &c.; again
+multiplied by 16 feet in each case.
+
+[Illustration: FIG. 35.--Curve described by a projectile, showing how it
+drops from the line of fire, _O D_, in successive seconds, the same
+distances _AP_, _BQ_, _CR_, &c., as are stated above for a dropped
+body.]
+
+All this was experimentally established by Galileo.
+
+A body takes half a second to drop 4 feet; and a quarter of a second to
+drop 1 foot. The easiest way of estimating a quarter of a second with
+some accuracy is to drop a bullet one foot.
+
+A bullet thrown or shot in any direction falls just as much as if merely
+dropped; but instead of falling from the starting-point it drops
+vertically from the line of fire. (See fig. 35).
+
+The rate of fall depends on the intensity of gravity; if it could be
+doubled, a body would fall twice as far in the same time; but to make it
+fall a given distance in half the time the intensity of gravity would
+have to be quadrupled. At a place where the intensity of gravity is
+1/3600 of what it is here, a body would fall as far in a minute as it
+now falls in a second. Such a place occurs at about the distance of the
+moon (_cf._ page 177).
+
+The fact that the height fallen through is proportional to the square
+of the time proves that the attraction of the earth or the intensity of
+gravity is sensibly constant throughout ordinary small ranges. Over
+great distances of fall, gravity cannot be considered constant; so for
+things falling through great spaces the Galilean law of the square of
+the time does not hold.
+
+The fact that things near the earth fall 16 feet in the first second
+proves that the intensity of ordinary terrestrial gravity is 32 British
+units of force per pound of matter.
+
+The fact that all bodies fall at the same rate (when the resistance of
+the air is eliminated), proves that weight is proportional to mass; or
+more explicitly, that the gravitative attraction of the earth on matter
+near its surface depends on the amount of that matter, as estimated by
+its inertia, and on nothing else.
+
+
+
+
+LECTURE IV
+
+GALILEO AND THE INVENTION OF THE TELESCOPE
+
+
+Contemporary with the life of Kepler, but overlapping it at both ends,
+comes the great and eventful life of Galileo Galilei,[5] a man whose
+influence on the development of human thought has been greater than that
+of any man whom we have yet considered, and upon whom, therefore, it is
+necessary for us, in order to carry out the plan of these lectures, to
+bestow much time. A man of great and wide culture, a so-called universal
+genius, it is as an experimental philosopher that he takes the first
+rank. In this capacity he must be placed alongside of Archimedes, and it
+is pretty certain that between the two there was no man of magnitude
+equal to either in experimental philosophy. It is perhaps too bold a
+speculation, but I venture to doubt whether in succeeding generations we
+find his equal in the domain of purely experimental science until we
+come to Faraday. Faraday was no doubt his superior, but I know of no
+other of whom the like can unhesitatingly be said. In mathematical and
+deductive science, of course, it is quite otherwise. Kepler, for
+instance, and many men before and since, have far excelled Galileo in
+mathematical skill and power, though at the same time his achievements
+in this department are by no means to be despised.
+
+Born at Pisa three centuries ago, on the very day that Michael Angelo
+lay dying in Rome, he inherited from his father a noble name, cultivated
+tastes, a keen love of truth, and an impoverished patrimony. Vincenzo de
+Galilei, a descendant of the important Bonajuti family, was himself a
+mathematician and a musician, and in a book of his still extant he
+declares himself in favour of free and open inquiry into scientific
+matters, unrestrained by the weight of authority and tradition.
+
+In all probability the son imbibed these precepts: certainly he acted on
+them.
+
+Vincenzo, having himself experienced the unremunerative character of
+scientific work, had a horror of his son's taking to it, especially as
+in his boyhood he was always constructing ingenious mechanical toys, and
+exhibiting other marks of precocity. So the son was destined for
+business--to be, in fact, a cloth-dealer. But he was to receive a good
+education first, and was sent to an excellent convent school.
+
+Here he made rapid progress, and soon excelled in all branches of
+classics and literature. He delighted in poetry, and in later years
+wrote several essays on Dante, Tasso, and Ariosto, besides composing
+some tolerable poems himself. He played skilfully on several musical
+instruments, especially on the lute, of which indeed he became a master,
+and on which he solaced himself when quite an old man. Besides this he
+seems to have had some skill as an artist, which was useful afterwards
+in illustrating his discoveries, and to have had a fine sensibility as
+an art critic, for we find several eminent painters of that day
+acknowledging the value of the opinion of the young Galileo.
+
+Perceiving all this display of ability, the father wisely came to the
+conclusion that the selling of woollen stuffs would hardly satisfy his
+aspirations for long, and that it was worth a sacrifice to send him to
+the University. So to the University of his native town he went, with
+the avowed object of studying medicine, that career seeming the most
+likely to be profitable. Old Vincenzo's horror of mathematics or science
+as a means of obtaining a livelihood is justified by the fact that while
+the University Professor of Medicine received 2,000 scudi a year, the
+Professor of Mathematics had only 60, that is L13 a year, or 7-1/2_d._ a
+day.
+
+So the son had been kept properly ignorant of such poverty-stricken
+subjects, and to study medicine he went.
+
+But his natural bent showed itself even here. For praying one day in the
+Cathedral, like a good Catholic as he was all his life, his attention
+was arrested by the great lamp which, after lighting it, the verger had
+left swinging to and fro. Galileo proceeded to time its swings by the
+only watch he possessed--viz., his own pulse. He noticed that the time
+of swing remained as near as he could tell the same, notwithstanding the
+fact that the swings were getting smaller and smaller.
+
+By subsequent experiment he verified the law, and the isochronism of the
+pendulum was discovered. An immensely important practical discovery
+this, for upon it all modern clocks are based; and Huyghens soon applied
+it to the astronomical clock, which up to that time had been a crude and
+quite untrustworthy instrument.
+
+The best clock which Tycho Brahe could get for his observatory was
+inferior to one that may now be purchased for a few shillings; and this
+change is owing to the discovery of the pendulum by Galileo. Not that he
+applied it to clocks; he was not thinking of astronomy, he was thinking
+of medicine, and wanted to count people's pulses. The pendulum served;
+and "pulsilogies," as they were called, were thus introduced to and used
+by medical practitioners.
+
+The Tuscan Court came to Pisa for the summer months, for it was then a
+seaside place, and among the suite was Ostillio Ricci, a distinguished
+mathematician and old friend of the Galileo family. The youth visited
+him, and one day, it is said, heard a lesson in Euclid being given by
+Ricci to the pages while he stood outside the door entranced. Anyhow he
+implored Ricci to help him into some knowledge of mathematics, and the
+old man willingly consented. So he mastered Euclid and passed on to
+Archimedes, for whom he acquired a great veneration.
+
+His father soon heard of this obnoxious proclivity, and did what he
+could to divert him back to medicine again. But it was no use.
+Underneath his Galen and Hippocrates were secreted copies of Euclid and
+Archimedes, to be studied at every available opportunity. Old Vincenzo
+perceived the bent of genius to be too strong for him, and at last gave
+way.
+
+[Illustration: FIG. 36.--Two forms of pulsilogy. The string is wound up
+till the swinging weight keeps time with the pulse, and the position of
+a bead or of an index connected with the string is then read on a scale
+or dial.]
+
+With prodigious rapidity the released philosopher now assimilated the
+elements of mathematics and physics, and at twenty-six we find him
+appointed for three years to the University Chair of Mathematics, and
+enjoying the paternally dreaded stipend of 7-1/2_d._ a day.
+
+Now it was that he pondered over the laws of falling bodies. He
+verified, by experiment, the fact that the velocity acquired by falling
+down any slope of given height was independent of the angle of slope.
+Also, that the height fallen through was proportional to the square of
+the time.
+
+Another thing he found experimentally was that all bodies, heavy and
+light, fell at the same rate, striking the ground at the same time.[6]
+
+Now this was clean contrary to what he had been taught. The physics of
+those days were a simple reproduction of statements in old books.
+Aristotle had asserted certain things to be true, and these were
+universally believed. No one thought of trying the thing to see if it
+really were so. The idea of making an experiment would have savoured of
+impiety, because it seemed to tend towards scepticism, and cast a doubt
+on a reverend authority.
+
+Young Galileo, with all the energy and imprudence of youth (what a
+blessing that youth has a little imprudence and disregard of
+consequences in pursuing a high ideal!), as soon as he perceived that
+his instructors were wrong on the subject of falling bodies, instantly
+informed them of the fact. Whether he expected them to be pleased or not
+is a question. Anyhow, they were not pleased, but were much annoyed by
+his impertinent arrogance.
+
+It is, perhaps, difficult for us now to appreciate precisely their
+position. These doctrines of antiquity, which had come down hoary with
+age, and the discovery of which had reawakened learning and quickened
+intellectual life, were accepted less as a science or a philosophy, than
+as a religion. Had they regarded Aristotle as a verbally inspired
+writer, they could not have received his statements with more
+unhesitating conviction. In any dispute as to a question of fact, such
+as the one before us concerning the laws of falling bodies, their method
+was not to make an experiment, but to turn over the pages of Aristotle;
+and he who could quote chapter and verse of this great writer was held
+to settle the question and raise it above the reach of controversy.
+
+It is very necessary for us to realize this state of things clearly,
+because otherwise the attitude of the learned of those days towards
+every new discovery seems stupid and almost insane. They had a
+crystallized system of truth, perfect, symmetrical--it wanted no
+novelty, no additions; every addition or growth was an imperfection, an
+excrescence, a deformity. Progress was unnecessary and undesired. The
+Church had a rigid system of dogma, which must be accepted in its
+entirety on pain of being treated as a heretic. Philosophers had a
+cast-iron system of truth to match--a system founded upon Aristotle--and
+so interwoven with the great theological dogmas that to question one was
+almost equivalent to casting doubt upon the other.
+
+In such an atmosphere true science was impossible. The life-blood of
+science is growth, expansion, freedom, development. Before it could
+appear it must throw off these old shackles of centuries. It must burst
+its old skin, and emerge, worn with the struggle, weakly and
+unprotected, but free and able to grow and to expand. The conflict was
+inevitable, and it was severe. Is it over yet? I fear not quite, though
+so nearly as to disturb science hardly at all. Then it was different; it
+was terrible. Honour to the men who bore the first shock of the battle!
+
+Now Aristotle had said that bodies fell at rates depending on their
+weight.
+
+A 5 lb. weight would fall five times as quick as a 1 lb. weight; a 50
+lb. weight fifty times as quick, and so on.
+
+Why he said so nobody knows. He cannot have tried. He was not above
+trying experiments, like his smaller disciples; but probably it never
+occurred to him to doubt the fact. It seems so natural that a heavy body
+should fall quicker than a light one; and perhaps he thought of a stone
+and a feather, and was satisfied.
+
+Galileo, however, asserted that the weight did not matter a bit, that
+everything fell at the same rate (even a stone and a feather, but for
+the resistance of the air), and would reach the ground in the same time.
+
+And he was not content to be pooh-poohed and snubbed. He knew he was
+right, and he was determined to make every one see the facts as he saw
+them. So one morning, before the assembled University, he ascended the
+famous leaning tower, taking with him a 100 lb. shot and a 1 lb. shot.
+He balanced them on the edge of the tower, and let them drop together.
+Together they fell, and together they struck the ground.
+
+The simultaneous clang of those two weights sounded the death-knell of
+the old system of philosophy, and heralded the birth of the new.
+
+But was the change sudden? Were his opponents convinced? Not a jot.
+Though they had seen with their eyes, and heard with their ears, the
+full light of heaven shining upon them, they went back muttering and
+discontented to their musty old volumes and their garrets, there to
+invent occult reasons for denying the validity of the observation, and
+for referring it to some unknown disturbing cause.
+
+They saw that if they gave way on this one point they would be letting
+go their anchorage, and henceforward would be liable to drift along with
+the tide, not knowing whither. They dared not do this. No; they _must_
+cling to the old traditions; they could not cast away their rotting
+ropes and sail out on to the free ocean of God's truth in a spirit of
+fearless faith.
+
+[Illustration: FIG. 37.--Tower of Pisa.]
+
+Yet they had received a shock: as by a breath of fresh salt breeze and
+a dash of spray in their faces, they had been awakened out of their
+comfortable lethargy. They felt the approach of a new era.
+
+Yes, it was a shock; and they hated the young Galileo for giving it
+them--hated him with the sullen hatred of men who fight for a lost and
+dying cause.
+
+We need scarcely blame these men; at least we need not blame them
+overmuch. To say that they acted as they did is to say that they were
+human, were narrow-minded, and were the apostles of a lost cause. But
+_they_ could not know this; _they_ had no experience of the past to
+guide them; the conditions under which they found themselves were novel,
+and had to be met for the first time. Conduct which was excusable then
+would be unpardonable now, in the light of all this experience to guide
+us. Are there any now who practically repeat their error, and resist new
+truth? who cling to any old anchorage of dogma, and refuse to rise with
+the tide of advancing knowledge? There may be some even now.
+
+Well, the unpopularity of Galileo smouldered for a time, until, by
+another noble imprudence, he managed to offend a semi-royal personage,
+Giovanni de Medici, by giving his real opinion, when consulted, about a
+machine which de Medici had invented for cleaning out the harbour of
+Leghorn. He said it was as useless as it in fact turned out to be.
+Through the influence of the mortified inventor he lost favour at Court;
+and his enemies took advantage of the fact to render his chair
+untenable. He resigned before his three years were up, and retired to
+Florence.
+
+His father at this time died, and the family were left in narrow
+circumstances. He had a brother and three sisters to provide for.
+
+He was offered a professorship at Padua for six years by the Senate of
+Venice, and willingly accepted it.
+
+Now began a very successful career. His introductory address was marked
+by brilliant eloquence, and his lectures soon acquired fame. He wrote
+for his pupils on the laws of motion, on fortifications, on sundials, on
+mechanics, and on the celestial globe: some of these papers are now
+lost, others have been printed during the present century.
+
+Kepler sent him a copy of his new book, _Mysterium Cosmographicum_, and
+Galileo in thanking him for it writes him the following letter:--[7]
+
+     "I count myself happy, in the search after truth, to have so great
+     an ally as yourself, and one who is so great a friend of the truth
+     itself. It is really pitiful that there are so few who seek truth,
+     and who do not pursue a perverse method of philosophising. But this
+     is not the place to mourn over the miseries of our times, but to
+     congratulate you on your splendid discoveries in confirmation of
+     truth. I shall read your book to the end, sure of finding much that
+     is excellent in it. I shall do so with the more pleasure, because
+     _I have been for many years an adherent of the Copernican system_,
+     and it explains to me the causes of many of the appearances of
+     nature which are quite unintelligible on the commonly accepted
+     hypothesis. _I have collected many arguments for the purpose of
+     refuting the latter_; but I do not venture to bring them to the
+     light of publicity, for fear of sharing the fate of our master,
+     Copernicus, who, although he has earned immortal fame with some,
+     yet with very many (so great is the number of fools) has become an
+     object of ridicule and scorn. I should certainly venture to publish
+     my speculations if there were more people like you. But this not
+     being the case, I refrain from such an undertaking."
+
+Kepler urged him to publish his arguments in favour of the Copernican
+theory, but he hesitated for the present, knowing that his declaration
+would be received with ridicule and opposition, and thinking it wiser to
+get rather more firmly seated in his chair before encountering the
+storm of controversy.
+
+The six years passed away, and the Venetian Senate, anxious not to lose
+so bright an ornament, renewed his appointment for another six years at
+a largely increased salary.
+
+Soon after this appeared a new star, the stella nova of 1604, not the
+one Tycho had seen--that was in 1572--but the same that Kepler was so
+much interested in.
+
+Galileo gave a course of three lectures upon it to a great audience. At
+the first the theatre was over-crowded, so he had to adjourn to a hall
+holding 1000 persons. At the next he had to lecture in the open air.
+
+He took occasion to rebuke his hearers for thronging to hear about an
+ephemeral novelty, while for the much more wonderful and important
+truths about the permanent stars and facts of nature they had but deaf
+ears.
+
+But the main point he brought out concerning the new star was that it
+upset the received Aristotelian doctrine of the immutability of the
+heavens. According to that doctrine the heavens were unchangeable,
+perfect, subject neither to growth nor to decay. Here was a body, not a
+meteor but a real distant star, which had not been visible and which
+would shortly fade away again, but which meanwhile was brighter than
+Jupiter.
+
+The staff of petrified professorial wisdom were annoyed at the
+appearance of the star, still more at Galileo's calling public attention
+to it; and controversy began at Padua. However, he accepted it; and now
+boldly threw down the gauntlet in favour of the Copernican theory,
+utterly repudiating the old Ptolemaic system which up to that time he
+had taught in the schools according to established custom.
+
+The earth no longer the only world to which all else in the firmament
+were obsequious attendants, but a mere insignificant speck among the
+host of heaven! Man no longer the centre and cynosure of creation, but,
+as it were, an insect crawling on the surface of this little speck! All
+this not set down in crabbed Latin in dry folios for a few learned
+monks, as in Copernicus's time, but promulgated and argued in rich
+Italian, illustrated by analogy, by experiment, and with cultured wit;
+taught not to a few scholars here and there in musty libraries, but
+proclaimed in the vernacular to the whole populace with all the energy
+and enthusiasm of a recent convert and a master of language! Had a
+bombshell been exploded among the fossilized professors it had been less
+disturbing.
+
+But there was worse in store for them.
+
+A Dutch optician, Hans Lippershey by name, of Middleburg, had in his
+shop a curious toy, rigged up, it is said, by an apprentice, and made
+out of a couple of spectacle lenses, whereby, if one looked through it,
+the weather-cock of a neighbouring church spire was seen nearer and
+upside down.
+
+The tale goes that the Marquis Spinola, happening to call at the shop,
+was struck with the toy and bought it. He showed it to Prince Maurice of
+Nassau, who thought of using it for military reconnoitring. All this is
+trivial. What is important is that some faint and inaccurate echo of
+this news found its way to Padua, and into the ears of Galileo.
+
+The seed fell on good soil. All that night he sat up and pondered. He
+knew about lenses and magnifying glasses. He had read Kepler's theory of
+the eye, and had himself lectured on optics. Could he not hit on the
+device and make an instrument capable of bringing the heavenly bodies
+nearer? Who knew what marvels he might not so perceive! By morning he
+had some schemes ready to try, and one of them was successful.
+Singularly enough it was not the same plan as the Dutch optician's, it
+was another mode of achieving the same end.
+
+He took an old small organ pipe, jammed a suitably chosen spectacle
+glass into either end, one convex the other concave, and behold, he had
+the half of a wretchedly bad opera glass capable of magnifying three
+times. It was better than the Dutchman's, however; it did not invert.
+
+     It is easy to understand the general principle of a telescope. A
+     general knowledge of the common magnifying glass may be assumed.
+     Roger Bacon knew about lenses; and the ancients often refer to
+     them, though usually as burning glasses. The magnifying power of
+     globes of water must have been noticed soon after the discovery of
+     glass and the art of working it.
+
+     A magnifying glass is most simply thought of as an additional lens
+     to the eye. The eye has a lens by which ordinary vision is
+     accomplished, an extra glass lens strengthens it and enables
+     objects to be seen nearer and therefore apparently bigger. But to
+     apply a magnifying glass to distant objects is impossible. In order
+     to magnify distant objects, another function of lenses has also to
+     be employed, viz., their power of forming real images, the power on
+     which their use as burning-glasses depends: for the best focus is
+     an image of the sun. Although the object itself is inaccessible,
+     the image of it is by no means so, and to the image a magnifier can
+     be applied. This is exactly what is done in the telescope; the
+     object glass or large lens forms an image, which is then looked at
+     through a magnifying glass or eye-piece.
+
+     Of course the image is nothing like so big as the object. For
+     astronomical objects it is almost infinitely less; still it is an
+     exact representation at an accessible place, and no one expects a
+     telescope to show distant bodies as big as they really are. All it
+     does is to show them bigger than they could be seen without it.
+
+     But if the objects are not distant, the same principle may still be
+     applied, and two lenses may be used, one to form an image, the
+     other to magnify it; only if the object can be put where we please,
+     we can easily place it so that its image is already much bigger
+     than the object even before magnification by the eye lens. This is
+     the compound microscope, the invention of which soon followed the
+     telescope. In fact the two instruments shade off into one another,
+     so that the reading telescope or reading microscope of a laboratory
+     (for reading thermometers, and small divisions generally) goes by
+     either name at random.
+
+     The arrangement so far described depicts things on the retina the
+     unaccustomed way up. By using a concave glass instead of a convex,
+     and placing it so as to prevent any image being formed, except on
+     the retina direct, this inconvenience is avoided.
+
+[Illustration: FIG. 38.--View of the half-moon in small telescope. The
+darker regions, or plains, used to be called "seas."]
+
+Such a thing as Galileo made may now be bought at a toy-shop for I
+suppose half a crown, and yet what a potentiality lay in that "glazed
+optic tube," as Milton called it. Away he went with it to Venice and
+showed it to the Signoria, to their great astonishment. "Many noblemen
+and senators," says Galileo, "though of advanced age, mounted to the top
+of one of the highest towers to watch the ships, which were visible
+through my glass two hours before they were seen entering the harbour,
+for it makes a thing fifty miles off as near and clear as if it were
+only five." Among the people too the instrument excited the greatest
+astonishment and interest, so that he was nearly mobbed. The Senate
+hinted to him that a present of the instrument would not be
+unacceptable, so Galileo took the hint and made another for them.
+
+[Illustration: FIG. 39.--Portion of the lunar surface more highly
+magnified, showing the shadows of a mountain range, deep pits, and other
+details.]
+
+They immediately doubled his salary at Padua, making it 1000 florins,
+and confirmed him in the enjoyment of it for life.
+
+He now eagerly began the construction of a larger and better instrument.
+Grinding the lenses with his own hands with consummate skill, he
+succeeded in making a telescope magnifying thirty times. Thus equipped
+he was ready to begin a survey of the heavens.
+
+[Illustration: FIG. 40.--Another portion of the lunar surface, showing a
+so-called crater or vast lava pool and other evidences of ancient heat
+unmodified by water.]
+
+The first object he carefully examined was naturally the moon. He found
+there everything at first sight very like the earth, mountains and
+valleys, craters and plains, rocks, and apparently seas. You may imagine
+the hostility excited among the Aristotelian philosophers, especially no
+doubt those he had left behind at Pisa, on the ground of his spoiling
+the pure, smooth, crystalline, celestial face of the moon as they had
+thought it, and making it harsh and rugged and like so vile and ignoble
+a body as the earth.
+
+[Illustration: FIG. 41.--Lunar landscape showing earth. The earth would
+be a stationary object in the moon's sky: its only apparent motion being
+a slow oscillation as of a pendulum (the result of the moon's
+libration).]
+
+He went further, however, into heterodoxy than this--he not only made
+the moon like the earth, but he made the earth shine like the moon. The
+visibility of "the old moon in the new moon's arms" he explained by
+earth-shine. Leonardo had given the same explanation a century before.
+Now one of the many stock arguments against Copernican theory of the
+earth being a planet like the rest was that the earth was dull and dark
+and did not shine. Galileo argued that it shone just as much as the moon
+does, and in fact rather more--especially if it be covered with clouds.
+One reason of the peculiar brilliancy of Venus is that she is a very
+cloudy planet.[8] Seen from the moon the earth would look exactly as the
+moon does to us, only a little brighter and sixteen times as big (four
+times the diameter).
+
+[Illustration: FIG. 42.--Galileo's method of estimating the height of
+lunar mountain.
+
+_AB'BC_ is the illuminated half of the moon. _SA_ is a solar ray just
+catching the peak of the mountain _M_. Then by geometry, as _MN_ is to
+_MA_, so is _MA_ to _MB'_; whence the height of the mountain, _MN_, can
+be determined. The earth and spectator are supposed to be somewhere in
+the direction _BA_ produced, _i.e._ towards the top of the page.]
+
+     Galileo made a very good estimate of the height of lunar mountains,
+     of which many are five miles high and some as much as seven. He did
+     this simply by measuring from the half-moon's straight edge the
+     distance at which their peaks caught the rising or setting sun. The
+     above simple diagram shows that as this distance is to the diameter
+     of the moon, so is the height of the sun-tipped mountain to the
+     aforesaid distance.
+
+Wherever Galileo turned his telescope new stars appeared. The Milky Way,
+which had so puzzled the ancients, was found to be composed of stars.
+Stars that appeared single to the eye were some of them found to be
+double; and at intervals were found hazy nebulous wisps, some of which
+seemed to be star clusters, while others seemed only a fleecy cloud.
+
+[Illustration: FIG. 43.--Some clusters and nebulae.]
+
+[Illustration: FIG. 44.--Jupiter's satellites, showing the stages of
+their discovery.]
+
+Now we come to his most brilliant, at least his most sensational,
+discovery. Examining Jupiter minutely on January 7, 1610, he noticed
+three little stars near it, which he noted down as fixing its then
+position. On the following night Jupiter had moved to the other side of
+the three stars. This was natural enough, but was it moving the right
+way? On examination it appeared not. Was it possible the tables were
+wrong? The next evening was cloudy, and he had to curb his feverish
+impatience. On the 10th there were only two, and those on the other
+side. On the 11th two again, but one bigger than the other. On the 12th
+the three re-appeared, and on the 13th there were four. No more
+appeared.
+
+Jupiter then had moons like the earth, four of them in fact, and they
+revolved round him in periods which were soon determined.
+
+     The reason why they were not all visible at first, and why their
+     visibility so rapidly changes, is because they revolve round him
+     almost in the plane of our vision, so that sometimes they are in
+     front and sometimes behind him, while again at other times they
+     plunge into his shadow and are thus eclipsed from the light of the
+     sun which enables us to see them. A large modern telescope will
+     show the moons when in front of Jupiter, but small telescopes will
+     only show them when clear of the disk and shadow. Often all four
+     can be thus seen, but three or two is a very common amount of
+     visibility. Quite a small telescope, such as a ship's telescope, if
+     held steadily, suffices to show the satellites of Jupiter, and very
+     interesting objects they are. They are of habitable size, and may
+     be important worlds for all we know to the contrary.
+
+The news of the discovery soon spread and excited the greatest interest
+and astonishment. Many of course refused to believe it. Some there were
+who having been shown them refused to believe their eyes, and asserted
+that although the telescope acted well enough for terrestrial objects,
+it was altogether false and illusory when applied to the heavens. Others
+took the safer ground of refusing to look through the glass. One of
+these who would not look at the satellites happened to die soon
+afterwards. "I hope," says Galileo, "that he saw them on his way to
+heaven."
+
+The way in which Kepler received the news is characteristic, though by
+adding four to the supposed number of planets it might have seemed to
+upset his notions about the five regular solids.
+
+     He says,[9] "I was sitting idle at home thinking of you, most
+     excellent Galileo, and your letters, when the news was brought me
+     of the discovery of four planets by the help of the double
+     eye-glass. Wachenfels stopped his carriage at my door to tell me,
+     when such a fit of wonder seized me at a report which seemed so
+     very absurd, and I was thrown into such agitation at seeing an old
+     dispute between us decided in this way, that between his joy, my
+     colouring, and the laughter of us both, confounded as we were by
+     such a novelty, we were hardly capable, he of speaking, or I of
+     listening....
+
+     "On our separating, I immediately fell to thinking how there could
+     be any addition to the number of planets without overturning my
+     _Mysterium Cosmographicon_, published thirteen years ago, according
+     to which Euclid's five regular solids do not allow more than six
+     planets round the sun.
+
+     "But I am so far from disbelieving the existence of the four
+     circumjovial planets that I long for a telescope to anticipate you
+     if possible in discovering two round Mars (as the proportion seems
+     to me to require) six or eight round Saturn, and one each round
+     Mercury and Venus."
+
+[Illustration: FIG. 45.--Eclipses of Jupiter's satellites. The diagram
+shows the first (_i.e._ the nearest) moon in Jupiter's shadow, the
+second as passing between earth and Jupiter, and appearing to transit
+his disk, the third as on the verge of entering his shadow, and the
+fourth quite plainly and separately visible.]
+
+As an illustration of the opposite school, I will take the following
+extract from Francesco Sizzi, a Florentine astronomer, who argues
+against the discovery thus:--
+
+     "There are seven windows in the head, two nostrils, two eyes, two
+     ears, and a mouth; so in the heavens there are two favourable
+     stars, two unpropitious, two luminaries, and Mercury alone
+     undecided and indifferent. From which and many other similar
+     phenomena of nature, such as the seven metals, &c., which it were
+     tedious to enumerate, we gather that the number of planets is
+     necessarily seven.
+
+     "Moreover, the satellites are invisible to the naked eye, and
+     therefore can have no influence on the earth, and therefore would
+     be useless, and therefore do not exist.
+
+     "Besides, the Jews and other ancient nations as well as modern
+     Europeans have adopted the division of the week into seven days,
+     and have named them from the seven planets: now if we increase the
+     number of the planets this whole system falls to the ground."
+
+To these arguments Galileo replied that whatever their force might be as
+a reason for believing beforehand that no more than seven planets would
+be discovered, they hardly seemed of sufficient weight to destroy the
+new ones when actually seen.
+
+Writing to Kepler at this time, Galileo ejaculates:
+
+     "Oh, my dear Kepler, how I wish that we could have one hearty laugh
+     together! Here, at Padua, is the principal professor of philosophy
+     whom I have repeatedly and urgently requested to look at the moon
+     and planets through my glass, which he pertinaciously refuses to
+     do. Why are you not here? What shouts of laughter we should have at
+     this glorious folly! And to hear the professor of philosophy at
+     Pisa labouring before the grand duke with logical arguments, as if
+     with magical incantations, to charm the new planets out of the
+     sky."
+
+A young German _protege_ of Kepler, Martin Horkey, was travelling in
+Italy, and meeting Galileo at Bologna was favoured with a view through
+his telescope. But supposing that Kepler must necessarily be jealous of
+such great discoveries, and thinking to please him, he writes, "I cannot
+tell what to think about these observations. They are stupendous, they
+are wonderful, but whether they are true or false I cannot tell." He
+concludes, "I will never concede his four new planets to that Italian
+from Padua though I die for it." So he published a pamphlet asserting
+that reflected rays and optical illusions were the sole cause of the
+appearance, and that the only use of the imaginary planets was to
+gratify Galileo's thirst for gold and notoriety.
+
+When after this performance he paid a visit to his old instructor
+Kepler, he got a reception which astonished him. However, he pleaded so
+hard to be forgiven that Kepler restored him to partial favour, on this
+condition, that he was to look again at the satellites, and this time to
+see them and own that they were there.
+
+By degrees the enemies of Galileo were compelled to confess to the truth
+of the discovery, and the next step was to outdo him. Scheiner counted
+five, Rheiter nine, and others went as high as twelve. Some of these
+were imaginary, some were fixed stars, and four satellites only are
+known to this day.[10]
+
+Here, close to the summit of his greatness, we must leave him for a
+time. A few steps more and he will be on the brow of the hill; a short
+piece of table-land, and then the descent begins.
+
+
+
+
+LECTURE V
+
+GALILEO AND THE INQUISITION
+
+
+One sinister event occurred while Galileo was at Padua, some time before
+the era we have now arrived at, before the invention of the
+telescope--two years indeed after he had first gone to Padua; an event
+not directly concerning Galileo, but which I must mention because it
+must have shadowed his life both at the time and long afterwards. It was
+the execution of Giordano Bruno for heresy. This eminent philosopher had
+travelled largely, had lived some time in England, had acquired new and
+heterodox views on a variety of subjects, and did not hesitate to
+propound them even after he had returned to Italy.
+
+The Copernican doctrine of the motion of the earth was one of his
+obnoxious heresies. Being persecuted to some extent by the Church, Bruno
+took refuge in Venice--a free republic almost independent of the
+Papacy--where he felt himself safe. Galileo was at Padua hard by: the
+University of Padua was under the government of the Senate of Venice:
+the two men must in all probability have met.
+
+Well, the Inquisition at Rome sent messengers to Venice with a demand
+for the extradition of Bruno--they wanted him at Rome to try him for
+heresy.
+
+In a moment of miserable weakness the Venetian republic gave him up, and
+Bruno was taken to Rome. There he was tried, and cast into the dungeons
+for six years, and because he entirely refused to recant, was at length
+delivered over to the secular arm and burned at the stake on 16th
+February, Anno Domini 1600.
+
+This event could not but have cast a gloom over the mind of lovers and
+expounders of truth, and the lesson probably sank deep into Galileo's
+soul.
+
+In dealing with these historic events will you allow me to repudiate
+once for all the slightest sectarian bias or meaning. I have nothing to
+do with Catholic or Protestant as such. I have nothing to do with the
+Church of Rome as such. I am dealing with the history of science. But
+historically at one period science and the Church came into conflict. It
+was not specially one Church rather than another--it was the Church in
+general, the only one that then existed in those countries.
+Historically, I say, they came into conflict, and historically the
+Church was the conqueror. It got its way; and science, in the persons of
+Bruno, Galileo, and several others, was vanquished.
+
+Such being the facts, there is no help but to mention them in dealing
+with the history of science.
+
+Doubtless _now_ the Church regards it as an unhappy victory, and gladly
+would ignore this painful struggle. This, however, is impossible. With
+their creed the Churchmen of that day could act in no other way. They
+were bound to prosecute heresy, and they were bound to conquer in the
+struggle or be themselves shattered.
+
+But let me insist on the fact that no one accuses the ecclesiastical
+courts of crime or evil motives. They attacked heresy after their
+manner, as the civil courts attacked witchcraft after _their_ manner.
+Both erred grievously, but both acted with the best intentions.
+
+We must remember, moreover, that his doctrines were scientifically
+heterodox, and the University Professors of that day were probably quite
+as ready to condemn them as the Church was. To realise the position we
+must think of some subjects which _to-day_ are scientifically
+heterodox, and of the customary attitude adopted towards them by
+persons of widely differing creeds.
+
+If it be contended now, as it is, that the ecclesiastics treated Galileo
+well, I admit it freely: they treated him as well as they possibly
+could. They overcame him, and he recanted; but if he had not recanted,
+if he had persisted in his heresy, they would--well, they would still
+have treated his soul well, but they would have set fire to his body.
+Their mistake consisted not in cruelty, but in supposing themselves the
+arbiters of eternal truth; and by no amount of slurring and glossing
+over facts can they evade the responsibility assumed by them on account
+of this mistaken attitude.
+
+I am not here attacking the dogma of Papal Infallibility: it is
+historically, I believe, quite unaffected by the controversy respecting
+the motion of the earth, no Papal edict _ex cathedra_ having been
+promulgated on the subject.
+
+We left Galileo standing at his telescope and beginning his survey of
+the heavens. We followed him indeed through a few of his first great
+discoveries--the discovery of the mountains and other variety of surface
+in the moon, of the nebulae and a multitude of faint stars, and lastly of
+the four satellites of Jupiter.
+
+This latter discovery made an immense sensation, and contributed its
+share to his removal from Padua, which quickly followed it, as I shall
+shortly narrate; but first I think it will be best to continue our
+survey of his astronomical discoveries without regard to the place
+whence they were made.
+
+Before the end of the year Galileo had made another discovery--this time
+on Saturn. But to guard against the host of plagiarists and impostors,
+he published it in the form of an anagram, which, at the request of the
+Emperor Rudolph (a request probably inspired by Kepler), he interpreted;
+it ran thus: The furthest planet is triple.
+
+Very soon after he found that Venus was changing from a full moon to a
+half moon appearance. He announced this also by an anagram, and waited
+till it should become a crescent, which it did.
+
+This was a dreadful blow to the anti-Copernicans, for it removed the
+last lingering difficulty to the reception of the Copernican doctrine.
+
+[Illustration: FIG. 46.--Old drawings of Saturn by different observers,
+with the imperfect instruments of that day. The first is Galileo's idea
+of what he saw.]
+
+Copernicus had predicted, indeed, a hundred years before, that, if ever
+our powers of sight were sufficiently enhanced, Venus and Mercury would
+be seen to have phases like the moon. And now Galileo with his
+telescope verifies the prediction to the letter.
+
+Here was a triumph for the grand old monk, and a bitter morsel for his
+opponents.
+
+     Castelli writes: "This must now convince the most obstinate." But
+     Galileo, with more experience, replies:--"You almost make me laugh
+     by saying that these clear observations are sufficient to convince
+     the most obstinate; it seems you have yet to learn that long ago
+     the observations were enough to convince those who are capable of
+     reasoning, and those who wish to learn the truth; but that to
+     convince the obstinate, and those who care for nothing beyond the
+     vain applause of the senseless vulgar, not even the testimony of
+     the stars would suffice, were they to descend on earth to speak for
+     themselves. Let us, then, endeavour to procure some knowledge for
+     ourselves, and rest contented with this sole satisfaction; but of
+     advancing in popular opinion, or of gaining the assent of the
+     book-philosophers, let us abandon both the hope and the desire."
+
+[Illustration: FIG. 47.--Phases of Venus. Showing also its apparent
+variations in size by reason of its varying distance from the earth.
+When fully illuminated it is necessarily most distant. It looks
+brightest to us when a broad crescent.]
+
+What a year's work it had been!
+
+In twelve months observational astronomy had made such a bound as it has
+never made before or since.
+
+Why did not others make any of these observations? Because no one could
+make telescopes like Galileo.
+
+He gathered pupils round him however, and taught them how to work the
+lenses, so that gradually these instruments penetrated Europe, and
+astronomers everywhere verified his splendid discoveries.
+
+But still he worked on, and by March in the very next year, he saw
+something still more hateful to the Aristotelian philosophers, viz.
+spots on the sun.
+
+[Illustration: FIG. 48.]
+
+If anything was pure and perfect it was the sun, they said. Was this
+impostor going to blacken its face too?
+
+Well, there they were. They slowly formed and changed, and by moving all
+together showed him that the sun rotated about once a month.
+
+Before taking leave of Galileo's astronomical researches, I must
+mention an observation made at the end of 1612, that the apparent
+triplicity of Saturn (Fig. 46) had vanished.
+
+[Illustration: FIG. 49.--A portion of the sun's disk as seen in a
+powerful modern telescope.]
+
+     "Looking on Saturn within these few days, I found it solitary,
+     without the assistance of its accustomed stars, and in short
+     perfectly round and defined, like Jupiter, and such it still
+     remains. Now what can be said of so strange a metamorphosis? Are
+     perhaps the two smaller stars consumed like spots on the sun? Have
+     they suddenly vanished and fled? Or has Saturn devoured his own
+     children? Or was the appearance indeed fraud and illusion, with
+     which the glasses have so long time mocked me and so many others
+     who have so often observed with me? Now perhaps the time is come to
+     revive the withering hopes of those, who, guided by more profound
+     contemplations, have fathomed all the fallacies of the new
+     observations and recognized their impossibility! I cannot resolve
+     what to say in a chance so strange, so new, so unexpected. The
+     shortness of time, the unexampled occurrence, the weakness of my
+     intellect, the terror of being mistaken, have greatly confounded
+     me."
+
+However, he plucked up courage, and conjectured that the two attendants
+would reappear, by revolving round the planet.
+
+[Illustration: FIG. 50.--Saturn and his rings, as seen under the most
+favourable circumstances.]
+
+The real reason of their disappearance is well known to us now. The
+plane of Saturn's rings oscillates slowly about our line of sight, and
+so we sometimes see them edgeways and sometimes with a moderate amount
+of obliquity. The rings are so thin that, when turned precisely
+edgeways, they become invisible. The two imaginary attendants were the
+most conspicuous portions of the ring, subsequently called _ansae_.
+
+I have thought it better not to interrupt this catalogue of brilliant
+discoveries by any biographical details; but we must now retrace our
+steps to the years 1609 and 1610, the era of the invention of the
+telescope.
+
+By this time Galileo had been eighteen years at Padua, and like many
+another man in like case, was getting rather tired of continual
+lecturing. Moreover, he felt so full of ideas that he longed to have a
+better opportunity of following them up, and more time for thinking them
+out.
+
+Now in the holidays he had been accustomed to return to his family home
+at Pisa, and there to come a good deal into contact with the Grand-Ducal
+House of Tuscany. Young Cosmo di Medici became in fact his pupil, and
+arrived at man's estate with the highest opinion of the philosopher.
+This young man had now come to the throne as Cosmo II., and to him
+Galileo wrote saying how much he should like more time and leisure, how
+full he was of discoveries if he only had the chance of a reasonable
+income without the necessity of consuming so large a portion of his time
+in elementary teaching, and practically asking to be removed to some
+position in the Court. Nothing was done for a time, but negotiations
+proceeded, and soon after the discovery of Jupiter's satellites Cosmo
+wrote making a generous offer, which Galileo gladly and enthusiastically
+accepted, and at once left Padua for Florence. All his subsequent
+discoveries date from Florence.
+
+Thus closed his brilliant and happy career as a professor at the
+University of Padua. He had been treated well: his pay had become larger
+than that of any Professor of Mathematics up to that time; and, as you
+know, immediately after his invention of the telescope the Venetian
+Senate, in a fit of enthusiasm, had doubled it and secured it to him for
+life wherever he was. To throw up his chair and leave the place the very
+next year scarcely seems a strictly honourable procedure. It was legal
+enough no doubt, and it is easy for small men to criticize a great one,
+but nevertheless I think we must admit that it is a step such as a man
+with a keen sense of honour would hardly have taken.
+
+One quite feels and sympathizes with the temptation. Not emolument, but
+leisure; freedom from harassing engagements and constant teaching, and
+liberty to prosecute his studies day and night without interference:
+this was the golden prospect before him. He yielded, but one cannot help
+wishing he had not.
+
+As it turned out it was a false step--the first false step of his public
+career. When made it was irretrievable, and it led to great misery.
+
+At first it seemed brilliant enough. The great philosopher of the Tuscan
+Court was courted and flattered by princes and nobles, he enjoyed a
+world-wide reputation, lived as luxuriously as he cared for, had his
+time all to himself, and lectured but very seldom, on great occasions or
+to a few crowned heads.
+
+His position was in fact analogous to that of Tycho Brahe in his island
+of Huen.
+
+Misfortune overtook both. In Tycho's case it arose mainly from the death
+of his patron. In Galileo's it was due to a more insidious cause, to
+understand which cause aright we must remember the political divisions
+of Italy at that date.
+
+Tuscany was a Papal State, and thought there was by no means free.
+Venice was a free republic, and was even hostile to the Papacy. In 1606
+the Pope had placed it under an interdict. In reply it had ejected every
+Jesuit.
+
+Out of this atmosphere of comparative enlightenment and freedom into
+that hotbed of mediaevalism and superstition went Galileo with his eyes
+open. Keen was the regret of his Paduan and Venetian friends; bitter
+were their remonstrances and exhortations. But he was determined to go,
+and, not without turning some of his old friends into enemies, he went.
+
+Seldom has such a man made so great a mistake: never, I suppose, has one
+been so cruelly punished for it.
+
+[Illustration: FIG. 51.--Map of Italy.]
+
+We must remember, however, that Galileo, though by no means a saint, was
+yet a really religious man, a devout Catholic and thorough adherent of
+the Church, so that he would have no dislike to place himself under her
+sway. Moreover, he had been born a Tuscan, his family had lived at
+Florence or Pisa, and it felt like going home. His theological attitude
+is worthy of notice, for he was not in the least a sceptic. He quite
+acquiesces in the authority of the Bible, especially in all matters
+concerning faith and conduct; as to its statements in scientific
+matters, he argues that we are so liable to misinterpret their meaning
+that it is really easier to examine Nature for truth in scientific
+matters, and that when direct observation and Scripture seem to clash,
+it is because of our fallacious interpretation of one or both of them.
+He is, in fact, what one now calls a "reconciler."
+
+It is curious to find such a man prosecuted for heresy, when to-day his
+opinions are those of the orthodox among the orthodox. But so it ever
+is, and the heresy of one generation becomes the commonplace of the
+next.
+
+He accepts Joshua's miracle, for instance, not as a striking poem, but
+as a literal fact; and he points out how much more simply it could be
+done on the Copernican system by stopping the earth's rotation for a
+short time, than by stopping the sun and moon and all the host of heaven
+as on the old Ptolemaic system, or again by stopping only the sun and
+not any of the other bodies, and so throwing astronomy all wrong.
+
+This reads to us like satire, but no doubt it was his genuine opinion.
+
+These Scriptural reconciliations of his, however, angered the religious
+authorities still more. They said it was bad enough for this heretic to
+try and upset old _scientific_ beliefs, and to spoil the face of
+_Nature_ with his infidel discoveries, but at least he might leave the
+Bible alone; and they addressed an indignant remonstrance to Rome, to
+protect it from the hands of ignorant laymen.
+
+Thus, wherever he turned he encountered hostility. Of course he had many
+friends--some of them powerful like Cosmo, all of them faithful and
+sincere. But against the power of Rome what could they do? Cosmo dared
+no more than remonstrate, and ultimately his successor had to refrain
+from even this, so enchained and bound was the spirit of the rulers of
+those days; and so when his day of tribulation came he stood alone and
+helpless in the midst of his enemies.
+
+You may wonder, perhaps, why this man should excite so much more
+hostility than many another man who was suffered to believe and teach
+much the same doctrines unmolested. But no other man had made such
+brilliant and exciting discoveries. No man stood so prominently forward
+in the eyes of all Christendom as the champion of the new doctrines. No
+other man stated them so clearly and forcibly, nor drove them home with
+such brilliant and telling illustrations.
+
+And again, there was the memory of his early conflict with the
+Aristotelians at Pisa, of his scornful and successful refutation of
+their absurdities. All this made him specially obnoxious to the
+Aristotelian Jesuits in their double capacity both of priests and of
+philosophers, and they singled him out for relentless official
+persecution.
+
+Not yet, however, is he much troubled by them. The chief men at Rome
+have not yet moved. Messages, however, keep going up from Tuscany to
+Rome respecting the teachings of this man, and of the harm he is doing
+by his pertinacious preaching of the Copernican doctrine that the earth
+moves.
+
+At length, in 1615, Pope Paul V. wrote requesting him to come to Rome to
+explain his views. He went, was well received, made a special friend of
+Cardinal Barberino--an accomplished man in high position, who became in
+fact the next Pope. Galileo showed cardinals and others his telescope,
+and to as many as would look through it he showed Jupiter's satellites
+and his other discoveries. He had a most successful visit. He talked, he
+harangued, he held forth in the midst of fifteen or twenty disputants at
+once, confounding his opponents and putting them to shame.
+
+His method was to let the opposite arguments be stated as fully and
+completely as possible, himself aiding, and often adducing the most
+forcible and plausible arguments against his own views; and then, all
+having been well stated, he would proceed to utterly undermine and
+demolish the whole fabric, and bring out the truth in such a way as to
+convince all honest minds. It was this habit that made him such a
+formidable antagonist. He never shrank from meeting an opposing
+argument, never sought to ignore it, or cloak it in a cloud of words.
+Every hostile argument he seemed to delight in, as a foe to be crushed,
+and the better and stronger they sounded the more he liked them. He knew
+many of them well, he invented a number more, and had he chosen could
+have out-argued the stoutest Aristotelian on his own grounds. Thus did
+he lead his adversaries on, almost like Socrates, only to ultimately
+overwhelm them in a more hopeless rout. All this in Rome too, in the
+heart of the Catholic world. Had he been worldly-wise, he would
+certainly have kept silent and unobtrusive till he had leave to go away
+again. But he felt like an apostle of the new doctrines, whose mission
+it was to proclaim them even in this centre of the world and of the
+Church.
+
+Well, he had an audience with the Pope--a chat an hour long--and the two
+parted good friends, mutually pleased with each other.
+
+He writes that he is all right now, and might return home when he liked.
+But the question began to be agitated whether the whole system of
+Copernicus ought not to be condemned as impious and heretical. This view
+was persistently urged upon the Pope and College of Cardinals, and it
+was soon to be decided upon.
+
+Had Galileo been unfaithful to the Church he could have left them to
+stultify themselves in any way they thought proper, and himself have
+gone; but he felt supremely interested in the result, and he stayed. He
+writes:--
+
+     "So far as concerns the clearing of my own character, I might
+     return home immediately; but although this new question regards me
+     no more than all those who for the last eighty years have supported
+     those opinions both in public and private, yet, as perhaps I may be
+     of some assistance in that part of the discussion which depends on
+     the knowledge of truths ascertained by means of the sciences which
+     I profess, I, as a zealous and Catholic Christian, neither can nor
+     ought to withhold that assistance which my knowledge affords, and
+     this business keeps me sufficiently employed."
+
+It is possible that his stay was the worst thing for the cause he had at
+heart. Anyhow, the result was that the system was condemned, and both
+the book of Copernicus and the epitome of it by Kepler were placed on
+the forbidden list,[11] and Galileo himself was formally ordered never
+to teach or to believe the motion of the earth.
+
+He quitted Rome in disgust, which before long broke out in satire. The
+only way in which he could safely speak of these views now was as if
+they were hypothetical and uncertain, and so we find him writing to the
+Archduke Leopold, with a presentation copy of his book on the tides, the
+following:--
+
+     "This theory occurred to me when in Rome whilst the theologians
+     were debating on the prohibition of Copernicus's book, and of the
+     opinion maintained in it of the motion of the earth, which I at
+     that time believed: until it pleased those gentlemen to suspend the
+     book, and declare the opinion false and repugnant to the Holy
+     Scriptures. Now, as I know how well it becomes me to obey and
+     believe the decisions of my superiors, which proceed out of more
+     knowledge than the weakness of my intellect can attain to, this
+     theory which I send you, which is founded on the motion of the
+     earth, I now look upon as a fiction and a dream, and beg your
+     highness to receive it as such. But as poets often learn to prize
+     the creations of their fancy, so in like manner do I set some value
+     on this absurdity of mine. It is true that when I sketched this
+     little work I did hope that Copernicus would not, after eighty
+     years, be convicted of error; and I had intended to develop and
+     amplify it further, but a voice from heaven suddenly awakened me,
+     and at once annihilated all my confused and entangled fancies."
+
+This sarcasm, if it had been in print, would probably have been
+dangerous. It was safe in a private letter, but it shows us his real
+feelings.
+
+However, he was left comparatively quiet for a time. He was getting an
+old man now, and passed the time studiously enough, partly at his house
+in Florence, partly at his villa in Arcetri, a mile or so out of the
+town.
+
+Here was a convent, and in it his two daughters were nuns. One of them,
+who passed under the name of Sister Maria Celeste, seems to have been a
+woman of considerable capacity--certainly she was of a most affectionate
+disposition--and loved and honoured her father in the most dutiful way.
+
+This was a quiet period of his life, spoiled only by occasional fits of
+illness and severe rheumatic pains, to which the old man was always
+liable. Many little circumstances are known of this peaceful time. For
+instance, the convent clock won't go, and Galileo mends it for them. He
+is always doing little things for them, and sending presents to the Lady
+Superior and his two daughters.
+
+He was occupied now with problems in hydrostatics, and on other matters
+unconnected with astronomy: a large piece of work which I must pass
+over. Most interesting and acute it is, however.
+
+In 1623, when the old Pope died, there was elected to the Papal throne,
+as Urban VIII., Cardinal Barberino, a man of very considerable
+enlightenment, and a personal friend of Galileo's, so that both he and
+his daughters rejoice greatly, and hope that things will come all right,
+and the forbidding edict be withdrawn.
+
+The year after this election he manages to make another journey to Rome
+to compliment his friend on his elevation to the Pontifical chair. He
+had many talks with Urban, and made himself very agreeable.
+
+Urban wrote to the Grand Duke Ferdinand, son of Cosmo:--
+
+     "For We find in him not only literary distinction but also love of
+     piety, and he is strong in those qualities by which Pontifical good
+     will is easily obtainable. And now, when he has been brought to
+     this city to congratulate Us on Our elevation, We have very
+     lovingly embraced him; nor can We suffer him to return to the
+     country whither your liberality recalls him without an ample
+     provision of Pontifical love. And that you may know how dear he is
+     to Us, We have willed to give him this honourable testimonial of
+     virtue and piety. And We further signify that every benefit which
+     you shall confer upon him, imitating or even surpassing your
+     father's liberality, will conduce to Our gratification."
+
+Encouraged, doubtless, by these marks of approbation, and reposing too
+much confidence in the individual good will of the Pope, without heeding
+the crowd of half-declared enemies who were seeking to undermine his
+reputation, he set about, after his return to Florence, his greatest
+literary and most popular work, _Dialogues on the Ptolemaic and
+Copernican Systems_. This purports to be a series of four conversations
+between three characters: Salviati, a Copernican philosopher; Sagredo, a
+wit and scholar, not specially learned, but keen and critical, and who
+lightens the talk with chaff; Simplicio, an Aristotelian philosopher,
+who propounds the stock absurdities which served instead of arguments to
+the majority of men.
+
+The conversations are something between Plato's _Dialogues_ and Sir
+Arthur Helps's _Friends in Council_. The whole is conducted with great
+good temper and fairness; and, discreetly enough, no definite conclusion
+is arrived at, the whole being left in abeyance as if for a fifth and
+decisive dialogue, which, however, was never written, and perhaps was
+only intended in case the reception was favourable.
+
+The preface also sets forth that the object of the writer is to show
+that the Roman edict forbidding the Copernican doctrine was not issued
+in ignorance of the facts of the case, as had been maliciously reported,
+and that he wishes to show how well and clearly it was all known
+beforehand. So he says the dialogue on the Copernican side takes up the
+question purely as a mathematical hypothesis or speculative figment, and
+gives it every artificial advantage of which the theory is capable.
+
+This piece of caution was insufficient to blind the eyes of the
+Cardinals; for in it the arguments in favour of the earth's motion are
+so cogent and unanswerable, and are so popularly stated, as to do more
+in a few years to undermine the old system than all that he had written
+and spoken before. He could not get it printed for two years after he
+had written it, and then only got consent through a piece of
+carelessness or laziness on the part of the ecclesiastical censor
+through whose hands the manuscript passed--for which he was afterwards
+dismissed.
+
+However, it did appear, and was eagerly read; the more, perhaps, as the
+Church at once sought to suppress it.
+
+The Aristotelians were furious, and represented to the Pope that he
+himself was the character intended by Simplicio, the philosopher whose
+opinions get alternately refuted and ridiculed by the other two, till he
+is reduced to an abject state of impotence.
+
+The idea that Galileo had thus cast ridicule upon his friend and patron
+is no doubt a gratuitous and insulting libel: there is no telling
+whether or not Urban believed it, but certainly his countenance changed
+to Galileo henceforward, and whether overruled by his Cardinals, or
+actuated by some other motive, his favour was completely withdrawn.
+
+The infirm old man was instantly summoned to Rome. His friends pleaded
+his age--he was now seventy--his ill-health, the time of year, the state
+of the roads, the quarantine existing on account of the plague. It was
+all of no avail, to Rome he must go, and on the 14th of February he
+arrived.
+
+[Illustration: FIG. 52.--Portrait of Galileo.]
+
+His daughter at Arcetri was in despair; and anxiety and fastings and
+penances self-inflicted on his account, dangerously reduced her health.
+
+At Rome he was not imprisoned, but he was told to keep indoors, and show
+himself as little as possible. He was allowed, however, to stay at the
+house of the Tuscan Ambassador instead of in gaol.
+
+By April he was removed to the chambers of the Inquisition, and examined
+several times. Here, however, the anxiety was too much, and his health
+began to give way seriously; so, before long, he was allowed to return
+to the Ambassador's house; and, after application had been made, was
+allowed to drive in the public garden in a half-closed carriage. Thus in
+every way the Inquisition dealt with him as leniently as they could. He
+was now their prisoner, and they might have cast him into their
+dungeons, as many another had been cast. By whatever they were
+influenced--perhaps the Pope's old friendship, perhaps his advanced age
+and infirmities--he was not so cruelly used.
+
+Still, they had their rules; he _must_ be made to recant and abjure his
+heresy; and, if necessary, torture must be applied. This he knew well
+enough, and his daughter knew it, and her distress may be imagined.
+Moreover, it is not as if they had really been heretics, as if they
+hated or despised the Church of Rome. On the contrary, they loved and
+honoured the Church. They were sincere and devout worshippers, and only
+on a few scientific matters did Galileo presume to differ from his
+ecclesiastical superiors: his disagreement with them occasioned him real
+sorrow; and his dearest hope was that they could be brought to his way
+of thinking and embrace the truth.
+
+Every time he was sent for by the Inquisition he was in danger of
+torture unless he recanted. All his friends urged him repeatedly to
+submit. They said resistance was hopeless and fatal. Within the memory
+of men still young, Giordano Bruno had been burnt alive for a similar
+heresy. This had happened while Galileo was at Padua. Venice was full of
+it. And since that, only eight years ago indeed, Antonio de Dominis,
+Archbishop of Salpetria, had been sentenced to the same fate: "to be
+handed over to the secular arm to be dealt with as mercifully as
+possible without the shedding of blood." So ran the hideous formula
+condemning a man to the stake. After his sentence, this unfortunate man
+died in the dungeons in which he had been incarcerated six years--died
+what is called a "natural" death; but the sentence was carried out,
+notwithstanding, on his lifeless body and his writings. His writings for
+which he had been willing to die!
+
+These were the tender mercies of the Inquisition; and this was the kind
+of meaning lurking behind many of their well-sounding and merciful
+phrases. For instance, what they call "rigorous examination," we call
+"torture." Let us, however, remember in our horror at this mode of
+compelling a prisoner to say anything they wished, that they were a
+legally constituted tribunal; that they acted with well established
+rules, and not in passion; and that torture was a recognized mode of
+extracting evidence, not only in ecclesiastical but in civil courts, at
+that date.
+
+All this, however, was but poor solace to the pitiable old philosopher,
+thus ruthlessly haled up and down, questioned and threatened, threatened
+and questioned, receiving agonizing letters from his daughter week by
+week, and trying to keep up a little spirit to reply as happily and
+hopefully as he could.
+
+This condition of things could not go on. From February to June the
+suspense lasted. On the 20th of June he was summoned again, and told he
+would be wanted all next day for a rigorous examination. Early in the
+morning of the 21st he repaired thither, and the doors were shut. Out of
+those chambers of horror he did not reappear till the 24th. What went on
+all those three days no one knows. He himself was bound to secrecy. No
+outsider was present. The records of the Inquisition are jealously
+guarded. That he was technically tortured is certain; that he actually
+underwent the torment of the rack is doubtful. Much learning has been
+expended upon the question, especially in Germany. Several eminent
+scholars have held the fact of actual torture to be indisputable
+(geometrically certain, one says), and they confirm it by the hernia
+from which he afterwards suffered, this being a well-known and frequent
+consequence.
+
+Other equally learned commentators, however, deny that the last stage
+was reached. For there are five stages all laid down in the rules of the
+Inquisition, and steadily adhered to in a rigorous examination, at each
+stage an opportunity being given for recantation, every utterance,
+groan, or sigh being strictly recorded. The recantation so given has to
+be confirmed a day or two later, under pain of a precisely similar
+ordeal.
+
+The five stages are:--1st. The official threat in the court. 2nd. The
+taking to the door of the torture chamber and renewing the official
+threat. 3rd. The taking inside and showing the instruments. 4th.
+Undressing and binding upon the rack. 5th. _Territio realis._
+
+Through how many of these ghastly acts Galileo passed I do not know. I
+hope and believe not the last.
+
+There are those who lament that he did not hold out, and accept the
+crown of martyrdom thus offered to him. Had he done so we know his
+fate--a few years' languishing in the dungeons, and then the flames.
+
+Whatever he ought to have done, he did not hold out--he gave way. At one
+stage or another of the dread ordeal he said: "I am in your hands. I
+will say whatever you wish." Then was he removed to a cell while his
+special form of perjury was drawn up.
+
+The next day, clothed as a penitent, the venerable old man was taken to
+the Convent of Minerva, where the Cardinals and prelates were assembled
+for the purpose of passing judgment upon him.
+
+The text of the judgment I have here, but it is too long to read. It
+sentences him--1st. To the abjuration. 2nd. To formal imprisonment for
+life. 3rd. To recite the seven penitential psalms every week.
+
+Ten Cardinals were present; but, to their honour be it said, three
+refused to sign; and this blasphemous record of intolerance and bigoted
+folly goes down the ages with the names of seven Cardinals immortalized
+upon it.
+
+This having been read, he next had to read word for word the abjuration
+which had been drawn up for him, and then sign it.
+
+
+THE ABJURATION OF GALILEO.
+
+     "I, Galileo Galilei, son of the late Vincenzo Galilei, of Florence,
+     aged seventy years, being brought personally to judgment, and
+     kneeling before you Most Eminent and Most Reverend Lords Cardinals,
+     General Inquisitors of the universal Christian republic against
+     heretical depravity, having before my eyes the Holy Gospels, which
+     I touch with my own hands, swear that I have always believed, and
+     now believe, and with the help of God will in future believe, every
+     article which the Holy Catholic and Apostolic Church of Rome holds,
+     teaches, and preaches. But because I have been enjoined by this
+     Holy Office altogether to abandon the false opinion which maintains
+     that the sun is the centre and immovable, and forbidden to hold,
+     defend, or teach the said false doctrine in any manner, and after
+     it hath been signified to me that the said doctrine is repugnant
+     with the Holy Scripture, I have written and printed a book, in
+     which I treat of the same doctrine now condemned, and adduce
+     reasons with great force in support of the same, without giving any
+     solution, and therefore have been judged grievously suspected of
+     heresy; that is to say, that I held and believed that the sun is
+     the centre of the universe and is immovable, and that the earth is
+     not the centre and is movable; willing, therefore, to remove from
+     the minds of your Eminences, and of every Catholic Christian, this
+     vehement suspicion rightfully entertained towards me, with a
+     sincere heart and unfeigned faith, I abjure, curse, and detest the
+     said errors and heresies, and generally every other error and sect
+     contrary to Holy Church; and I swear that I will never more in
+     future say or assert anything verbally, or in writing, which may
+     give rise to a similar suspicion of me; but if I shall know any
+     heretic, or any one suspected of heresy, that I will denounce him
+     to this Holy Office, or to the Inquisitor or Ordinary of the place
+     where I may be; I swear, moreover, and promise, that I will fulfil
+     and observe fully, all the penances which have been or shall be
+     laid on me by this Holy Office. But if it shall happen that I
+     violate any of my said promises, oaths, and protestations (which
+     God avert!), I subject myself to all the pains and punishments
+     which have been decreed and promulgated by the sacred canons, and
+     other general and particular constitutions, against delinquents of
+     this description. So may God help me, and his Holy Gospels which I
+     touch with my own hands. I, the above-named Galileo Galilei, have
+     abjured, sworn, promised, and bound myself as above, and in witness
+     thereof with my own hand have subscribed this present writing of my
+     abjuration, which I have recited word for word. At Rome, in the
+     Convent of Minerva, 22nd June, 1633. I, Galileo Galilei, have
+     abjured as above with my own hand."
+
+Those who believe the story about his muttering to a friend, as he rose
+from his knees, "e pur si muove," do not realize the scene.
+
+1st. There was no friend in the place.
+
+2nd. It would have been fatally dangerous to mutter anything before such
+an assemblage.
+
+3rd. He was by this time an utterly broken and disgraced old man;
+wishful, of all things, to get away and hide himself and his miseries
+from the public gaze; probably with his senses deadened and stupefied by
+the mental sufferings he had undergone, and no longer able to think or
+care about anything--except perhaps his daughter,--certainly not about
+any motion of this wretched earth.
+
+Far and wide the news of the recantation spread. Copies of the
+abjuration were immediately sent to all Universities, with instructions
+to the professors to read it publicly.
+
+At Florence, his home, it was read out in the Cathedral church, all his
+friends and adherents being specially summoned to hear it.
+
+For a short time more he was imprisoned in Rome; but at length was
+permitted to depart, never more of his own will to return.
+
+He was allowed to go to Siena. Here his daughter wrote consolingly,
+rejoicing at his escape, and saying how joyfully she already recited the
+penitential psalms for him, and so relieved him of that part of his
+sentence.
+
+But the poor girl was herself, by this time, ill--thoroughly worn out
+with anxiety and terror; she lay, in fact, on what proved to be her
+death-bed. Her one wish was to see her dearest lord and father, so she
+calls him, once more. The wish was granted. His prison was changed, by
+orders from Rome, from Siena to Arcetri, and once more father and
+daughter embraced. Six days after this she died.
+
+The broken-hearted old man now asks for permission to go to live in
+Florence, but is met with the stern answer that he is to stay at
+Arcetri, is not to go out of the house, is not to receive visitors, and
+that if he asks for more favours, or transgresses the commands laid upon
+him, he is liable to be haled back to Rome and cast into a dungeon.
+These harsh measures were dictated, not by cruelty, but by the fear of
+his still spreading heresy by conversation, and so he was to be kept
+isolated.
+
+Idle, however, he was not and could not be. He often complains that his
+head is too busy for his body. In the enforced solitude of Arcetri he
+was composing those dialogues on motion which are now reckoned his
+greatest and most solid achievement. In these the true laws of motion
+are set forth for the first time (see page 167). One more astronomical
+discovery also he was to make--that of the moon's libration.
+
+And then there came one more crushing blow. His eyes became inflamed and
+painful--the sight of one of them failed, the other soon went; he
+became totally blind. But this, being a heaven-sent infliction, he could
+bear with resignation, though it must have been keenly painful to a
+solitary man of his activity. "Alas!" says he, in one of his letters,
+"your dear friend and servant is totally blind. Henceforth this heaven,
+this universe, which by wonderful observations I had enlarged a hundred
+and a thousand times beyond the conception of former ages, is shrunk for
+me into the narrow space which I myself fill in it. So it pleases God;
+it shall therefore please me also."
+
+He was now allowed an amanuensis, and the help of his pupils Torricelli,
+Castelli, and Viviani, all devotedly attached to him, and Torricelli
+very famous after him. Visitors also were permitted, after approval by a
+Jesuit supervisor; and under these circumstances many visited him, among
+them a man as immortal as himself--John Milton, then only twenty-nine,
+travelling in Italy. Surely a pathetic incident, this meeting of these
+two great men--the one already blind, the other destined to become so.
+No wonder that, as in his old age he dictated his masterpiece, the
+thoughts of the English poet should run on the blind sage of Tuscany,
+and the reminiscence of their conversation should lend colour to the
+poem.
+
+Well, it were tedious to follow the petty annoyances and troubles to
+which Galileo was still subject--how his own son was set to see that no
+unauthorized procedure took place, and that no heretic visitors were
+admitted; how it was impossible to get his new book printed till long
+afterwards; and how one form of illness after another took possession of
+him. The merciful end came at last, and at the age of seventy-eight he
+was released from the Inquisition.
+
+They wanted to deny him burial--they did deny him a monument; they
+threatened to cart his bones away from Florence if his friends attempted
+one. And so they hoped that he and his work might be forgotten.
+
+Poor schemers! Before the year was out an infant was born in
+Lincolnshire, whose destiny it was to round and complete and carry
+forward the work of their victim, so that, until man shall cease from
+the planet, neither the work nor its author shall have need of a
+monument.
+
+*      *      *      *      *
+
+Here might I end, were it not that the same kind of struggle as went on
+fiercely in the seventeenth century is still smouldering even now. Not
+in astronomy indeed, as then; nor yet in geology, as some fifty years
+ago; but in biology mainly--perhaps in other subjects. I myself have
+heard Charles Darwin spoken of as an atheist and an infidel, the theory
+of evolution assailed as unscriptural, and the doctrine of the ascent of
+man from a lower state of being, as opposed to the fall of man from some
+higher condition, denied as impious and un-Christian.
+
+Men will not learn by the past; still they brandish their feeble weapons
+against the truths of Nature, as if assertions one way or another could
+alter fact, or make the thing other than it really is. As Galileo said
+before his spirit was broken, "In these and other positions certainly no
+man doubts but His Holiness the Pope hath always an absolute power of
+admitting or condemning them; but it is not in the power of any creature
+to make them to be true or false, or otherwise than of their own nature
+and in fact they are."
+
+I know nothing of the views of any here present; but I have met educated
+persons who, while they might laugh at the men who refused to look
+through a telescope lest they should learn something they did not like,
+yet also themselves commit the very same folly. I have met persons who
+utterly refuse to listen to any view concerning the origin of man other
+than that of a perfect primaeval pair in a garden, and I am constrained
+to say this much: Take heed lest some prophet, after having excited your
+indignation at the follies and bigotry of a bygone generation, does not
+turn upon you with the sentence, "Thou art the man."
+
+
+
+
+SUMMARY OF FACTS FOR LECTURE VI
+
+_Science before Newton_
+
+
+_Dr. Gilbert_, of Colchester, Physician to Queen Elizabeth, was an
+excellent experimenter, and made many discoveries in magnetism and
+electricity. He was contemporary with Tycho Brahe, and lived from 1540
+to 1603.
+
+_Francis Bacon_, Lord Verulam, 1561-1626, though a brilliant writer, is
+not specially important as regards science. He was not a scientific man,
+and his rules for making discoveries, or methods of induction, have
+never been consciously, nor often indeed unconsciously, followed by
+discoverers. They are not in fact practical rules at all, though they
+were so intended. His really strong doctrines are that phenomena must be
+studied direct, and that variations in the ordinary course of nature
+must be induced by aid of experiment; but he lacked the scientific
+instinct for pursuing these great truths into detail and special cases.
+He sneered at the work and methods of both Gilbert and Galileo, and
+rejected the Copernican theory as absurd. His literary gifts have
+conferred on him an artificially high scientific reputation, especially
+in England; at the same time his writings undoubtedly helped to make
+popular the idea of there being new methods for investigating Nature,
+and, by insisting on the necessity for freedom from preconceived ideas
+and opinions, they did much to release men from the bondage of
+Aristotelian authority and scholastic tradition.
+
+The greatest name between Galileo and Newton is that of Descartes.
+
+_Rene Descartes_ was born at La Haye in Touraine, 1596, and died at
+Stockholm in 1650. He did important work in mathematics, physics,
+anatomy, and philosophy. Was greatest as a philosopher and
+mathematician. At the age of twenty-one he served as a volunteer under
+Prince Maurice of Nassau, but spent most of his later life in Holland.
+His famous _Discourse on Method_ appeared at Leyden in 1637, and his
+_Principia_ at Amsterdam in 1644; great pains being taken to avoid the
+condemnation of the Church.
+
+Descartes's main scientific achievement was the application of algebra
+to geometry; his most famous speculation was the "theory of vortices,"
+invented to account for the motion of planets. He also made many
+discoveries in optics and physiology. His best known immediate pupils
+were the Princess Elizabeth of Bohemia, and Christina, Queen of Sweden.
+
+He founded a distinct school of thought (the Cartesian), and was the
+precursor of the modern mathematical method of investigating science,
+just as Galileo and Gilbert were the originators of the modern
+experimental method.
+
+
+
+
+LECTURE VI
+
+DESCARTES AND HIS THEORY OF VORTICES
+
+
+After the dramatic life we have been considering in the last two
+lectures, it is well to have a breathing space, to look round on what
+has been accomplished, and to review the state of scientific thought,
+before proceeding to the next great era. For we are still in the early
+morning of scientific discovery: the dawn of the modern period, faintly
+heralded by Copernicus, brought nearer by the work of Tycho and Kepler,
+and introduced by the discoveries of Galileo--the dawn has occurred, but
+the sun is not yet visible. It is hidden by the clouds and mists of the
+long night of ignorance and prejudice. The light is sufficient, indeed,
+to render these earth-born vapours more visible: it is not sufficient to
+dispel them. A generation of slow and doubtful progress must pass,
+before the first ray of sunlight can break through the eastern clouds
+and the full orb of day itself appear.
+
+It is this period of hesitating progress and slow leavening of men's
+ideas that we have to pass through in this week's lecture. It always
+happens thus: the assimilation of great and new ideas is always a slow
+and gradual process: there is no haste either here or in any other
+department of Nature. _Die Zeit ist unendlich lang._ Steadily the forces
+work, sometimes seeming to accomplish nothing; sometimes even the
+motion appears retrograde; but in the long run the destined end is
+reached, and the course, whether of a planet or of men's thoughts about
+the universe, is permanently altered. Then, the controversy was about
+the _earth's_ place in the universe; now, if there be any controversy of
+the same kind, it is about _man's_ place in the universe; but the
+process is the same: a startling statement by a great genius or prophet,
+general disbelief, and, it may be, an attitude of hostility, gradual
+acceptance by a few, slow spreading among the many, ending in universal
+acceptance and faith often as unquestioning and unreasoning as the old
+state of unfaith had been. Now the process is comparatively speedy:
+twenty years accomplishes a great deal: then it was tediously slow, and
+a century seemed to accomplish very little. Periodical literature may be
+responsible for some waste of time, but it certainly assists the rapid
+spread of ideas. The rate with which ideas are assimilated by the
+general public cannot even now be considered excessive, but how much
+faster it is than it was a few centuries ago may be illustrated by the
+attitude of the public to Darwinism now, twenty-five years after _The
+Origin of Species_, as compared with their attitude to the Copernican
+system a century after _De Revolutionibus_. By the way, it is, I know,
+presumptuous for me to have an opinion, but I cannot hear Darwin
+compared to or mentioned along with Newton without a shudder. The stage
+in which he found biology seems to me far more comparable with the
+Ptolemaic era in astronomy, and he himself to be quite fairly comparable
+to Copernicus.
+
+Let us proceed to summarize the stage at which the human race had
+arrived at the epoch with which we are now dealing.
+
+The Copernican view of the solar system had been stated, restated,
+fought, and insisted on; a chain of brilliant telescopic discoveries had
+made it popular and accessible to all men of any intelligence:
+henceforth it must be left to slowly percolate and sink into the minds
+of the people. For the nations were waking up now, and were accessible
+to new ideas. England especially was, in some sort, at the zenith of its
+glory; or, if not at the zenith, was in that full flush of youth and
+expectation and hope which is stronger and more prolific of great deeds
+and thoughts than a maturer period.
+
+A common cause against a common and detested enemy had roused in the
+hearts of Englishmen a passion of enthusiasm and patriotism; so that the
+mean elements of trade, their cheating yard-wands, were forgotten for a
+time; the Armada was defeated, and the nation's true and conscious adult
+life began. Commerce was now no mere struggle for profit and hard
+bargains; it was full of the spirit of adventure and discovery; a new
+world had been opened up; who could tell what more remained unexplored?
+Men awoke to the splendour of their inheritance, and away sailed Drake
+and Frobisher and Raleigh into the lands of the West.
+
+For literature, you know what a time it was. The author of _Hamlet_ and
+_Othello_ was alive: it is needless to say more. And what about science?
+The atmosphere of science is a more quiet and less stirring one; it
+thrives best when the fever of excitement is allayed; it is necessarily
+a later growth than literature. Already, however, our second great man
+of science was at work in a quiet country town--second in point of time,
+I mean, Roger Bacon being the first. Dr. Gilbert, of Colchester, was the
+second in point of time, and the age was ripening for the time when
+England was to be honoured with such a galaxy of scientific
+luminaries--Hooke and Boyle and Newton--as the world had not yet known.
+
+Yes, the nations were awake. "In all directions," as Draper says,
+"Nature was investigated: in all directions new methods of examination
+were yielding unexpected and beautiful results. On the ruins of its
+ivy-grown cathedrals Ecclesiasticism [or Scholasticism], surprised and
+blinded by the breaking day, sat solemnly blinking at the light and life
+about it, absorbed in the recollection of the night that had passed,
+dreaming of new phantoms and delusions in its wished-for return, and
+vindictively striking its talons at any derisive assailant who
+incautiously approached too near."
+
+Of the work of Gilbert there is much to say; so there is also of Roger
+Bacon, whose life I am by no means sure I did right in omitting. But
+neither of them had much to do with astronomy, and since it is in
+astronomy that the most startling progress was during these centuries
+being made, I have judged it wiser to adhere mainly to the pioneers in
+this particular department.
+
+Only for this reason do I pass Gilbert with but slight mention. He knew
+of the Copernican theory and thoroughly accepted it (it is convenient to
+speak of it as the Copernican theory, though you know that it had been
+considerably improved in detail since the first crude statement by
+Copernicus), but he made in it no changes. He was a cultivated
+scientific man, and an acute experimental philosopher; his main work lay
+in the domain of magnetism and electricity. The phenomena connected with
+the mariner's compass had been studied somewhat by Roger Bacon; and they
+were now examined still more thoroughly by Gilbert, whose treatise _De
+Magnete_, marks the beginning of the science of magnetism.
+
+As an appendix to that work he studied the phenomenon of amber, which
+had been mentioned by Thales. He resuscitated this little fact after its
+burial of 2,200 years, and greatly extended it. He it was who invented
+the name electricity--I wish it had been a shorter one. Mankind invents
+names much better than do philosophers. What can be better than "heat,"
+"light," "sound"? How favourably they compare with electricity,
+magnetism, galvanism, electro-magnetism, and magneto-electricity! The
+only long-established monosyllabic name I know invented by a philosopher
+is "gas"--an excellent attempt, which ought to be imitated.[12]
+
+Of Lord Bacon, who flourished about the same time (a little later), it
+is necessary to say something, because many persons are under the
+impression that to him and his _Novum Organon_ the reawakening of the
+world, and the overthrow of Aristotelian tradition, are mainly due. His
+influence, however, has been exaggerated. I am not going to enter into a
+discussion of the _Novum Organon_, and the mechanical methods which he
+propounded as certain to evolve truth if patiently pursued; for this is
+what he thought he was doing--giving to the world an infallible recipe
+for discovering truth, with which any ordinarily industrious man could
+make discoveries by means of collection and discrimination of instances.
+You will take my statement for what it is worth, but I assert this: that
+many of the methods which Bacon lays down are not those which the
+experience of mankind has found to be serviceable; nor are they such as
+a scientific man would have thought of devising.
+
+True it is that a real love and faculty for science are born in a man,
+and that to the man of scientific capacity rules of procedure are
+unnecessary; his own intuition is sufficient, or he has mistaken his
+vocation,--but that is not my point. It is not that Bacon's methods are
+useless because the best men do not need them; if they had been founded
+on a careful study of the methods actually employed, though it might be
+unconsciously employed, by scientific men--as the methods of induction,
+stated long after by John Stuart Mill, were founded--then, no doubt,
+their statement would have been a valuable service and a great thing to
+accomplish. But they were not this. They are the ideas of a brilliant
+man of letters, writing in an age when scientific research was almost
+unknown, about a subject in which he was an amateur. I confess I do not
+see how he, or John Stuart Mill, or any one else, writing in that age,
+could have formulated the true rules of philosophizing; because the
+materials and information were scarcely to hand. Science and its methods
+were only beginning to grow. No doubt it was a brilliant attempt. No
+doubt also there are many good and true points in the statement,
+especially in his insistence on the attitude of free and open candour
+with which the investigation of Nature should be approached. No doubt
+there was much beauty in his allegories of the errors into which men
+were apt to fall--the _idola_ of the market-place, of the tribe, of the
+theatre, and of the den; but all this is literature, and on the solid
+progress of science may be said to have had little or no effect.
+Descartes's _Discourse on Method_ was a much more solid production.
+
+You will understand that I speak of Bacon purely as a scientific man. As
+a man of letters, as a lawyer, a man of the world, and a statesman, he
+is beyond any criticism of mine. I speak only of the purely scientific
+aspect of the _Novum Organon_. _The Essays_ and _The Advancement of
+Learning_ are masterly productions; and as a literary man he takes high
+rank.
+
+The over-praise which, in the British Isles, has been lavished upon his
+scientific importance is being followed abroad by what may be an
+unnecessary amount of detraction. This is always the worst of setting up
+a man on too high a pinnacle; some one has to undertake the ungrateful
+task of pulling him down again. Justus von Liebig addressed himself to
+this task with some vigour in his _Reden und Abhandlung_ (Leipzig,
+1874), where he quotes from Bacon a number of suggestions for absurd
+experimentation.[13]
+
+The next paragraph I read, not because I endorse it, but because it is
+always well to hear both sides of a question. You have probably been
+long accustomed to read over-estimates of Bacon's importance, and
+extravagant laudation of his writings as making an epoch in science;
+hear what Draper says on the opposite side:--[14]
+
+     "The more closely we examine the writings of Lord Bacon, the more
+     unworthy does he seem to have been of the great reputation which
+     has been awarded to him. The popular delusion to which he owes so
+     much originated at a time when the history of science was unknown.
+     They who first brought him into notice knew nothing of the old
+     school of Alexandria. This boasted founder of a new philosophy
+     could not comprehend, and would not accept, the greatest of all
+     scientific doctrines when it was plainly set before his eyes.
+
+     "It has been represented that the invention of the true method of
+     physical science was an amusement of Bacon's hours of relaxation
+     from the more laborious studies of law, and duties of a Court.
+
+     "His chief admirers have been persons of a literary turn, who have
+     an idea that scientific discoveries are accomplished by a
+     mechanico-mental operation. Bacon never produced any great
+     practical result himself, no great physicist has ever made any use
+     of his method. He has had the same to do with the development of
+     modern science that the inventor of the orrery has had to do with
+     the discovery of the mechanism of the world. Of all the important
+     physical discoveries, there is not one which shows that its author
+     made it by the Baconian instrument.
+
+     "Newton never seems to have been aware that he was under any
+     obligation to Bacon. Archimedes, and the Alexandrians, and the
+     Arabians, and Leonardo da Vinci did very well before he was born;
+     the discovery of America by Columbus and the circumnavigation by
+     Magellan can hardly be attributed to him, yet they were the
+     consequences of a truly philosophical reasoning. But the
+     investigation of Nature is an affair of genius, not of rules. No
+     man can invent an _organon_ for writing tragedies and epic poems.
+     Bacon's system is, in its own terms, an idol of the theatre. It
+     would scarcely guide a man to a solution of the riddle of AElia
+     Laelia Crispis, or to that of the charade of Sir Hilary.
+
+     "Few scientific pretenders have made more mistakes than Lord Bacon.
+     He rejected the Copernican system, and spoke insolently of its
+     great author; he undertook to criticize adversely Gilbert's
+     treatise _De Magnete_; he was occupied in the condemnation of any
+     investigation of final causes, while Harvey was deducing the
+     circulation of the blood from Aquapendente's discovery of the
+     valves in the veins; he was doubtful whether instruments were of
+     any advantage, while Galileo was investigating the heavens with the
+     telescope. Ignorant himself of every branch of mathematics, he
+     presumed that they were useless in science but a few years before
+     Newton achieved by their aid his immortal discoveries.
+
+     "It is time that the sacred name of philosophy should be severed
+     from its long connection with that of one who was a pretender in
+     science, a time-serving politician, an insidious lawyer, a corrupt
+     judge, a treacherous friend, a bad man."
+
+This seems to me a depreciation as excessive as are the eulogies
+commonly current. The truth probably lies somewhere between the two
+extremes. It is unfair to judge Bacon's methods by thinking of physical
+science in its present stage. To realise his position we must think of a
+subject still in its very early infancy, one in which the advisability
+of applying experimental methods is still doubted; one which has been
+studied by means of books and words and discussion of normal instances,
+instead of by collection and observation of the unusual and irregular,
+and by experimental production of variety. If we think of a subject
+still in this infantile and almost pre-scientific stage, Bacon's words
+and formulae are far from inapplicable; they are, within their
+limitations, quite necessary and wholesome. A subject in this stage,
+strange to say, exists,--psychology; now hesitatingly beginning to
+assume its experimental weapons amid a stifling atmosphere of distrust
+and suspicion. Bacon's lack of the modern scientific instinct must be
+admitted, but he rendered humanity a powerful service in directing it
+from books to nature herself, and his genius is indubitable. A judicious
+account of his life and work is given by Prof. Adamson, in the
+_Encyclopaedia Britannica_, and to this article I now refer you.
+
+*      *      *      *      *
+
+Who, then, was the man of first magnitude filling up the gap in
+scientific history between the death of Galileo and the maturity of
+Newton? Unknown and mysterious are the laws regulating the appearance of
+genius. We have passed in review a Pole, a Dane, a German, and an
+Italian,--the great man is now a Frenchman, Rene Descartes, born in
+Touraine, on the 31st of March, 1596.
+
+His mother died at his birth; the father was of no importance, save as
+the owner of some landed property. The boy was reared luxuriously, and
+inherited a fair fortune. Nearly all the men of first rank, you notice,
+were born well off. Genius born to poverty might, indeed, even then
+achieve name and fame--as we see in the case of Kepler--but it was
+terribly handicapped. Handicapped it is still, but far less than of old;
+and we may hope it will become gradually still less so as enlightenment
+proceeds, and the tremendous moment of great men to a nation is more
+clearly and actively perceived.
+
+It is possible for genius, when combined with strong character, to
+overcome all obstacles, and reach the highest eminence, but the
+struggle must be severe; and the absence of early training and
+refinement during the receptive years of youth must be a lifelong
+drawback.
+
+Descartes had none of these drawbacks; life came easily to him, and, as
+a consequence perhaps, he never seems to have taken it quite seriously.
+Great movements and stirring events were to him opportunities for the
+study of men and manners; he was not the man to court persecution, nor
+to show enthusiasm for a losing or struggling cause.
+
+In this, as in many other things, he was imbued with a very modern
+spirit, a cynical and sceptical spirit, which, to an outside and
+superficial observer like myself, seems rather rife just now.
+
+He was also imbued with a phase of scientific spirit which you sometimes
+still meet with, though I believe it is passing away, viz. an uncultured
+absorption in his own pursuits, and some feeling of contempt for
+classical and literary and aesthetic studies.
+
+In politics, art, and history he seems to have had no interest. He was a
+spectator rather than an actor on the stage of the world; and though he
+joined the army of that great military commander Prince Maurice of
+Nassau, he did it not as a man with a cause at heart worth fighting for,
+but precisely in the spirit in which one of our own gilded youths would
+volunteer in a similar case, as a good opportunity for frolic and for
+seeing life.
+
+He soon tired of it and withdrew--at first to gay society in Paris. Here
+he might naturally have sunk into the gutter with his companions, but
+for a great mental shock which became the main epoch and turning-point
+of his life, the crisis which diverted him from frivolity to
+seriousness. It was a purely intellectual emotion, not excited by
+anything in the visible or tangible world; nor could it be called
+conversion in the common acceptation of that term. He tells us that on
+the 10th of November, 1619, at the age of twenty-four, a brilliant idea
+flashed upon him--the first idea, namely, of his great and powerful
+mathematical method, of which I will speak directly; and in the flush of
+it he foresaw that just as geometers, starting with a few simple and
+evident propositions or axioms, ascend by a long and intricate ladder of
+reasoning to propositions more and more abstruse, so it might be
+possible to ascend from a few data, to all the secrets and facts of the
+universe, by a process of mathematical reasoning.
+
+"Comparing the mysteries of Nature with the laws of mathematics, he
+dared to hope that the secrets of both could be unlocked with the same
+key."
+
+That night he lapsed gradually into a state of enthusiasm, in which he
+saw three dreams or visions, which he interpreted at the time, even
+before waking, to be revelations from the Spirit of Truth to direct his
+future course, as well as to warn him from the sins he had already
+committed.
+
+His account of the dreams is on record, but is not very easy to follow;
+nor is it likely that a man should be able to convey to others any
+adequate idea of the deepest spiritual or mental agitation which has
+shaken him to his foundations.
+
+His associates in Paris were now abandoned, and he withdrew, after some
+wanderings, to Holland, where he abode the best part of his life and did
+his real work.
+
+Even now, however, he took life easily. He recommends idleness as
+necessary to the production of good mental work. He worked and meditated
+but a few hours a day: and most of those in bed. He used to think best
+in bed, he said. The afternoon he devoted to society and recreation.
+After supper he wrote letters to various persons, all plainly intended
+for publication, and scrupulously preserved. He kept himself free from
+care, and was most cautious about his health, regarding himself, no
+doubt, as a subject of experiment, and wishful to see how long he could
+prolong his life. At one time he writes to a friend that he shall be
+seriously disappointed if he does not manage to see 100 years.
+
+[Illustration: FIG. 53.--Descartes.]
+
+This plan of not over-working himself, and limiting the hours devoted to
+serious thought, is one that might perhaps advantageously be followed by
+some over-laborious students of the present day. At any rate it conveys
+a lesson; for the amount of ground covered by Descartes, in a life not
+very long, is extraordinary. He must, however, have had a singular
+aptitude for scientific work; and the judicious leaven of selfishness
+whereby he was able to keep himself free from care and embarrassments
+must have been a great help to him.
+
+And what did his versatile genius accomplish during his fifty-four years
+of life?
+
+In philosophy, using the term as meaning mental or moral philosophy and
+metaphysics, as opposed to natural philosophy or physics, he takes a
+very high rank, and it is on this that perhaps his greatest fame rests.
+(He is the author, you may remember, of the famous aphorism, "_Cogito,
+ergo sum_.")
+
+In biology I believe he may be considered almost equally great:
+certainly he spent a great deal of time in dissecting, and he made out a
+good deal of what is now known of the structure of the body, and of the
+theory of vision. He eagerly accepted the doctrine of the circulation of
+the blood, then being taught by Harvey, and was an excellent anatomist.
+
+You doubtless know Professor Huxley's article on Descartes in the _Lay
+Sermons_, and you perceive in what high estimation he is there held.
+
+He originated the hypothesis that animals are automata, for which indeed
+there is much to be said from some points of view; but he unfortunately
+believed that they were unconscious and non-sentient automata, and this
+belief led his disciples into acts of abominable cruelty. Professor
+Huxley lectured on this hypothesis and partially upheld it not many
+years since. The article is included in his volume called _Science and
+Culture_.
+
+Concerning his work in mathematics and physics I can speak with more
+confidence. He is the author of the Cartesian system of algebraic or
+analytic geometry, which has been so powerful an engine of research, far
+easier to wield than the old synthetic geometry. Without it Newton could
+never have written the _Principia_, or made his greatest discoveries.
+He might indeed have invented it for himself, but it would have consumed
+some of his life to have brought it to the necessary perfection.
+
+     The principle of it is the specification of the position of a point
+     in a plane by two numbers, indicating say its distance from two
+     lines of reference in the plane; like the latitude and longitude of
+     a place on the globe. For instance, the two lines of reference
+     might be the bottom edge and the left-hand vertical edge of a wall;
+     then a point on the wall, stated as being for instance 6 feet along
+     and 2 feet up, is precisely determined. These two distances are
+     called co-ordinates; horizontal ones are usually denoted by _x_,
+     and vertical ones by _y_.
+
+     If, instead of specifying two things, only one statement is made,
+     such as _y_ = 2, it is satisfied by a whole row of points, all the
+     points in a horizontal line 2 feet above the ground. Hence _y_ = 2
+     may be said to represent that straight line, and is called the
+     equation to that straight line. Similarly _x_ = 6 represents a
+     vertical straight line 6 feet (or inches or some other unit) from
+     the left-hand edge. If it is asserted that _x_ = 6 and _y_ = 2,
+     only one point can be found to satisfy both conditions, viz. the
+     crossing point of the above two straight lines.
+
+     Suppose an equation such as _x_ = _y_ to be given. This also is
+     satisfied by a row of points, viz. by all those that are
+     equidistant from bottom and left-hand edges. In other words, _x_ =
+     _y_ represents a straight line slanting upwards at 45 deg.. The
+     equation _x_ = 2_y_ represents another straight line with a
+     different angle of slope, and so on. The equation x^2 + y^2
+     = 36 represents a circle of radius 6. The equation 3x^2 +
+     4y^2 = 25 represents an ellipse; and in general every algebraic
+     equation that can be written down, provided it involve only two
+     variables, _x_ and _y_, represents some curve in a plane; a curve
+     moreover that can be drawn, or its properties completely
+     investigated without drawing, from the equation. Thus algebra is
+     wedded to geometry, and the investigation of geometric relations by
+     means of algebraic equations is called analytical geometry, as
+     opposed to the old Euclidian or synthetic mode of treating the
+     subject by reasoning consciously directed to the subject by help of
+     figures.
+
+     If there be three variables--_x_, _y_, and _z_,--instead of only
+     two, an equation among them represents not a curve in a plane but a
+     surface in space; the three variables corresponding to the three
+     dimensions of space: length, breadth, and thickness.
+
+     An equation with four variables usually requires space of four
+     dimensions for its geometrical interpretation, and so on.
+
+     Thus geometry can not only be reasoned about in a more mechanical
+     and therefore much easier, manner, but it can be extended into
+     regions of which we have and can have no direct conception, because
+     we are deficient in sense organs for accumulating any kind of
+     experience in connexion with such ideas.
+
+[Illustration: FIG. 54.--The eye diagram. [From Descartes' _Principia_.]
+Three external points are shown depicted on the retina: the image being
+appreciated by a representation of the brain.]
+
+In physics proper Descartes' tract on optics is of considerable
+historical interest. He treats all the subjects he takes up in an able
+and original manner.
+
+In Astronomy he is the author of that famous and long upheld theory, the
+doctrine of vortices.
+
+He regarded space as a plenum full of an all-pervading fluid. Certain
+portions of this fluid were in a state of whirling motion, as in a
+whirlpool or eddy of water; and each planet had its own eddy, in which
+it was whirled round and round, as a straw is caught and whirled in a
+common whirlpool. This idea he works out and elaborates very fully,
+applying it to the system of the world, and to the explanation of all
+the motions of the planets.
+
+[Illustration: FIG. 55.--Descartes's diagram of vortices, from his
+_Principia_.]
+
+This system evidently supplied a void in men's minds, left vacant by the
+overthrow of the Ptolemaic system, and it was rapidly accepted. In the
+English Universities it held for a long time almost undisputed sway; it
+was in this faith that Newton was brought up.
+
+Something was felt to be necessary to keep the planets moving on their
+endless round; the _primum mobile_ of Ptolemy had been stopped; an angel
+was sometimes assigned to each planet to carry it round, but though a
+widely diffused belief, this was a fantastic and not a serious
+scientific one. Descartes's vortices seemed to do exactly what was
+wanted.
+
+It is true they had no connexion with the laws of Kepler. I doubt
+whether he knew about the laws of Kepler; he had not much opinion of
+other people's work; he read very little--found it easier to think. (He
+travelled through Florence once when Galileo was at the height of his
+renown without calling upon or seeing him.) In so far as the motion of a
+planet was not circular, it had to be accounted for by the jostling and
+crowding and distortion of the vortices.
+
+Gravitation he explained by a settling down of bodies toward the centre
+of each vortex; and cohesion by an absence of relative motion tending to
+separate particles of matter. He "can imagine no stronger cement."
+
+The vortices, as Descartes imagined them, are not now believed in. Are
+we then to regard the system as absurd and wholly false? I do not see
+how we can do this, when to this day philosophers are agreed in
+believing space to be completely full of fluid, which fluid is certainly
+capable of vortex motion, and perhaps everywhere does possess that
+motion. True, the now imagined vortices are not the large whirls of
+planetary size, they are rather infinitesimal whirls of less than atomic
+dimensions; still a whirling fluid is believed in to this day, and many
+are seeking to deduce all the properties of matter (rigidity,
+elasticity, cohesion gravitation, and the rest) from it.
+
+Further, although we talk glibly about gravitation and magnetism, and so
+on, we do not really know what they are. Progress is being made, but we
+do not yet properly know. Much, overwhelmingly much, remains to be
+discovered, and it ill-behoves us to reject any well-founded and
+long-held theory as utterly and intrinsically false and absurd. The more
+one gets to know, the more one perceives a kernel of truth even in the
+most singular statements; and scientific men have learned by experience
+to be very careful how they lop off any branch of the tree of knowledge,
+lest as they cut away the dead wood they lose also some green shoot,
+some healthy bud of unperceived truth.
+
+However, it may be admitted that the idea of a Cartesian vortex in
+connexion with the solar system applies, if at all, rather to an
+earlier--its nebulous--stage, when the whole thing was one great whirl,
+ready to split or shrink off planetary rings at their appropriate
+distances.
+
+Soon after he had written his great work, the _Principia Mathematica_,
+and before he printed it, news reached him of the persecution and
+recantation of Galileo. "He seems to have been quite thunderstruck at
+the tidings," says Mr. Mahaffy, in his _Life of Descartes_.[15] "He had
+started on his scientific journeys with the firm determination to enter
+into no conflict with the Church, and to carry out his system of pure
+mathematics and physics without ever meddling with matters of faith. He
+was rudely disillusioned as to the possibility of this severance. He
+wrote at once--apparently, November 20th, 1633--to Mersenne to say he
+would on no account publish his work--nay, that he had at first resolved
+to burn all his papers, for that he would never prosecute philosophy at
+the risk of being censured by his Church. 'I could hardly have
+believed,' he says, 'that an Italian, and in favour with the Pope as I
+hear, could be considered criminal for nothing else than for seeking to
+establish the earth's motion; though I know it has formerly been
+censured by some Cardinals. But I thought I had heard that since then it
+was constantly being taught, even at Rome; and I confess that if the
+opinion of the earth's movement is false, all the foundations of my
+philosophy are so also, because it is demonstrated clearly by them. It
+is so bound up with every part of my treatise that I could not sever it
+without making the remainder faulty; and although I consider all my
+conclusions based on very certain and clear demonstrations, I would not
+for all the world sustain them against the authority of the Church.'"
+
+Ten years later, however, he did publish the book, for he had by this
+time hit on an ingenious compromise. He formally denied that the earth
+moved, and only asserted that it was carried along with its water and
+air in one of those larger motions of the celestial ether which produce
+the diurnal and annual revolutions of the solar system. So, just as a
+passenger on the deck of a ship might be called stationary, so was the
+earth. He gives himself out therefore as a follower of Tycho rather than
+of Copernicus, and says if the Church won't accept this compromise he
+must return to the Ptolemaic system; but he hopes they won't compel him
+to do that, seeing that it is manifestly untrue.
+
+This elaborate deference to the powers that be did not indeed save the
+work from being ultimately placed upon the forbidden list by the Church,
+but it saved himself, at any rate, from annoying persecution. He was
+not, indeed, at all willing to be persecuted, and would no doubt have at
+once withdrawn anything they wished. I should be sorry to call him a
+time-server, but he certainly had plenty of that worldly wisdom in which
+some of his predecessors had been so lamentably deficient. Moreover, he
+was really a sceptic, and cared nothing at all about the Church or its
+dogmas. He knew the Church's power, however, and the advisability of
+standing well with it: he therefore professed himself a Catholic, and
+studiously kept his science and his Christianity distinct.
+
+In saying that he was a sceptic you must not understand that he was in
+the least an atheist. Very few men are; certainly Descartes never
+thought of being one. The term is indeed ludicrously inapplicable to
+him, for a great part of his philosophy is occupied with what he
+considers a rigorous proof of the existence of the Deity.
+
+At the age of fifty-three he was sent for to Stockholm by Christina,
+Queen of Sweden, a young lady enthusiastically devoted to study of all
+kinds and determined to surround her Court with all that was most famous
+in literature and science. Thither, after hesitation, Descartes went. He
+greatly liked royalty, but he dreaded the cold climate. Born in
+Touraine, a Swedish winter was peculiarly trying to him, especially as
+the energetic Queen would have lessons given her at five o'clock in the
+morning. She intended to treat him well, and was immensely taken with
+him; but this getting up at five o'clock on a November morning, to a man
+accustomed all his life to lie in bed till eleven, was a cruel hardship.
+He was too much of a courtier, however, to murmur, and the early morning
+audience continued. His health began to break down: he thought of
+retreating, but suddenly he gave way and became delirious. The Queen's
+physician attended him, and of course wanted to bleed him. This, knowing
+all he knew of physiology, sent him furious, and they could do nothing
+with him. After some days he became quiet, was bled twice, and gradually
+sank, discoursing with great calmness on his approaching death, and duly
+fortified with all the rites of the Catholic Church.
+
+His general method of research was as nearly as possible a purely
+deductive one:--_i.e._, after the manner of Euclid he starts with a few
+simple principles, and then, by a chain of reasoning, endeavours to
+deduce from them their consequences, and so to build up bit by bit an
+edifice of connected knowledge. In this he was the precursor of Newton.
+This method, when rigorously pursued, is the most powerful and
+satisfactory of all, and results in an ordered province of science far
+superior to the fragmentary conquests of experiment. But few indeed are
+the men who can handle it safely and satisfactorily: and none without
+continual appeals to experiment for verification. It was through not
+perceiving the necessity for verification that he erred. His importance
+to science lies not so much in what he actually discovered as in his
+anticipation of the right conditions for the solution of problems in
+physical science. He in fact made the discovery that Nature could after
+all be interrogated mathematically--a fact that was in great danger of
+remaining unknown. For, observe, that the mathematical study of Nature,
+the discovery of truth with a piece of paper and a pen, has a perilous
+similarity at first sight to the straw-thrashing subtleties of the
+Greeks, whose methods of investigating nature by discussing the meaning
+of words and the usage of language and the necessities of thought, had
+proved to be so futile and unproductive.
+
+A reaction had set in, led by Galileo, Gilbert, and the whole modern
+school of experimental philosophers, lasting down to the present
+day:--men who teach that the only right way of investigating Nature is
+by experiment and observation.
+
+It is indeed a very right and an absolutely necessary way; but it is not
+the only way. A foundation of experimental fact there must be; but upon
+this a great structure of theoretical deduction can be based, all
+rigidly connected together by pure reasoning, and all necessarily as
+true as the premises, provided no mistake is made. To guard against the
+possibility of mistake and oversight, especially oversight, all
+conclusions must sooner or later be brought to the test of experiment;
+and if disagreeing therewith, the theory itself must be re-examined,
+and the flaw discovered, or else the theory must be abandoned.
+
+Of this grand method, quite different from the gropings in the dark of
+Kepler--this method, which, in combination with experiment, has made
+science what it now is--this which in the hands of Newton was to lead to
+such stupendous results, we owe the beginning and early stages to Rene
+Descartes.
+
+
+
+
+SUMMARY OF FACTS FOR LECTURES VII AND VIII
+
+  Otto Guericke          1602-1686
+  Hon. Robert Boyle      1626-1691
+  Huyghens               1629-1695
+  Christopher Wren       1632-1723
+  Robert Hooke           1635-1702
+  NEWTON                 1642-1727
+  Edmund Halley          1656-1742
+  James Bradley          1692-1762
+
+_Chronology of Newton's Life._
+
+
+Isaac Newton was born at Woolsthorpe, near Grantham, Lincolnshire, on
+Christmas Day, 1642. His father, a small freehold farmer, also named
+Isaac, died before his birth. His mother, _nee_ Hannah Ayscough, in two
+years married a Mr. Smith, rector of North Witham, but was again left a
+widow in 1656. His uncle, W. Ayscough, was rector of a near parish and a
+graduate of Trinity College, Cambridge. At the age of fifteen Isaac was
+removed from school at Grantham to be made a farmer of, but as it seemed
+he would not make a good one his uncle arranged for him to return to
+school and thence to Cambridge, where he entered Trinity College as a
+sub-sizar in 1661. Studied Descartes's geometry. Found out a method of
+infinite series in 1665, and began the invention of Fluxions. In the
+same year and the next he was driven from Cambridge by the plague. In
+1666, at Woolsthorpe, the apple fell. In 1667 he was elected a fellow of
+his college, and in 1669 was specially noted as possessing an
+unparalleled genius by Dr. Barrow, first Lucasian Professor of
+Mathematics. The same year Dr. Barrow retired from his chair in favour
+of Newton, who was thus elected at the age of twenty-six. He lectured
+first on optics with great success. Early in 1672 he was elected a
+Fellow of the Royal Society, and communicated his researches in optics,
+his reflecting telescope, and his discovery of the compound nature of
+white light. Annoying controversies arose; but he nevertheless
+contributed a good many other most important papers in optics, including
+observations in diffraction, and colours of thin plates. He also
+invented the modern sextant. In 1672 a letter from Paris was read at the
+Royal Society concerning a new and accurate determination of the size of
+the earth by Picard. When Newton heard of it he began the _Principia_,
+working in silence. In 1684 arose a discussion between Wren, Hooke, and
+Halley concerning the law of inverse square as applied to gravity and
+the path it would cause the planets to describe. Hooke asserted that he
+had a solution, but he would not produce it. After waiting some time for
+it Halley went to Cambridge to consult Newton on the subject, and thus
+discovered the existence of the first part of the _Principia_, wherein
+all this and much more was thoroughly worked out. On his representations
+to the Royal Society the manuscript was asked for, and when complete was
+printed and published in 1687 at Halley's expense. While it was being
+completed Newton and seven others were sent to uphold the dignity of the
+University, before the Court of High Commission and Judge Jeffreys,
+against a high-handed action of James II. In 1682 he was sent to
+Parliament, and was present at the coronation of William and Mary. Made
+friends with Locke. In 1694 Montague, Lord Halifax, made him Warden, and
+in 1697 Master, of the Mint. Whiston succeeded him as Lucasian
+Professor. In 1693 the method of fluxions was published. In 1703 Newton
+was made President of the Royal Society, and held the office to the end
+of his life. In 1705 he was knighted by Anne. In 1713 Cotes helped him
+to bring out a new edition of the _Principia_, completed as we now have
+it. On the 20th of March 1727, he died: having lived from Charles I. to
+George II.
+
+
+THE LAWS OF MOTION, DISCOVERED BY GALILEO, STATED BY NEWTON.
+
+_Law 1._--If no force acts on a body in motion, it continues to move
+uniformly in a straight line.
+
+_Law 2._--If force acts on a body, it produces a change of motion
+proportional to the force and in the same direction.
+
+_Law 3._--When one body exerts force on another, that other reacts with
+equal force upon the one.
+
+
+
+
+LECTURE VII
+
+SIR ISAAC NEWTON
+
+
+The little hamlet of Woolsthorpe lies close to the village of
+Colsterworth, about six miles south of Grantham, in the county of
+Lincoln. In the manor house of Woolsthorpe, on Christmas Day, 1642, was
+born to a widowed mother a sickly infant who seemed not long for this
+world. Two women who were sent to North Witham to get some medicine for
+him scarcely expected to find him alive on their return. However, the
+child lived, became fairly robust, and was named Isaac, after his
+father. What sort of a man this father was we do not know. He was what
+we may call a yeoman, that most wholesome and natural of all classes. He
+owned the soil he tilled, and his little estate had already been in the
+family for some hundred years. He was thirty-six when he died, and had
+only been married a few months.
+
+Of the mother, unfortunately, we know almost as little. We hear that she
+was recommended by a parishioner to the Rev. Barnabas Smith, an old
+bachelor in search of a wife, as "the widow Newton--an extraordinary
+good woman:" and so I expect she was, a thoroughly sensible, practical,
+homely, industrious, middle-class, Mill-on-the-Floss sort of woman.
+However, on her second marriage she went to live at North Witham, and
+her mother, old Mrs. Ayscough, came to superintend the farm at
+Woolsthorpe, and take care of young Isaac.
+
+By her second marriage his mother acquired another piece of land, which
+she settled on her first son; so Isaac found himself heir to two little
+properties, bringing in a rental of about L80 a year.
+
+[Illustration: FIG. 56.--Manor-house of Woolsthorpe.]
+
+He had been sent to a couple of village schools to acquire the ordinary
+accomplishments taught at those places, and for three years to the
+grammar school at Grantham, then conducted by an old gentleman named Mr.
+Stokes. He had not been very industrious at school, nor did he feel
+keenly the fascinations of the Latin Grammar, for he tells us that he
+was the last boy in the lowest class but one. He used to pay much more
+attention to the construction of kites and windmills and waterwheels,
+all of which he made to work very well. He also used to tie paper
+lanterns to the tail of his kite, so as to make the country folk fancy
+they saw a comet, and in general to disport himself as a boy should.
+
+It so happened, however, that he succeeded in thrashing, in fair fight,
+a bigger boy who was higher in the school, and who had given him a
+kick. His success awakened a spirit of emulation in other things than
+boxing, and young Newton speedily rose to be top of the school.
+
+Under these circumstances, at the age of fifteen, his mother, who had
+now returned to Woolsthorpe, which had been rebuilt, thought it was time
+to train him for the management of his land, and to make a farmer and
+grazier of him. The boy was doubtless glad to get away from school, but
+he did not take kindly to the farm--especially not to the marketing at
+Grantham. He and an old servant were sent to Grantham every week to buy
+and sell produce, but young Isaac used to leave his old mentor to do all
+the business, and himself retire to an attic in the house he had lodged
+in when at school, and there bury himself in books.
+
+After a time he didn't even go through the farce of visiting Grantham at
+all; but stopped on the road and sat under a hedge, reading or making
+some model, until his companion returned.
+
+We hear of him now in the great storm of 1658, the storm on the day
+Cromwell died, measuring the force of the wind by seeing how far he
+could jump with it and against it. He also made a water-clock and set it
+up in the house at Grantham, where it kept fairly good time so long as
+he was in the neighbourhood to look after it occasionally.
+
+At his own home he made a couple of sundials on the side of the wall (he
+began by marking the position of the sun by the shadow of a peg driven
+into the wall, but this gradually developed into a regular dial) one of
+which remained of use for some time; and was still to be seen in the
+same place during the first half of the present century, only with the
+gnomon gone. In 1844 the stone on which it was carved was carefully
+extracted and presented to the Royal Society, who preserve it in their
+library. The letters WTON roughly carved on it are barely visible.
+
+All these pursuits must have been rather trying to his poor mother, and
+she probably complained to her brother, the rector of Burton Coggles:
+at any rate this gentleman found master Newton one morning under a hedge
+when he ought to have been farming. But as he found him working away at
+mathematics, like a wise man he persuaded his sister to send the boy
+back to school for a short time, and then to Cambridge. On the day of
+his finally leaving school old Mr. Stokes assembled the boys, made them
+a speech in praise of Newton's character and ability, and then dismissed
+him to Cambridge.
+
+At Trinity College a new world opened out before the country-bred lad.
+He knew his classics passably, but of mathematics and science he was
+ignorant, except through the smatterings he had picked up for himself.
+He devoured a book on logic, and another on Kepler's Optics, so fast
+that his attendance at lectures on these subjects became unnecessary. He
+also got hold of a Euclid and of Descartes's Geometry. The Euclid seemed
+childishly easy, and was thrown aside, but the Descartes baffled him for
+a time. However, he set to it again and again and before long mastered
+it. He threw himself heart and soul into mathematics, and very soon made
+some remarkable discoveries. First he discovered the binomial theorem:
+familiar now to all who have done any algebra, unintelligible to others,
+and therefore I say nothing about it. By the age of twenty-one or two he
+had begun his great mathematical discovery of infinite series and
+fluxions--now known by the name of the Differential Calculus. He wrote
+these things out and must have been quite absorbed in them, but it never
+seems to have occurred to him to publish them or tell any one about
+them.
+
+In 1664 he noticed some halos round the moon, and, as his manner was, he
+measured their angles--the small ones 3 and 5 degrees each, the larger
+one 22 deg..35. Later he gave their theory.
+
+     Small coloured halos round the moon are often seen, and are said to
+     be a sign of rain. They are produced by the action of minute
+     globules of water or cloud particles upon light, and are brightest
+     when the particles are nearly equal in size. They are not like the
+     rainbow, every part of which is due to light that has entered a
+     raindrop, and been refracted and reflected with prismatic
+     separation of colours; a halo is caused by particles so small as to
+     be almost comparable with the size of waves of light, in a way
+     which is explained in optics under the head "diffraction." It may
+     be easily imitated by dusting an ordinary piece of window-glass
+     over with lycopodium, placing a candle near it, and then looking at
+     the candle-flame through the dusty glass from a fair distance. Or
+     you may look at the image of a candle in a dusted looking-glass.
+     Lycopodium dust is specially suitable, for its granules are
+     remarkably equal in size. The large halo, more rarely seen, of
+     angular radius 22 deg..35, is due to another cause again, and is a
+     prismatic effect, although it exhibits hardly any colour. The angle
+     22-1/2 deg. is characteristic of refraction in crystals with angles of
+     60 deg. and refractive index about the same as water; in other words
+     this halo is caused by ice crystals in the higher regions of the
+     atmosphere.
+
+He also the same year observed a comet, and sat up so late watching it
+that he made himself ill. By the end of the year he was elected to a
+scholarship and took his B.A. degree. The order of merit for that year
+never existed or has not been kept. It would have been interesting, not
+as a testimony to Newton, but to the sense or non-sense of the
+examiners. The oldest Professorship of Mathematics at the University of
+Cambridge, the Lucasian, had not then been long founded, and its first
+occupant was Dr. Isaac Barrow, an eminent mathematician, and a kind old
+man. With him Newton made good friends, and was helpful in preparing a
+treatise on optics for the press. His help is acknowledged by Dr. Barrow
+in the preface, which states that he had corrected several errors and
+made some capital additions of his own. Thus we see that, although the
+chief part of his time was devoted to mathematics, his attention was
+already directed to both optics and astronomy. (Kepler, Descartes,
+Galileo, all combined some optics with astronomy. Tycho and the old ones
+combined alchemy; Newton dabbled in this also.)
+
+Newton reached the age of twenty-three in 1665, the year of the Great
+Plague. The plague broke out in Cambridge as well as in London, and the
+whole college was sent down. Newton went back to Woolsthorpe, his mind
+teeming with ideas, and spent the rest of this year and part of the next
+in quiet pondering. Somehow or other he had got hold of the notion of
+centrifugal force. It was six years before Huyghens discovered and
+published the laws of centrifugal force, but in some quiet way of his
+own Newton knew about it and applied the idea to the motion of the
+planets.
+
+We can almost follow the course of his thoughts as he brooded and
+meditated on the great problem which had taxed so many previous
+thinkers,--What makes the planets move round the sun? Kepler had
+discovered how they moved, but why did they so move, what urged them?
+
+Even the "how" took a long time--all the time of the Greeks, through
+Ptolemy, the Arabs, Copernicus, Tycho: circular motion, epicycles, and
+excentrics had been the prevailing theory. Kepler, with his marvellous
+industry, had wrested from Tycho's observations the secret of their
+orbits. They moved in ellipses with the sun in one focus. Their rate of
+description of area, not their speed, was uniform and proportional to
+time.
+
+Yes, and a third law, a mysterious law of unintelligible import, had
+also yielded itself to his penetrating industry--a law the discovery of
+which had given him the keenest delight, and excited an outburst of
+rapture--viz. that there was a relation between the distances and the
+periodic times of the several planets. The cubes of the distances were
+proportional to the squares of the times for the whole system. This law,
+first found true for the six primary planets, he had also extended,
+after Galileo's discovery, to the four secondary planets, or satellites
+of Jupiter (p. 81).
+
+But all this was working in the dark--it was only the first step--this
+empirical discovery of facts; the facts were so, but how came they so?
+What made the planets move in this particular way? Descartes's vortices
+was an attempt, a poor and imperfect attempt, at an explanation. It had
+been hailed and adopted throughout Europe for want of a better, but it
+did not satisfy Newton. No, it proceeded on a wrong tack, and Kepler had
+proceeded on a wrong tack in imagining spokes or rays sticking out from
+the sun and driving the planets round like a piece of mechanism or mill
+work. For, note that all these theories are based on a wrong idea--the
+idea, viz., that some force is necessary to maintain a body in motion.
+But this was contrary to the laws of motion as discovered by Galileo.
+You know that during his last years of blind helplessness at Arcetri,
+Galileo had pondered and written much on the laws of motion, the
+foundation of mechanics. In his early youth, at Pisa, he had been
+similarly occupied; he had discovered the pendulum, he had refuted the
+Aristotelians by dropping weights from the leaning tower (which we must
+rejoice that no earthquake has yet injured), and he had returned to
+mechanics at intervals all his life; and now, when his eyes were useless
+for astronomy, when the outer world has become to him only a prison to
+be broken by death, he returns once more to the laws of motion, and
+produces the most solid and substantial work of his life.
+
+For this is Galileo's main glory--not his brilliant exposition of the
+Copernican system, not his flashes of wit at the expense of a moribund
+philosophy, not his experiments on floating bodies, not even his
+telescope and astronomical discoveries--though these are the most taking
+and dazzling at first sight. No; his main glory and title to immortality
+consists in this, that he first laid the foundation of mechanics on a
+firm and secure basis of experiment, reasoning, and observation. He
+first discovered the true Laws of Motion.
+
+I said little of this achievement in my lecture on him; for the work was
+written towards the end of his life, and I had no time then. But I knew
+I should have to return to it before we came to Newton, and here we are.
+
+You may wonder how the work got published when so many of his
+manuscripts were destroyed. Horrible to say, Galileo's own son destroyed
+a great bundle of his father's manuscripts, thinking, no doubt, thereby
+to save his own soul. This book on mechanics was not burnt, however. The
+fact is it was rescued by one or other of his pupils, Toricelli or
+Viviani, who were allowed to visit him in his last two or three years;
+it was kept by them for some time, and then published surreptitiously in
+Holland. Not that there is anything in it bearing in any visible way on
+any theological controversy; but it is unlikely that the Inquisition
+would have suffered it to pass notwithstanding.
+
+I have appended to the summary preceding this lecture (p. 160) the three
+axioms or laws of motion discovered by Galileo. They are stated by
+Newton with unexampled clearness and accuracy, and are hence known as
+Newton's laws, but they are based on Galileo's work. The first is the
+simplest; though ignorance of it gave the ancients a deal of trouble. It
+is simply a statement that force is needed to change the motion of a
+body; _i.e._ that if no force act on a body it will continue to move
+uniformly both in speed and direction--in other words, steadily, in a
+straight line. The old idea had been that some force was needed to
+maintain motion. On the contrary, the first law asserts, some force is
+needed to destroy it. Leave a body alone, free from all friction or
+other retarding forces, and it will go on for ever. The planetary motion
+through empty space therefore wants no keeping up; it is not the motion
+that demands a force to maintain it, it is the curvature of the path
+that needs a force to produce it continually. The motion of a planet is
+approximately uniform so far as speed is concerned, but it is not
+constant in direction; it is nearly a circle. The real force needed is
+not a propelling but a deflecting force.
+
+The second law asserts that when a force acts, the motion changes,
+either in speed or in direction, or both, at a pace proportional to the
+magnitude of the force, and in the same direction as that in which the
+force acts. Now since it is almost solely in direction that planetary
+motion alters, a deflecting force only is needed; a force at right
+angles to the direction of motion, a force normal to the path.
+Considering the motion as circular, a force along the radius, a radial
+or centripetal force, must be acting continually. Whirl a weight round
+and round by a bit of elastic, the elastic is stretched; whirl it
+faster, it is stretched more. The moving mass pulls at the elastic--that
+is its centrifugal force; the hand at the centre pulls also--that is
+centripetal force.
+
+The third law asserts that these two forces are equal, and together
+constitute the tension in the elastic. It is impossible to have one
+force alone, there must be a pair. You can't push hard against a body
+that offers no resistance. Whatever force you exert upon a body, with
+that same force the body must react upon you. Action and reaction are
+always equal and opposite.
+
+Sometimes an absurd difficulty is felt with respect to this, even by
+engineers. They say, "If the cart pulls against the horse with precisely
+the same force as the horse pulls the cart, why should the cart move?"
+Why on earth not? The cart moves because the horse pulls it, and because
+nothing else is pulling it back. "Yes," they say, "the cart is pulling
+back." But what is it pulling back? Not itself, surely? "No, the horse."
+Yes, certainly the cart is pulling at the horse; if the cart offered no
+resistance what would be the good of the horse? That is what he is for,
+to overcome the pull-back of the cart; but nothing is pulling the cart
+back (except, of course, a little friction), and the horse is pulling it
+forward, hence it goes forward. There is no puzzle at all when once you
+realise that there are two bodies and two forces acting, and that one
+force acts on each body.[16]
+
+If, indeed, two balanced forces acted on one body that would be in
+equilibrium, but the two equal forces contemplated in the third law act
+on two different bodies, and neither is in equilibrium.
+
+So much for the third law, which is extremely simple, though it has
+extraordinarily far-reaching consequences, and when combined with a
+denial of "action at a distance," is precisely the principle of the
+Conservation of Energy. Attempts at perpetual motion may all be regarded
+as attempts to get round this "third law."
+
+[Illustration: FIG. 57.]
+
+     On the subject of the _second_ law a great deal more has to be said
+     before it can be in any proper sense even partially appreciated,
+     but a complete discussion of it would involve a treatise on
+     mechanics. It is _the_ law of mechanics. One aspect of it we must
+     attend to now in order to deal with the motion of the planets, and
+     that is the fact that the change of motion of a body depends solely
+     and simply on the force acting, and not at all upon what the body
+     happens to be doing at the time it acts. It may be stationary, or
+     it may be moving in any direction; that makes no difference.
+
+     Thus, referring back to the summary preceding Lecture IV, it is
+     there stated that a dropped body falls 16 feet in the first second,
+     that in two seconds it falls 64 feet, and so on, in proportion to
+     the square of the time. So also will it be the case with a thrown
+     body, but the drop must be reckoned from its line of motion--the
+     straight line which, but for gravity, it would describe.
+
+     Thus a stone thrown from _O_ with the velocity _OA_ would in one
+     second find itself at _A_, in two seconds at _B_, in three seconds
+     at _C_, and so on, in accordance with the first law of motion, if
+     no force acted. But if gravity acts it will have fallen 16 feet by
+     the time it would have got to _A_, and so will find itself at _P_.
+     In two seconds it will be at _Q_, having fallen a vertical height
+     of 64 feet; in three seconds it will be at _R_, 144 feet below _C_;
+     and so on. Its actual path will be a curve, which in this case is a
+     parabola. (Fig. 57.)
+
+     If a cannon is pointed horizontally over a level plain, the cannon
+     ball will be just as much affected by gravity as if it were
+     dropped, and so will strike the plain at the same instant as
+     another which was simply dropped where it started. One ball may
+     have gone a mile and the other only dropped a hundred feet or so,
+     but the time needed by both for the vertical drop will be the same.
+     The horizontal motion of one is an extra, and is due to the powder.
+
+     As a matter of fact the path of a projectile in vacuo is only
+     approximately a parabola. It is instructive to remember that it is
+     really an ellipse with one focus very distant, but not at infinity.
+     One of its foci is the centre of the earth. A projectile is really
+     a minute satellite of the earth's, and in vacuo it accurately obeys
+     all Kepler's laws. It happens not to be able to complete its orbit,
+     because it was started inconveniently close to the earth, whose
+     bulk gets in its way; but in that respect the earth is to be
+     reckoned as a gratuitous obstruction, like a target, but a target
+     that differs from most targets in being hard to miss.
+
+[Illustration: FIG. 58.]
+
+     Now consider circular motion in the same way, say a ball whirled
+     round by a string. (Fig. 58.)
+
+     Attending to the body at _O_, it is for an instant moving towards
+     _A_, and if no force acted it would get to _A_ in a time which for
+     brevity we may call a second. But a force, the pull of the string,
+     is continually drawing it towards _S_, and so it really finds
+     itself at _P_, having described the circular arc _OP_, which may
+     be considered to be compounded of, and analyzable into the
+     rectilinear motion _OA_ and the drop _AP_. At _P_ it is for an
+     instant moving towards _B_, and the same process therefore carries
+     it to _Q_; in the third second it gets to _R_; and so on: always
+     falling, so to speak, from its natural rectilinear path, towards
+     the centre, but never getting any nearer to the centre.
+
+     The force with which it has thus to be constantly pulled in towards
+     the centre, or, which is the same thing, the force with which it is
+     tugging at whatever constraint it is that holds it in, is
+     _mv^2/r_; where _m_ is the mass of the particle, _v_ its
+     velocity, and _r_ the radius of its circle of movement. This is the
+     formula first given by Huyghens for centrifugal force.
+
+     We shall find it convenient to express it in terms of the time of
+     one revolution, say _T_. It is easily done, since plainly T =
+     circumference/speed = _2[pi]r/v_; so the above expression for
+     centrifugal force becomes _4[pi]^2mr/T^2_.
+
+     As to the fall of the body towards the centre every microscopic
+     unit of time, it is easily reckoned. For by Euclid III. 36, and
+     Fig. 58, _AP.AA' = AO^2_. Take _A_ very near _O_, then _OA = vt_,
+     and _AA' = 2r_; so _AP = v^2t^2/2r = 2[pi]^2r
+     t^2/T^2_; or the fall per second is _2[pi]^2r/T^2_,
+     _r_ being its distance from the centre, and _T_ its time of going
+     once round.
+
+     In the case of the moon for instance, _r_ is 60 earth radii; more
+     exactly 60.2; and _T_ is a lunar month, or more precisely 27 days,
+     7 hours, 43 minutes, and 11-1/2 seconds. Hence the moon's
+     deflection from the tangential or rectilinear path every minute
+     comes out as very closely 16 feet (the true size of the earth being
+     used).
+
+Returning now to the case of a small body revolving round a big one, and
+assuming a force directly proportional to the mass of both bodies, and
+inversely proportional to the square of the distance between them:
+_i.e._ assuming the known force of gravity, it is
+
+  _V Mm/r^2_
+
+where _V_ is a constant, called the gravitation constant, to be
+determined by experiment.
+
+If this is the centripetal force pulling a planet or satellite in, it
+must be equal to the centrifugal force of this latter, viz. (see above).
+
+  _4[pi]^2mr/T^2
+
+Equate the two together, and at once we get
+
+  _r^3/T^2 = V/4[pi]^2M;_
+
+or, in words, the cube of the distance divided by the square of the
+periodic time for every planet or satellite of the system under
+consideration, will be constant and proportional to the mass of the
+central body.
+
+This is Kepler's third law, with a notable addition. It is stated above
+for circular motion only, so as to avoid geometrical difficulties, but
+even so it is very instructive. The reason of the proportion between
+_r^3_ and _T^2_ is at once manifest; and as soon as the constant _V_
+became known, _the mass of the central body_, the sun in the case of a
+planet, the earth in the case of the moon, Jupiter in the case of his
+satellites, was at once determined.
+
+Newton's reasoning at this time might, however, be better displayed
+perhaps by altering the order of the steps a little, as thus:--
+
+The centrifugal force of a body is proportional to _r^3/T^2_, but by
+Kepler's third law _r^3/T^2_ is constant for all the planets,
+reckoning _r_ from the sun. Hence the centripetal force needed to hold
+in all the planets will be a single force emanating from the sun and
+varying inversely with the square of the distance from that body.
+
+Such a force is at once necessary and sufficient. Such a force would
+explain the motion of the planets.
+
+But then all this proceeds on a wrong assumption--that the planetary
+motion is circular. Will it hold for elliptic orbits? Will an inverse
+square law of force keep a body moving in an elliptic orbit about the
+sun in one focus? This is a far more difficult question. Newton solved
+it, but I do not believe that even he could have solved it, except that
+he had at his disposal two mathematical engines of great power--the
+Cartesian method of treating geometry, and his own method of Fluxions.
+One can explain the elliptic motion now mathematically, but hardly
+otherwise; and I must be content to state that the double fact is
+true--viz., that an inverse square law will move the body in an ellipse
+or other conic section with the sun in one focus, and that if a body so
+moves it _must_ be acted on by an inverse square law.
+
+[Illustration: FIG. 59.]
+
+This then is the meaning of the first and third laws of Kepler. What
+about the second? What is the meaning of the equable description of
+areas? Well, that rigorously proves that a planet is acted on by a force
+directed to the centre about which the rate of description of areas is
+equable. It proves, in fact, that the sun is the attracting body, and
+that no other force acts.
+
+     For first of all if the first law of motion is obeyed, _i.e._ if no
+     force acts, and if the path be equally subdivided to represent
+     equal times, and straight lines be drawn from the divisions to any
+     point whatever, all these areas thus enclosed will be equal,
+     because they are triangles on equal base and of the same height
+     (Euclid, I). See Fig. 59; _S_ being any point whatever, and _A_,
+     _B_, _C_, successive positions of a body.
+
+     Now at each of the successive instants let the body receive a
+     sudden blow in the direction of that same point _S_, sufficient to
+     carry it from _A_ to _D_ in the same time as it would have got to
+     _B_ if left alone. The result will be that there will be a
+     compromise, and it will really arrive at _P_, travelling along the
+     diagonal of the parallelogram _AP_. The area its radius vector
+     sweeps out is therefore _SAP_, instead of what it would have been,
+     _SAB_. But then these two areas are equal, because they are
+     triangles on the same base _AS_, and between the same parallels
+     _BP_, _AS_; for by the parallelogram law _BP_ is parallel to _AD_.
+     Hence the area that would have been described is described, and as
+     all the areas were equal in the case of no force, they remain equal
+     when the body receives a blow at the end of every equal interval of
+     time, _provided_ that every blow is actually directed to _S_, the
+     point to which radii vectores are drawn.
+
+[Illustration: FIG. 60.]
+
+[Illustration: FIG. 61.]
+
+     It is instructive to see that it does not hold if the blow is any
+     otherwise directed; for instance, as in Fig. 61, when the blow is
+     along _AE_, the body finds itself at _P_ at the end of the second
+     interval, but the area _SAP_ is by no means equal to _SAB_, and
+     therefore not equal to _SOA_, the area swept out in the first
+     interval.
+
+     In order to modify Fig. 60 so as to represent continuous motion and
+     steady forces, we have to take the sides of the polygon _OAPQ_,
+     &c., very numerous and very small; in the limit, infinitely
+     numerous and infinitely small. The path then becomes a curve, and
+     the series of blows becomes a steady force directed towards _S_.
+     About whatever point therefore the rate of description of areas is
+     uniform, that point and no other must be the centre of all the
+     force there is. If there be no force, as in Fig. 59, well and good,
+     but if there be any force however small not directed towards _S_,
+     then the rate of description of areas about _S_ cannot be uniform.
+     Kepler, however, says that the rate of description of areas of each
+     planet about the sun is, by Tycho's observations, uniform; hence
+     the sun is the centre of all the force that acts on them, and there
+     is no other force, not even friction. That is the moral of Kepler's
+     second law.
+
+     We may also see from it that gravity does not travel like light, so
+     as to take time on its journey from sun to planet; for, if it did,
+     there would be a sort of aberration, and the force on its arrival
+     could no longer be accurately directed to the centre of the sun.
+     (See _Nature_, vol. xlvi., p. 497.) It is a matter for accuracy of
+     observation, therefore, to decide whether the minutest trace of
+     such deviation can be detected, _i.e._ within what limits of
+     accuracy Kepler's second law is now known to be obeyed.
+
+     I will content myself by saying that the limits are extremely
+     narrow. [Reference may be made also to p. 208.]
+
+Thus then it became clear to Newton that the whole solar system depended
+on a central force emanating from the sun, and varying inversely with
+the square of the distance from him: for by that hypothesis all the laws
+of Kepler concerning these motions were completely accounted for; and,
+in fact, the laws necessitated the hypothesis and established it as a
+theory.
+
+Similarly the satellites of Jupiter were controlled by a force emanating
+from Jupiter and varying according to the same law. And again our moon
+must be controlled by a force from the earth, decreasing with the
+distance according to the same law.
+
+Grant this hypothetical attracting force pulling the planets towards
+the sun, pulling the moon towards the earth, and the whole mechanism of
+the solar system is beautifully explained.
+
+If only one could be sure there was such a force! It was one thing to
+calculate out what the effects of such a force would be: it was another
+to be able to put one's finger upon it and say, this is the force that
+actually exists and is known to exist. We must picture him meditating in
+his garden on this want--an attractive force towards the earth.
+
+If only such an attractive force pulling down bodies to the earth
+existed. An apple falls from a tree. Why, it does exist! There is
+gravitation, common gravity that makes bodies fall and gives them their
+weight.
+
+Wanted, a force tending towards the centre of the earth. It is to hand!
+
+It is common old gravity that had been known so long, that was perfectly
+familiar to Galileo, and probably to Archimedes. Gravity that regulates
+the motion of projectiles. Why should it only pull stones and apples?
+Why should it not reach as high as the moon? Why should it not be the
+gravitation of the sun that is the central force acting on all the
+planets?
+
+Surely the secret of the universe is discovered! But, wait a bit; is it
+discovered? Is this force of gravity sufficient for the purpose? It must
+vary inversely with the square of the distance from the centre of the
+earth. How far is the moon away? Sixty earth's radii. Hence the force of
+gravity at the moon's distance can only be 1/3600 of what it is on the
+earth's surface. So, instead of pulling it 16 ft. per second, it should
+pull it 16/3600 ft. per second, or 16 ft. a minute.[17] How can one
+decide whether such a force is able to pull the moon the actual amount
+required? To Newton this would seem only like a sum in arithmetic. Out
+with a pencil and paper and reckon how much the moon falls toward the
+earth in every second of its motion. Is it 16/3600? That is what it
+ought to be: but is it? The size of the earth comes into the
+calculation. Sixty miles make a degree, 360 degrees a circumference.
+This gives as the earth's diameter 6,873 miles; work it out.
+
+The answer is not 16 feet a minute, it is 13.9 feet.
+
+Surely a mistake of calculation?
+
+No, it is no mistake: there is something wrong in the theory, gravity is
+too strong.
+
+Instead of falling toward the earth 5-1/3 hundredths of an inch every
+second, as it would under gravity, the moon only falls 4-2/3 hundredths
+of an inch per second.
+
+With such a discovery in his grasp at the age of twenty-three he is
+disappointed--the figures do not agree, and he cannot make them agree.
+Either gravity is not the force in action, or else something interferes
+with it. Possibly, gravity does part of the work, and the vortices of
+Descartes interfere with it.
+
+He must abandon the fascinating idea for the time. In his own words, "he
+laid aside at that time any further thought of the matter."
+
+So far as is known, he never mentioned his disappointment to a soul. He
+might, perhaps, if he had been at Cambridge, but he was a shy and
+solitary youth, and just as likely he might not. Up in Lincolnshire, in
+the seventeenth century, who was there for him to consult?
+
+True, he might have rushed into premature publication, after our
+nineteenth century fashion, but that was not his method. Publication
+never seemed to have occurred to him.
+
+His reticence now is noteworthy, but later on it is perfectly
+astonishing. He is so absorbed in making discoveries that he actually
+has to be reminded to tell any one about them, and some one else always
+has to see to the printing and publishing for him.
+
+I have entered thus fully into what I conjecture to be the stages of
+this early discovery of the law of gravitation, as applicable to the
+heavenly bodies, because it is frequently and commonly misunderstood. It
+is sometimes thought that he discovered the force of gravity; I hope I
+have made it clear that he did no such thing. Every educated man long
+before his time, if asked why bodies fell, would reply just as glibly as
+they do now, "Because the earth attracts them," or "because of the force
+of gravity."
+
+His discovery was that the motions of the solar system were due to the
+action of a central force, directed to the body at the centre of the
+system, and varying inversely with the square of the distance from it.
+This discovery was based upon Kepler's laws, and was clear and certain.
+It might have been published had he so chosen.
+
+But he did not like hypothetical and unknown forces; he tried to see
+whether the known force of gravity would serve. This discovery at that
+time he failed to make, owing to a wrong numerical datum. The size of
+the earth he only knew from the common doctrine of sailors that 60 miles
+make a degree; and that threw him out. Instead of falling 16 feet a
+minute, as it ought under gravity, it only fell 13.9 feet, so he
+abandoned the idea. We do not find that he returned to it for sixteen
+years.
+
+
+
+
+LECTURE VIII
+
+NEWTON AND THE LAW OF GRAVITATION
+
+
+We left Newton at the age of twenty-three on the verge of discovering
+the mechanism of the solar system, deterred therefrom only by an error
+in the then imagined size of the earth. He had proved from Kepler's laws
+that a centripetal force directed to the sun, and varying as the inverse
+square of the distance from that body, would account for the observed
+planetary motions, and that a similar force directed to the earth would
+account for the lunar motion; and it had struck him that this force
+might be the very same as the familiar force of gravitation which gave
+to bodies their weight: but in attempting a numerical verification of
+this idea in the case of the moon he was led by the then received notion
+that sixty miles made a degree on the earth's surface into an erroneous
+estimate of the size of the moon's orbit. Being thus baffled in
+obtaining such verification, he laid the matter aside for a time.
+
+The anecdote of the apple we learn from Voltaire, who had it from
+Newton's favourite niece, who with her husband lived and kept house for
+him all his later life. It is very like one of those anecdotes which are
+easily invented and believed in, and very often turn out on scrutiny to
+have no foundation. Fortunately this anecdote is well authenticated, and
+moreover is intrinsically probable; I say fortunately, because it is
+always painful to have to give up these child-learnt anecdotes, like
+Alfred and the cakes and so on. This anecdote of the apple we need not
+resign. The tree was blown down in 1820 and part of its wood is
+preserved.
+
+I have mentioned Voltaire in connection with Newton's philosophy. This
+acute critic at a later stage did a good deal to popularise it
+throughout Europe and to overturn that of his own countryman Descartes.
+Cambridge rapidly became Newtonian, but Oxford remained Cartesian for
+fifty years or more. It is curious what little hold science and
+mathematics have ever secured in the older and more ecclesiastical
+University. The pride of possessing Newton has however no doubt been the
+main stimulus to the special pursuits of Cambridge.
+
+He now began to turn his attention to optics, and, as was usual with
+him, his whole mind became absorbed in this subject as if nothing else
+had ever occupied him. His cash-book for this time has been discovered,
+and the entries show that he is buying prisms and lenses and polishing
+powder at the beginning of 1667. He was anxious to improve telescopes by
+making more perfect lenses than had ever been used before. Accordingly
+he calculated out their proper curves, just as Descartes had also done,
+and then proceeded to grind them as near as he could to those figures.
+But the images did not please him; they were always blurred and rather
+indistinct.
+
+At length, it struck him that perhaps it was not the lenses but the
+light which was at fault. Perhaps light was so composed that it _could_
+not be focused accurately to a sharp and definite point. Perhaps the law
+of refraction was not quite accurate, but only an approximation. So he
+bought a prism to try the law. He let in sunlight through a small round
+hole in a window shutter, inserted the prism in the light, and received
+the deflected beam on a white screen; turning the prism about till it
+was deviated as little as possible. The patch on the screen was not a
+round disk, as it would have been without the prism, but was an
+elongated oval and was coloured at its extremities. Evidently
+refraction was not a simple geometrical deflection of a ray, there was a
+spreading out as well.
+
+[Illustration: FIG. 63.--A prism not only _deviates_ a beam of sunlight,
+but also spreads it out or _disperses_ it.]
+
+Why did the image thus spread out? If it were due to irregularities in
+the glass a second prism should rather increase them, but a second prism
+when held in appropriate position was able to neutralise the dispersion
+and to reproduce the simple round white spot without deviation.
+Evidently the spreading out of the beam was connected in some definite
+way with its refraction. Could it be that the light particles after
+passing through the prism travelled in variously curved lines, as
+spinning racquet balls do? To examine this he measured the length of the
+oval patch when the screen was at different distances from the prism,
+and found that the two things were directly proportional to each other.
+Doubling the distance of the screen doubled the length of the patch.
+Hence the rays travelled in straight lines from the prism, and the
+spreading out was due to something that occurred within its substance.
+Could it be that white light was compound, was a mixture of several
+constituents, and that its different constituents were differently bent?
+No sooner thought than tried. Pierce the screen to let one of the
+constituents through and interpose a second prism in its path. If the
+spreading out depended on the prism only it should spread out just as
+much as before, but if it depended on the complex character of white
+light, this isolated simple constituent should be able to spread out no
+more. It did not spread out any more: a prism had no more dispersive
+power over it; it was deflected by the appropriate amount, but it was
+not analysed into constituents. It differed from sunlight in being
+simple. With many ingenious and beautifully simple experiments, which
+are quoted in full in several books on optics, he clinched the argument
+and established his discovery. White light was not simple but compound.
+It could be sorted out by a prism into an infinite number of constituent
+parts which were differently refracted, and the most striking of which
+Newton named violet, indigo, blue, green, yellow, orange, and red.
+
+[Illustration: FIG. 64.--A single constituent of white light, obtained
+by the use of perforated screens is capable of no more dispersion.]
+
+At once the true nature of colour became manifest. Colour resided not in
+the coloured object as had till now been thought, but in the light which
+illuminated it. Red glass for instance adds nothing to sunlight. The
+light does not get dyed red by passing through the glass; all that the
+red glass does is to stop and absorb a large part of the sunlight; it is
+opaque to the larger portion, but it is transparent to that particular
+portion which affects our eyes with the sensation of red. The prism acts
+like a sieve sorting out the different kinds of light. Coloured media
+act like filters, stopping certain kinds but allowing the rest to go
+through. Leonardo's and all the ancient doctrines of colour had been
+singularly wrong; colour is not in the object but in the light.
+
+Goethe, in his _Farbenlehre_, endeavoured to controvert Newton, and to
+reinstate something more like the old views; but his failure was
+complete.
+
+Refraction analysed out the various constituents of white light and
+displayed them in the form of a series of overlapping images of the
+aperture, each of a different colour; this series of images we call a
+spectrum, and the operation we now call spectrum analysis. The reason of
+the defect of lenses was now plain: it was not so much a defect of the
+lens as a defect of light. A lens acts by refraction and brings rays to
+a focus. If light be simple it acts well, but if ordinary white light
+fall upon a lens, its different constituents have different foci; every
+bright object is fringed with colour, and nothing like a clear image can
+be obtained.
+
+[Illustration: FIG. 65.--Showing the boundary rays of a parallel beam
+passing through a lens.]
+
+A parallel beam passing through a lens becomes conical; but instead of a
+single cone it is a sheaf or nest of cones, all having the edge of the
+lens as base, but each having a different vertex. The violet cone is
+innermost, near the lens, the red cone outermost, while the others lie
+between. Beyond the crossing point or focus the order of cones is
+reversed, as the above figure shows. Only the two marginal rays of the
+beam are depicted.
+
+If a screen be held anywhere nearer the lens than the place marked 1
+there will be a whitish centre to the patch of light and a red and
+orange fringe or border. Held anywhere beyond the region 2, the border
+of the patch will be blue and violet. Held about 3 the colour will be
+less marked than elsewhere, but nowhere can it be got rid of. Each point
+of an object will be represented in the image not by a point but by a
+coloured patch: a fact which amply explains the observed blurring and
+indistinctness.
+
+Newton measured and calculated the distance between the violet and red
+foci--VR in the diagram--and showed that it was 1/50th the diameter of
+the lens. To overcome this difficulty (called chromatic aberration)
+telescope glasses were made small and of very long focus: some of them
+so long that they had no tube, all of them egregiously cumbrous. Yet it
+was with such instruments that all the early discoveries were made. With
+such an instrument, for instance, Huyghens discovered the real shape of
+Saturn's ring.
+
+The defects of refractors seemed irremediable, being founded in the
+nature of light itself. So he gave up his "glass works"; and proceeded
+to think of reflexion from metal specula. A concave mirror forms an
+image just as a lens does, but since it does so without refraction or
+transmission through any substance, there is no accompanying dispersion
+or chromatic aberration.
+
+The first reflecting telescope he made was 1 in. diameter and 6 in.
+long, and magnified forty times. It acted as well as a three or four
+feet refractor of that day, and showed Jupiter's moons. So he made a
+larger one, now in the library of the Royal Society, London, with an
+inscription:
+
+"The first reflecting telescope, invented by Sir Isaac Newton, and made
+with his own hands."
+
+This has been the parent of most of the gigantic telescopes of the
+present day. Fifty years elapsed before it was much improved on, and
+then, first by Hadley and afterwards by Herschel and others, large and
+good reflectors were constructed.
+
+The largest telescope ever made, that of Lord Rosse, is a Newtonian
+reflector, fifty feet long, six feet diameter, with a mirror weighing
+four tons. The sextant, as used by navigators, was also invented by
+Newton.
+
+The year after the plague, in 1667, Newton returned to Trinity College,
+and there continued his experiments on optics. It is specially to be
+noted that at this time, at the age of twenty-four, Newton had laid the
+foundations of all his greatest discoveries:--
+
+[Illustration: FIG. 66.--Newton's telescope.]
+
+The Theory of Fluxions; or, the Differential Calculus.
+
+The Law of Gravitation; or, the complete theory of astronomy.
+
+The compound nature of white light; or, the beginning of Spectrum
+Analysis.
+
+[Illustration: FIG. 67.--The sextant, as now made.]
+
+His later life was to be occupied in working these incipient discoveries
+out. But the most remarkable thing is that no one knew about any one of
+them. However, he was known as an accomplished young mathematician, and
+was made a fellow of his college. You remember that he had a friend
+there in the person of Dr. Isaac Barrow, first Lucasian Professor of
+Mathematics in the University. It happened, about 1669, that a
+mathematical discovery of some interest was being much discussed, and
+Dr. Barrow happened to mention it to Newton, who said yes, he had worked
+out that and a few other similar things some time ago. He accordingly
+went and fetched some papers to Dr. Barrow, who forwarded them to other
+distinguished mathematicians, and it thus appeared that Newton had
+discovered theorems much more general than this special case that was
+exciting so much interest. Dr. Barrow, being anxious to devote his time
+more particularly to theology, resigned his chair the same year in
+favour of Newton, who was accordingly elected to the Lucasian
+Professorship, which he held for thirty years. This chair is now the
+most famous in the University, and it is commonly referred to as the
+chair of Newton.
+
+Still, however, his method of fluxions was unknown, and still he did not
+publish it. He lectured first on optics, giving an account of his
+experiments. His lectures were afterwards published both in Latin and
+English, and are highly valued to this day.
+
+The fame of his mathematical genius came to the ears of the Royal
+Society, and a motion was made to get him elected a fellow of that body.
+The Royal Society, the oldest and most famous of all scientific
+societies with a continuous existence, took its origin in some private
+meetings, got up in London by the Hon. Robert Boyle and a few scientific
+friends, during all the trouble of the Commonwealth.
+
+After the restoration, Charles II. in 1662 incorporated it under Royal
+Charter; among the original members being Boyle, Hooke, Christopher
+Wren, and other less famous names. Boyle was a great experimenter, a
+worthy follower of Dr. Gilbert. Hooke began as his assistant, but being
+of a most extraordinary ingenuity he rapidly rose so as to exceed his
+master in importance. Fate has been a little unkind to Hooke in placing
+him so near to Newton; had he lived in an ordinary age he would
+undoubtedly have shone as a star of the first magnitude. With great
+ingenuity, remarkable scientific insight, and consummate experimental
+skill, he stands in many respects almost on a level with Galileo. But it
+is difficult to see stars even of the first magnitude when the sun is
+up, and thus it happens that the name and fame of this brilliant man are
+almost lost in the blaze of Newton. Of Christopher Wren I need not say
+much. He is well known as an architect, but he was a most accomplished
+all-round man, and had a considerable taste and faculty for science.
+
+These then were the luminaries of the Royal Society at the time we are
+speaking of, and to them Newton's first scientific publication was
+submitted. He communicated to them an account of his reflecting
+telescope, and presented them with the instrument.
+
+Their reception of it surprised him; they were greatly delighted with
+it, and wrote specially thanking him for the communication, and assuring
+him that all right should be done him in the matter of the invention.
+The Bishop of Salisbury (Bishop Burnet) proposed him for election as a
+fellow, and elected he was.
+
+In reply, he expressed his surprise at the value they set on the
+telescope, and offered, if they cared for it, to send them an account of
+a discovery which he doubts not will prove much more grateful than the
+communication of that instrument, "being in my judgment the oddest, if
+not the most considerable detection that has recently been made into the
+operations of Nature."
+
+So he tells them about his optical researches and his discovery of the
+nature of white light, writing them a series of papers which were long
+afterwards incorporated and published as his _Optics_. A magnificent
+work, which of itself suffices to place its author in the first rank of
+the world's men of science.
+
+The nature of white light, the true doctrine of colour, and the
+differential calculus! besides a good number of minor results--binomial
+theorem, reflecting telescope, sextant, and the like; one would think it
+enough for one man's life-work, but the masterpiece remains still to be
+mentioned. It is as when one is considering Shakspeare: _King Lear_,
+_Macbeth_, _Othello_,--surely a sufficient achievement,--but the
+masterpiece remains.
+
+Comparisons in different departments are but little help perhaps,
+nevertheless it seems to me that in his own department, and considered
+simply as a man of science, Newton towers head and shoulders over, not
+only his contemporaries--that is a small matter--but over every other
+scientific man who has ever lived, in a way that we can find no parallel
+for in other departments. Other nations admit his scientific
+pre-eminence with as much alacrity as we do.
+
+Well, we have arrived at the year 1672 and his election to the Royal
+Society. During the first year of his membership there was read at one
+of the meetings a paper giving an account of a very careful
+determination of the length of a degree (_i.e._ of the size of the
+earth), which had been made by Picard near Paris. The length of the
+degree turned out to be not sixty miles, but nearly seventy miles. How
+soon Newton heard of this we do not learn--probably not for some
+years,--Cambridge was not so near London then as it is now, but
+ultimately it was brought to his notice. Armed with this new datum, his
+old speculation concerning gravity occurred to him. He had worked out
+the mechanics of the solar system on a certain hypothesis, but it had
+remained a hypothesis somewhat out of harmony with apparent fact. What
+if it should turn out to be true after all!
+
+He took out his old papers and began again the calculation. If gravity
+were the force keeping the moon in its orbit, it would fall toward the
+earth sixteen feet every minute. How far did it fall? The newly known
+size of the earth would modify the figures: with intense excitement he
+runs through the working, his mind leaps before his hand, and as he
+perceives the answer to be coming out right, all the infinite meaning
+and scope of his mighty discovery flashes upon him, and he can no longer
+see the paper. He throws down the pen; and the secret of the universe
+is, to one man, known.
+
+But of course it had to be worked out. The meaning might flash upon him,
+but its full detail required years of elaboration; and deeper and deeper
+consequences revealed themselves to him as he proceeded.
+
+For two years he devoted himself solely to this one object. During
+those years he lived but to calculate and think, and the most ludicrous
+stories are told concerning his entire absorption and inattention to
+ordinary affairs of life. Thus, for instance, when getting up in a
+morning he would sit on the side of the bed half-dressed, and remain
+like that till dinner time. Often he would stay at home for days
+together, eating what was taken to him, but without apparently noticing
+what he was doing.
+
+One day an intimate friend, Dr. Stukely, called on him and found on the
+table a cover laid for his solitary dinner. After waiting a long time,
+Dr. Stukely removed the cover and ate the chicken underneath it,
+replacing and covering up the bones again. At length Newton appeared,
+and after greeting his friend, sat down to dinner, but on lifting the
+cover he said in surprise, "Dear me, I thought I had not dined, but I
+see I have."
+
+It was by this continuous application that the _Principia_ was
+accomplished. Probably nothing of the first magnitude can be
+accomplished without something of the same absorbed unconsciousness and
+freedom from interruption. But though desirable and essential for the
+_work_, it was a severe tax upon the powers of the _man_. There is, in
+fact, no doubt that Newton's brain suffered temporary aberration after
+this effort for a short time. The attack was slight, and it has been
+denied; but there are letters extant which are inexplicable otherwise,
+and moreover after a year or two he writes to his friends apologizing
+for strange and disjointed epistles, which he believed he had written
+without understanding clearly what he wrote. The derangement was,
+however, both slight and temporary: and it is only instructive to us as
+showing at what cost such a work as the _Principia_ must be produced,
+even by so mighty a mind as that of Newton.
+
+The first part of the work having been done, any ordinary mortal would
+have proceeded to publish it; but the fact is that after he had sent to
+the Royal Society his papers on optics, there had arisen controversies
+and objections; most of them rather paltry, to which he felt compelled
+to find answers. Many men would have enjoyed this part of the work, and
+taken it as evidence of interest and success. But to Newton's shy and
+retiring disposition these discussions were merely painful. He writes,
+indeed, his answers with great patience and ability, and ultimately
+converts the more reasonable of his opponents, but he relieves his mind
+in the following letter to the secretary of the Royal Society: "I see I
+have made myself a slave to philosophy, but if I get free of this
+present business I will resolutely bid adieu to it eternally, except
+what I do for my private satisfaction or leave to come out after me; for
+I see a man must either resolve to put out nothing new, or to become a
+slave to defend it." And again in a letter to Leibnitz: "I have been so
+persecuted with discussions arising out of my theory of light that I
+blamed my own imprudence for parting with so substantial a blessing as
+my quiet to run after a shadow." This shows how much he cared for
+contemporary fame.
+
+So he locked up the first part of the _Principia_ in his desk, doubtless
+intending it to be published after his death. But fortunately this was
+not so to be.
+
+In 1683, among the leading lights of the Royal Society, the same sort of
+notions about gravity and the solar system began independently to be
+bruited. The theory of gravitation seemed to be in the air, and Wren,
+Hooke, and Halley had many a talk about it.
+
+Hooke showed an experiment with a pendulum, which he likened to a planet
+going round the sun. The analogy is more superficial than real. It does
+not obey Kepler's laws; still it was a striking experiment. They had
+guessed at a law of inverse squares, and their difficulty was to prove
+what curve a body subject to it would describe. They knew it ought to be
+an ellipse if it was to serve to explain the planetary motion, and Hooke
+said he could prove that an ellipse it was; but he was nothing of a
+mathematician, and the others scarcely believed him. Undoubtedly he had
+shrewd inklings of the truth, though his guesses were based on little
+else than a most sagacious intuition. He surmised also that gravity was
+the force concerned, and asserted that the path of an ordinary
+projectile was an ellipse, like the path of a planet--which is quite
+right. In fact the beginnings of the discovery were beginning to dawn
+upon him in the well-known way in which things do dawn upon ordinary men
+of genius: and had Newton not lived we should doubtless, by the labours
+of a long chain of distinguished men, beginning with Hooke, Wren, and
+Halley, have been now in possession of all the truths revealed by the
+_Principia_. We should never have had them stated in the same form, nor
+proved with the same marvellous lucidity and simplicity, but the facts
+themselves we should by this time have arrived at. Their developments
+and completions, due to such men as Clairaut, Euler, D'Alembert,
+Lagrange, Laplace, Airy, Leverrier, Adams, we should of course not have
+had to the same extent; because the lives and energies of these great
+men would have been partially consumed in obtaining the main facts
+themselves.
+
+The youngest of the three questioners at the time we are speaking of was
+Edmund Halley, an able and remarkable man. He had been at Cambridge,
+doubtless had heard Newton lecture, and had acquired a great veneration
+for him.
+
+In January, 1684, we find Wren offering Hooke and Halley a prize, in the
+shape of a book worth forty shillings, if they would either of them
+bring him within two months a demonstration that the path of a planet
+subject to an inverse square law would be an ellipse. Not in two months,
+nor yet in seven, was there any proof forthcoming. So at last, in
+August, Halley went over to Cambridge to speak to Newton about the
+difficult problem and secure his aid. Arriving at his rooms he went
+straight to the point. He said, "What path will a body describe if it
+be attracted by a centre with a force varying as the inverse square of
+the distance." To which Newton at once replied, "An ellipse." "How on
+earth do you know?" said Halley in amazement. "Why, I have calculated
+it," and began hunting about for the paper. He actually couldn't find it
+just then, but sent it him shortly by post, and with it much more--in
+fact, what appeared to be a complete treatise on motion in general.
+
+With his valuable burden Halley hastened to the Royal Society and told
+them what he had discovered. The Society at his representation wrote to
+Mr. Newton asking leave that it might be printed. To this he consented;
+but the Royal Society wisely appointed Mr. Halley to see after him and
+jog his memory, in case he forgot about it. However, he set to work to
+polish it up and finish it, and added to it a great number of later
+developments and embellishments, especially the part concerning the
+lunar theory, which gave him a deal of trouble--and no wonder; for in
+the way he has put it there never was a man yet living who could have
+done the same thing. Mathematicians regard the achievement now as men
+might stare at the work of some demigod of a bygone age, wondering what
+manner of man this was, able to wield such ponderous implements with
+such apparent ease.
+
+To Halley the world owes a great debt of gratitude--first, for
+discovering the _Principia_; second, for seeing it through the press;
+and third, for defraying the cost of its publication out of his own
+scanty purse. For though he ultimately suffered no pecuniary loss,
+rather the contrary, yet there was considerable risk in bringing out a
+book which not a dozen men living could at the time comprehend. It is no
+small part of the merit of Halley that he recognized the transcendent
+value of the yet unfinished work, that he brought it to light, and
+assisted in its becoming understood to the best of his ability.
+
+Though Halley afterwards became Astronomer-Royal, lived to the ripe old
+age of eighty-six, and made many striking observations, yet he would be
+the first to admit that nothing he ever did was at all comparable in
+importance with his discovery of the _Principia_; and he always used to
+regard his part in it with peculiar pride and pleasure.
+
+And how was the _Principia_ received? Considering the abstruse nature of
+its subject, it was received with great interest and enthusiasm. In less
+than twenty years the edition was sold out, and copies fetched large
+sums. We hear of poor students copying out the whole in manuscript in
+order to possess a copy--not by any means a bad thing to do, however
+many copies one may possess. The only useful way really to read a book
+like that is to pore over every sentence: it is no book to be skimmed.
+
+While the _Principia_ was preparing for the press a curious incident of
+contact between English history and the University occurred. It seems
+that James II., in his policy of Catholicising the country, ordered both
+Universities to elect certain priests to degrees without the ordinary
+oaths. Oxford had given way, and the Dean of Christ Church was a
+creature of James's choosing. Cambridge rebelled, and sent eight of its
+members, among them Mr. Newton, to plead their cause before the Court of
+High Commission. Judge Jeffreys presided over the Court, and threatened
+and bullied with his usual insolence. The Vice-Chancellor of Cambridge
+was deprived of office, the other deputies were silenced and ordered
+away. From the precincts of this court of justice Newton returned to
+Trinity College to complete the _Principia_.
+
+By this time Newton was only forty-five years old, but his main work was
+done. His method of fluxions was still unpublished; his optics was
+published only imperfectly; a second edition of the _Principia_, with
+additions and improvements, had yet to appear; but fame had now come
+upon him, and with fame worries of all kinds.
+
+By some fatality, principally no doubt because of the interest they
+excited, every discovery he published was the signal for an outburst of
+criticism and sometimes of attack. I shall not go into these matters:
+they are now trivial enough, but it is necessary to mention them,
+because to Newton they evidently loomed large and terrible, and
+occasioned him acute torment.
+
+[Illustration: FIG. 68.--Newton when young. (_From an engraving by B.
+Reading after Sir Peter Lely._)]
+
+No sooner was the _Principia_ put than Hooke put in his claims for
+priority. And indeed his claims were not altogether negligible; for
+vague ideas of the same sort had been floating in his comprehensive
+mind, and he doubtless felt indistinctly conscious of a great deal more
+than he could really state or prove.
+
+By indiscreet friends these two great men were set somewhat at
+loggerheads, and worse might have happened had they not managed to come
+to close quarters, and correspond privately in a quite friendly manner,
+instead of acting through the mischievous medium of third parties. In
+the next edition Newton liberally recognizes the claims of both Hooke
+and Wren. However, he takes warning betimes of what he has to expect,
+and writes to Halley that he will only publish the first two books,
+those containing general theorems on motion. The third book--concerning
+the system of the world, _i.e._ the application to the solar system--he
+says "I now design to suppress. Philosophy is such an impertinently
+litigious lady that a man had as good be engaged in law-suits as have to
+do with her. I found it so formerly, and now I am no sooner come near
+her again but she gives me warning. The two books without the third will
+not so well bear the title 'Mathematical Principles of Natural
+Philosophy,' and therefore I had altered it to this, 'On the Free Motion
+of Two Bodies'; but on second thoughts I retain the former title: 'twill
+help the sale of the book--which I ought not to diminish now 'tis
+yours."
+
+However, fortunately, Halley was able to prevail upon him to publish the
+third book also. It is, indeed, the most interesting and popular of the
+three, as it contains all the direct applications to astronomy of the
+truths established in the other two.
+
+Some years later, when his method of fluxions was published, another and
+a worse controversy arose--this time with Leibnitz, who had also
+independently invented the differential calculus. It was not so well
+recognized then how frequently it happens that two men independently
+and unknowingly work at the very same thing at the same time. The
+history of science is now full of such instances; but then the friends
+of each accused the other of plagiarism.
+
+I will not go into the controversy: it is painful and useless. It only
+served to embitter the later years of two great men, and it continued
+long after Newton's death--long after both their deaths. It can hardly
+be called ancient history even now.
+
+But fame brought other and less unpleasant distractions than
+controversies. We are a curious, practical, and rather stupid people,
+and our one idea of honouring a man is to _vote_ for him in some way or
+other; so they sent Newton to Parliament. He went, I believe, as a Whig,
+but it is not recorded that he spoke. It is, in fact, recorded that he
+was once expected to speak when on a Royal Commission about some
+question of chronometers, but that he would not. However, I dare say he
+made a good average member.
+
+Then a little later it was realized that Newton was poor, that he still
+had to teach for his livelihood, and that though the Crown had continued
+his fellowship to him as Lucasian Professor without the necessity of
+taking orders, yet it was rather disgraceful that he should not be
+better off. So an appeal was made to the Government on his behalf, and
+Lord Halifax, who exerted himself strongly in the matter, succeeding to
+office on the accession of William III., was able to make him ultimately
+Master of the Mint, with a salary of some L1,200 a year. I believe he
+made rather a good Master, and turned out excellent coins: certainly he
+devoted his attention to his work there in a most exemplary manner.
+
+But what a pitiful business it all is! Here is a man sent by Heaven to
+do certain things which no man else could do, and so long as he is
+comparatively unknown he does them; but so soon as he is found out, he
+is clapped into a routine office with a big salary: and there is,
+comparatively speaking, an end of him. It is not to be supposed that he
+had lost his power, for he frequently solved problems very quickly which
+had been given out by great Continental mathematicians as a challenge to
+the world.
+
+We may ask why Newton allowed himself to be thus bandied about instead
+of settling himself down to the work in which he was so pre-eminently
+great. Well, I expect your truly great man never realizes how great he
+is, and seldom knows where his real strength lies. Certainly Newton did
+not know it. He several times talks of giving up philosophy altogether;
+and though he never really does it, and perhaps the feeling is one only
+born of some temporary overwork, yet he does not sacrifice everything
+else to it as he surely must had he been conscious of his own greatness.
+No; self-consciousness was the last thing that affected him. It is for a
+great man's contemporaries to discover him, to make much of him, and to
+put him in surroundings where he may flourish luxuriantly in his own
+heaven-intended way.
+
+However, it is difficult for us to judge of these things. Perhaps if he
+had been maintained at the national expense to do that for which he was
+preternaturally fitted, he might have worn himself out prematurely;
+whereas by giving him routine work the scientific world got the benefit
+of his matured wisdom and experience. It was no small matter to the
+young Royal Society to be able to have him as their President for
+twenty-four years. His portrait has hung over the President's chair ever
+since, and there I suppose it will continue to hang until the Royal
+Society becomes extinct.
+
+The events of his later life I shall pass over lightly. He lived a calm,
+benevolent life, universally respected and beloved. His silver-white
+hair when he removed his peruke was a venerable spectacle. A lock of it
+is still preserved, with many other relics, in the library of Trinity
+College. He died quietly, after a painful illness, at the ripe age of
+eighty-five. His body lay in state in the Jerusalem Chamber, and he was
+buried in Westminster Abbey, six peers bearing the pall. These things
+are to be mentioned to the credit of the time and the country; for
+after we have seen the calamitous spectacle of the way Tycho and Kepler
+and Galileo were treated by their ungrateful and unworthy countries, it
+is pleasant to reflect that England, with all its mistakes, yet
+recognized _her_ great man when she received him, and honoured him with
+the best she knew how to give.
+
+[Illustration: FIG. 69.--Sir Isaac Newton.]
+
+Concerning his character, one need only say that it was what one would
+expect and wish. It was characterized by a modest, calm, dignified
+simplicity. He lived frugally with his niece and her husband, Mr.
+Conduit, who succeeded him as Master of the Mint. He never married, nor
+apparently did he ever think of so doing. The idea, perhaps, did not
+naturally occur to him, any more than the idea of publishing his work
+did.
+
+He was always a deeply religious man and a sincere Christian, though
+somewhat of the Arian or Unitarian persuasion--so, at least, it is
+asserted by orthodox divines who understand these matters. He studied
+theology more or less all his life, and towards the end was greatly
+interested in questions of Biblical criticism and chronology. By some
+ancient eclipse or other he altered the recognized system of dates a few
+hundred years; and his book on the prophecies of Daniel and the
+Revelation of St. John, wherein he identifies the beast with the Church
+of Rome in quite the orthodox way, is still by some admired.
+
+But in all these matters it is probable that he was a merely ordinary
+man, with natural acumen and ability doubtless, but nothing in the least
+superhuman. In science, the impression he makes upon me is only
+expressible by the words inspired, superhuman.
+
+And yet if one realizes his method of work, and the calm, uninterrupted
+flow of all his earlier life, perhaps his achievements become more
+intelligible. When asked how he made his discoveries, he replied: "By
+always thinking unto them. I keep the subject constantly before me, and
+wait till the first dawnings open slowly by little and little into a
+full and clear light." That is the way--quiet, steady, continuous
+thinking, uninterrupted and unharassed brooding. Much may be done under
+those conditions. Much ought to be sacrificed to obtain those
+conditions. All the best thinking work of the world has been thus
+done.[18] Buffon said: "Genius is patience." So says Newton: "If I have
+done the public any service this way, it is due to nothing but industry
+and patient thought." Genius patience? No, it is not quite that, or,
+rather, it is much more than that; but genius without patience is like
+fire without fuel--it will soon burn itself out.
+
+
+
+
+NOTES FOR LECTURE IX
+
+  The _Principia_ published 1687.
+  Newton died               1727.
+
+
+THE LAW OF GRAVITATION.--Every particle of matter attracts every other
+particle of matter with a force proportional to the mass of each and to
+the inverse square of the distance between them.
+
+
+SOME OF NEWTON'S DEDUCTIONS.
+
+1. Kepler's second law (equable description of areas) proves that each
+planet is acted on by a force directed towards the sun as a centre of
+force.
+
+2. Kepler's first law proves that this central force diminishes in the
+same proportion as the square of the distance increases.
+
+3. Kepler's third law proves that all the planets are acted on by the
+same kind of force; of an intensity depending on the mass of the
+sun.[19]
+
+4. So by knowing the length of year and distance of any planet from the
+sun, the sun's mass can be calculated, in terms of that of the earth.
+
+5. For the satellites, the force acting depends on the mass of _their_
+central body, a planet. Hence the mass of any planet possessing a
+satellite becomes known.
+
+6. The force constraining the moon in her orbit is the same gravity as
+gives terrestrial bodies their weight and regulates the motion of
+projectiles. [Because, while a stone drops 16 feet in a second, the
+moon, which is 60 times as far from the centre of the earth, drops 16
+feet in a minute.]
+
+*      *      *      *      *
+
+7. The moon is attracted not only by the earth, but by the sun also;
+hence its orbit is perturbed, and Newton calculated out the chief of
+these perturbations, viz.:--
+
+     (The equation of the centre, discovered by Hipparchus.)
+
+     (_a_) The evection, discovered by Hipparchus and Ptolemy.
+
+     (_b_) The variation, discovered by Tycho Brahe.
+
+     (_c_) The annual equation, discovered by Tycho Brahe.
+
+     (_d_) The retrogression of the nodes, then being observed at
+     Greenwich by Flamsteed.
+
+     (_e_) The variation of inclination, then being observed at
+     Greenwich by Flamsteed.
+
+     (_f_) The progression of the apses (with an error of one-half).
+
+     (_g_) The inequality of apogee, previously unknown.
+
+     (_h_) The inequality of nodes, previously unknown.
+
+8. Each planet is attracted not only by the sun but by the other
+planets, hence their orbits are slightly affected by each other. Newton
+began the theory of planetary perturbations.
+
+9. He recognized the comets as members of the solar system, obedient to
+the same law of gravity and moving in very elongated ellipses; so their
+return could be predicted (_e.g._ Halley's comet).
+
+10. Applying the idea of centrifugal force to the earth considered as a
+rotating body, he perceived that it could not be a true sphere, and
+calculated its oblateness, obtaining 28 miles greater equatorial than
+polar diameter.
+
+11. Conversely, from the observed shape of Jupiter, or any planet, the
+length of its day could be estimated.
+
+12. The so-calculated shape of the earth, in combination with
+centrifugal force, causes the weight of bodies to vary with latitude;
+and Newton calculated the amount of this variation. 194 lbs. at pole
+balance 195 lbs. at equator.
+
+13. A homogeneous sphere attracts as if its mass were concentrated at
+its centre. For any other figure, such as an oblate spheroid, this is
+not exactly true. A hollow concentric spherical shell exerts no force on
+small bodies inside it.
+
+14. The earth's equatorial protuberance, being acted on by the
+attraction of the sun and moon, must disturb its axis of rotation in a
+calculated manner; and thus is produced the precession of the equinoxes.
+[The attraction of the planets on the same protuberance causes a smaller
+and rather different kind of precession.]
+
+15. The waters of the ocean are attracted towards the sun and moon on
+one side, and whirled a little further away than the solid earth on the
+other side: hence Newton explained all the main phenomena of the tides.
+
+16. The sun's mass being known, he calculated the height of the solar
+tide.
+
+17. From the observed heights of spring and neap tides he determined the
+lunar tide, and thence made an estimate of the mass of the moon.
+
+REFERENCE TABLE OF NUMERICAL DATA.
+
+ +---------+---------------+----------------------+-----------------+
+ |         |Masses in Solar| Height dropped by a  | Length of Day or|
+ |         |    System.    |stone in first second.|time of rotation.|
+ +---------+---------------+----------------------+-----------------+
+ |Mercury  |         .065  |       7.0 feet       |   24    hours   |
+ |Venus    |         .885  |      15.8  "         |   23-1/2  "     |
+ |Earth    |        1.000  |      16.1  "         |   24      "     |
+ |Mars     |         .108  |       6.2  "         |   24-1/2  "     |
+ |Jupiter  |      300.8    |      45.0  "         |   10      "     |
+ |Saturn   |       89.7    |      18.4  "         |   10-1/2  "     |
+ |The Sun  |   316000.     |     436.0  "         |   608     "     |
+ |The Moon |   about .012  |       3.7  "         |   702     "     |
+ +---------+---------------+----------------------+-----------------+
+
+The mass of the earth, taken above as unity, is 6,000 trillion tons.
+
+_Observatories._--Uraniburg flourished from 1576 to 1597; the
+Observatory of Paris was founded in 1667; Greenwich Observatory in 1675.
+
+_Astronomers-Royal._--Flamsteed, Halley, Bradley, Bliss, Maskelyne,
+Pond, Airy, Christie.
+
+
+
+
+LECTURE IX
+
+NEWTON'S "PRINCIPIA"
+
+
+The law of gravitation, above enunciated, in conjunction with the laws
+of motion rehearsed at the end of the preliminary notes of Lecture VII.,
+now supersedes the laws of Kepler and includes them as special cases.
+The more comprehensive law enables us to criticize Kepler's laws from a
+higher standpoint, to see how far they are exact and how far they are
+only approximations. They are, in fact, not precisely accurate, but the
+reason for every discrepancy now becomes abundantly clear, and can be
+worked out by the theory of gravitation.
+
+We may treat Kepler's laws either as immediate consequences of the law
+of gravitation, or as the known facts upon which that law was founded.
+Historically, the latter is the more natural plan, and it is thus that
+they are treated in the first three statements of the above notes; but
+each proposition may be worked inversely, and we might state them
+thus:--
+
+1. The fact that the force acting on each planet is directed to the sun,
+necessitates the equable description of areas.
+
+2. The fact that the force varies as the inverse square of the distance,
+necessitates motion in an ellipse, or some other conic section, with the
+sun in one focus.
+
+3. The fact that one attracting body acts on all the planets with an
+inverse square law, causes the cubes of their mean distances to be
+proportional to the squares of their periodic times.
+
+Not only these but a multitude of other deductions follow rigorously
+from the simple datum that every particle of matter attracts every other
+particle with a force directly proportional to the mass of each and to
+the inverse square of their mutual distance. Those dealt with in the
+_Principia_ are summarized above, and it will be convenient to run over
+them in order, with the object of giving some idea of the general
+meaning of each, without attempting anything too intricate to be readily
+intelligible.
+
+[Illustration: FIG. 70.]
+
+No. 1. Kepler's second law (equable description of areas) proves that
+each planet is acted on by a force directed towards the sun as a centre
+of force.
+
+The equable description of areas about a centre of force has already
+been fully, though briefly, established. (p. 175.) It is undoubtedly of
+fundamental importance, and is the earliest instance of the serious
+discussion of central forces, _i.e._ of forces directed always to a
+fixed centre.
+
+We may put it afresh thus:--OA has been the motion of a particle in a
+unit of time; at A it receives a knock towards C, whereby in the next
+unit it travels along AD instead of AB. Now the area of the triangle
+CAD, swept out by the radius vector in unit time, is 1/2_bh_; _h_ being
+the perpendicular height of the triangle from the base AC. (Fig. 70.)
+Now the blow at A, being along the base, has no effect upon _h_; and
+consequently the area remains just what it would have been without the
+blow. A blow directed to any point other than C would at once alter the
+area of the triangle.
+
+One interesting deduction may at once be drawn. If gravity were a
+radiant force emitted from the sun with a velocity like that of light,
+the moving planet would encounter it at a certain apparent angle
+(aberration), and the force experienced would come from a point a little
+in advance of the sun. The rate of description of areas would thus tend
+to increase; whereas in reality it is constant. Hence the force of
+gravity, if it travel at all, does so with a speed far greater than that
+of light. It appears to be practically instantaneous. (Cf. "Modern Views
+of Electricity," Sec. 126, end of chap. xii.) Again, anything like a
+retarding effect of the medium through which the planets move would
+constitute a tangential force, entirely un-directed towards the sun.
+Hence no such frictional or retarding force can appreciably exist. It
+is, however, conceivable that both these effects might occur and just
+neutralize each other. The neutralization is unlikely to be exact for
+all the planets; and the fact is, that no trace of either effect has as
+yet been discovered. (See also p. 176.)
+
+The planets are, however, subject to forces not directed towards the
+sun, viz. their attractions for each other; and these perturbing forces
+do produce a slight discrepancy from Kepler's second law, but a
+discrepancy which is completely subject to calculation.
+
+No. 2. Kepler's first law proves that this central force diminishes in
+the same proportion as the square of the distance increases.
+
+To prove the connection between the inverse-square law of distance, and
+the travelling in a conic section with the centre of force in one focus
+(the other focus being empty), is not so simple. It obviously involves
+some geometry, and must therefore be left to properly armed students.
+But it may be useful to state that the inverse-square law of distance,
+although the simplest possible law for force emanating from a point or
+sphere, is not to be regarded as self-evident or as needing no
+demonstration. The force of a magnetic pole on a magnetized steel scrap,
+for instance, varies as the inverse cube of the distance; and the curve
+described by such a particle would be quite different from a conic
+section--it would be a definite class of spiral (called Cotes's spiral).
+Again, on an iron filing the force of a single pole might vary more
+nearly as the inverse fifth power; and so on. Even when the thing
+concerned is radiant in straight lines, like light, the law of inverse
+squares is not universally true. Its truth assumes, first, that the
+source is a point or sphere; next, that there is no reflection or
+refraction of any kind; and lastly, that the medium is perfectly
+transparent. The law of inverse squares by no means holds from a prairie
+fire for instance, or from a lighthouse, or from a street lamp in a fog.
+
+Mutual perturbations, especially the pull of Jupiter, prevent the path
+of a planet from being really and truly an ellipse, or indeed from being
+any simple re-entrant curve. Moreover, when a planet possesses a
+satellite, it is not the centre of the planet which ever attempts to
+describe the Keplerian ellipse, but it is the common centre of gravity
+of the two bodies. Thus, in the case of the earth and moon, the point
+which really does describe a close attempt at an ellipse is a point
+displaced about 3000 miles from the centre of the earth towards the
+moon, and is therefore only 1000 miles beneath the surface.
+
+No. 3. Kepler's third law proves that all the planets are acted on by
+the same kind of force; of an intensity depending on the mass of the
+sun.
+
+The third law of Kepler, although it requires geometry to state and
+establish it for elliptic motion (for which it holds just as well as it
+does for circular motion), is very easy to establish for circular
+motion, by any one who knows about centrifugal force. If _m_ is the mass
+of a planet, _v_ its velocity, _r_ the radius of its orbit, and _T_ the
+time of describing it; 2[pi]_r_ = _vT_, and the centripetal force
+needed to hold it in its orbit is
+
+       mv^2        4[pi]^2_mr_
+     --------  or  -----------
+       _r_             T^2
+
+Now the force of gravitative attraction between the planet and the sun
+is
+
+  _VmS_
+  -----,
+   r^2
+
+where _v_ is a fixed quantity called the gravitation-constant, to be
+determined if possible by experiment once for all. Now, expressing the
+fact that the force of gravitation _is_ the force holding the planet in,
+we write,
+
+  4[pi]^2_mr_    _VmS_
+  ----------- = ---------,
+     T^2          r^2
+
+whence, by the simplest algebra,
+
+   r^3      _VS_
+  ------ = ---------.
+   T^2      4[pi]^2
+
+The mass of the planet has been cancelled out; the mass of the sun
+remains, multiplied by the gravitation-constant, and is seen to be
+proportional to the cube of the distance divided by the square of the
+periodic time: a ratio, which is therefore the same for all planets
+controlled by the sun. Hence, knowing _r_ and _T_ for any single planet,
+the value of _VS_ is known.
+
+No. 4. So by knowing the length of year and distance of any planet from
+the sun, the sun's mass can be calculated, in terms of that of the
+earth.
+
+No. 5. For the satellites, the force acting depends on the mass of
+_their_ central body, a planet. Hence the mass of any planet possessing
+a satellite becomes known.
+
+The same argument holds for any other system controlled by a central
+body--for instance, for the satellites of Jupiter; only instead of _S_
+it will be natural to write _J_, as meaning the mass of Jupiter. Hence,
+knowing _r_ and _T_ for any one satellite of Jupiter, the value of _VJ_
+is known.
+
+Apply the argument also to the case of moon and earth. Knowing the
+distance and time of revolution of our moon, the value of _VE_ is at
+once determined; _E_ being the mass of the earth. Hence, _S_ and _J_,
+and in fact the mass of any central body possessing a visible satellite,
+are now known in terms of _E_, the mass of the earth (or, what is
+practically the same thing, in terms of _V_, the gravitation-constant).
+Observe that so far none of these quantities are known absolutely. Their
+relative values are known, and are tabulated at the end of the Notes
+above, but the finding of their absolute values is another matter, which
+we must defer.
+
+But, it may be asked, if Kepler's third law only gives us the mass of a
+_central_ body, how is the mass of a _satellite_ to be known? Well, it
+is not easy; the mass of no satellite is known with much accuracy. Their
+mutual perturbations give us some data in the case of the satellites of
+Jupiter; but to our own moon this method is of course inapplicable. Our
+moon perturbs at first sight nothing, and accordingly its mass is not
+even yet known with exactness. The mass of comets, again, is quite
+unknown. All that we can be sure of is that they are smaller than a
+certain limit, else they would perturb the planets they pass near.
+Nothing of this sort has ever been detected. They are themselves
+perturbed plentifully, but they perturb nothing; hence we learn that
+their mass is small. The mass of a comet may, indeed, be a few million
+or even billion tons; but that is quite small in astronomy.
+
+But now it may be asked, surely the moon perturbs the earth, swinging it
+round their common centre of gravity, and really describing its own
+orbit about this point instead of about the earth's centre? Yes, that is
+so; and a more precise consideration of Kepler's third law enables us to
+make a fair approximation to the position of this common centre of
+gravity, and thus practically to "weigh the moon," i.e. to compare its
+mass with that of the earth; for their masses will be inversely as their
+respective distances from the common centre of gravity or balancing
+point--on the simple steel-yard principle.
+
+Hitherto we have not troubled ourselves about the precise point about
+which the revolution occurs, but Kepler's third law is not precisely
+accurate unless it is attended to. The bigger the revolving body the
+greater is the discrepancy: and we see in the table preceding Lecture
+III., on page 57, that Jupiter exhibits an error which, though very
+slight, is greater than that of any of the other planets, when the sun
+is considered the fixed centre.
+
+     Let the common centre of gravity of earth and moon be displaced a
+     distance _x_ from the centre of the earth, then the moon's distance
+     from the real centre of revolution is not _r_, but _r-x_; and the
+     equation of centrifugal force to gravitative-attraction is strictly
+
+   4[pi]^2             _VE_
+  --------- (_r-x_) = ------,
+     T^2               r^2
+
+     instead of what is in the text above; and this gives a slightly
+     modified "third law." From this equation, if we have any distinct
+     method of determining _VE_ (and the next section gives such a
+     method), we can calculate _x_ and thus roughly weigh the moon,
+     since
+
+  _r-x_     E
+  ----- = -----,
+   _r_     E+M
+
+     but to get anything like a reasonable result the data must be very
+     precise.
+
+No. 6. The force constraining the moon in her orbit is the same gravity
+as gives terrestrial bodies their weight and regulates the motion of
+projectiles.
+
+Here we come to the Newtonian verification already several times
+mentioned; but because of its importance I will repeat it in other
+words. The hypothesis to be verified is that the force acting on the
+moon is the same kind of force as acts on bodies we can handle and
+weigh, and which gives them their weight. Now the weight of a mass _m_
+is commonly written _mg_, where _g_ is the intensity of terrestrial
+gravity, a thing easily measured; being, indeed, numerically equal to
+twice the distance a stone drops in the first second of free fall. [See
+table p. 205.] Hence, expressing that the weight of a body is due to
+gravity, and remembering that the centre of the earth's attraction is
+distant from us by one earth's radius (R), we can write
+
+         _Vm_E
+  _mg_ = ------,
+          R^2
+
+or
+
+_V_E = gR^2 = 95,522 cubic miles-per-second per second.
+
+But we already know _v_E, in terms of the moon's motion, as
+
+  4[pi]^2r^3
+  -----------
+      T^2
+
+approximately, [more accurately, see preceding note, this quantity is
+_V_(E + M)]; hence we can easily see if the two determinations of this
+quantity agree.[20]
+
+All these deductions are fundamental, and may be considered as the
+foundation of the _Principia_. It was these that flashed upon Newton
+during that moment of excitement when he learned the real size of the
+earth, and discovered his speculations to be true.
+
+The next are elaborations and amplifications of the theory, such as in
+ordinary times are left for subsequent generations of theorists to
+discover and work out.
+
+Newton did not work out these remoter consequences of his theory
+completely by any means: the astronomical and mathematical world has
+been working them out ever since; but he carried the theory a great way,
+and here it is that his marvellous power is most conspicuous.
+
+It is his treatment of No. 7, the perturbations of the moon, that
+perhaps most especially has struck all future mathematicians with
+amazement. No. 7, No. 14, No. 15, these are the most inspired of the
+whole.
+
+No. 7. The moon is attracted not only by the earth, but by the sun also;
+hence its orbit is perturbed, and Newton calculated out the chief of
+these perturbations.
+
+Now running through the perturbations (p. 203) in order:--The first is
+in parenthesis, because it is mere excentricity. It is not a true
+perturbation at all, and more properly belongs to Kepler.
+
+(_a_) The first true perturbation is what Ptolemy called "the evection,"
+the principal part of which is a periodic change in the ellipticity or
+excentricity of the moon's orbit, owing to the pull of the sun. It is a
+complicated matter, and Newton only partially solved it. I shall not
+attempt to give an account of it.
+
+(_b_) The next, "the variation," is a much simpler affair. It is caused
+by the fact that as the moon revolves round the earth it is half the
+time nearer to the sun than the earth is, and so gets pulled more than
+the average, while for the other fortnight it is further from the sun
+than the earth is, and so gets pulled less. For the week during which
+it is changing from a decreasing half to a new moon it is moving in the
+direction of the extra pull, and hence becomes new sooner than would
+have been expected. All next week it is moving against the same extra
+pull, and so arrives at quadrature (half moon) somewhat late. For the
+next fortnight it is in the region of too little pull, the earth gets
+pulled more than it does; the effect of this is to hurry it up for the
+third week, so that the full moon occurs a little early, and to retard
+it for the fourth week, so that the decreasing half moon like the
+increasing half occurs behind time again. Thus each syzygy (as new and
+full are technically called) is too early; each quadrature is too late;
+the maximum hurrying and slackening force being felt at the octants, or
+intermediate 45 deg. points.
+
+(_c_) The "annual equation" is a fluctuation introduced into the other
+perturbations by reason of the varying distance of the disturbing body,
+the sun, at different seasons of the year. Its magnitude plainly depends
+simply on the excentricity of the earth's orbit.
+
+Both these perturbations, (_b_) and (_c_), Newton worked out completely.
+
+(_d_) and (_e_) Next come the retrogression of the nodes and the
+variation of the inclination, which at the time were being observed at
+Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged
+for data that he might complete their theory while he had his mind upon
+it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see
+list at end of notes above], and then Newton would have no difficulty in
+gaining such information as the national Observatory could give.
+
+The "inclination" meant is the angle between the plane of the moon's
+orbit and that of the earth. The plane of the earth's orbit round the
+sun is called the ecliptic; the plane of the moon's orbit round the
+earth is inclined to it at a certain angle, which is slowly changing,
+though in a periodic manner. Imagine a curtain ring bisected by a sheet
+of paper, and tilted to a certain angle; it may be likened to the moon's
+orbit, cutting the plane of the ecliptic. The two points at which the
+plane is cut by the ring are called "nodes"; and these nodes are not
+stationary, but are slowly regressing, _i.e._ travelling in a direction
+opposite to that of the moon itself. Also the angle of tilt is varying
+slowly, oscillating up and down in the course of centuries.
+
+(_f_) The two points in the moon's elliptic orbit where it comes nearest
+to or farthest from the earth, _i.e._ the points at the extremity of the
+long axis of the ellipse, are called separately perigee and apogee, or
+together "the apses." Now the pull of the sun causes the whole orbit to
+slowly revolve in its own plane, and consequently these apses
+"progress," so that the true path is not quite a closed curve, but a
+sort of spiral with elliptic loops.
+
+But here comes in a striking circumstance. Newton states with reference
+to this perturbation that theory only accounts for 1-1/2 deg. per annum,
+whereas observation gives 3 deg., or just twice as much.
+
+This is published in the _Principia_ as a fact, without comment. It was
+for long regarded as a very curious thing, and many great mathematicians
+afterwards tried to find an error in the working. D'Alembert, Clairaut,
+and others attacked the problem, but were led to just the same result.
+It constituted the great outstanding difficulty in the way of accepting
+the theory of gravitation. It was suggested that perhaps the inverse
+square law was only a first approximation; that perhaps a more complete
+expression, such as
+
+   A       B
+  ---- + -----,
+  r^2     r^4
+
+must be given for it; and so on.
+
+Ultimately, Clairaut took into account a whole series of neglected
+terms, and it came out correct; thus verifying the theory.
+
+But the strangest part of this tale is to come. For only a few years
+ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was
+editing various old papers of Newton's, now in the possession of the
+Duke of Portland, and he found manuscripts bearing on this very point,
+and discovered that Newton had reworked out the calculations himself,
+had found the cause of the error, had taken into account the terms
+hitherto neglected, and so, fifty years before Clairaut, had completely,
+though not publicly, solved this long outstanding problem of the
+progression of the apses.
+
+(_g_) and (_h_) Two other inequalities he calculated out and predicted,
+viz. variation in the motions of the apses and the nodes. Neither of
+these had then been observed, but they were afterwards detected and
+verified.
+
+A good many other minor irregularities are now known--some thirty, I
+believe; and altogether the lunar theory, or problem of the moon's exact
+motion, is one of the most complicated and difficult in astronomy; the
+perturbations being so numerous and large, because of the enormous mass
+of the perturbing body.
+
+The disturbances experienced by the planets are much smaller, because
+they are controlled by the sun and perturbed by each other. The moon is
+controlled only by the earth, and perturbed by the sun. Planetary
+perturbations can be treated as a series of disturbances with some
+satisfaction: not so those of the moon. And yet it is the only way at
+present known of dealing with the lunar theory.
+
+To deal with it satisfactorily would demand the solution of such a
+problem as this:--Given three rigid spherical masses thrown into empty
+space with any initial motions whatever, and abandoned to gravity: to
+determine their subsequent motions. With two masses the problem is
+simple enough, being pretty well summed up in Kepler's laws; but with
+three masses, strange to say, it is so complicated as to be beyond the
+reach of even modern mathematics. It is a famous problem, known as that
+of "the three bodies," but it has not yet been solved. Even when it is
+solved it will be only a close approximation to the case of earth, moon,
+and sun, for these bodies are not spherical, and are not rigid. One may
+imagine how absurdly and hopelessly complicated a complete treatment of
+the motions of the entire solar system would be.
+
+No. 8. Each planet is attracted not only by the sun but by the other
+planets, hence their orbits are slightly affected by each other.
+
+The subject of planetary perturbation was only just begun by Newton.
+Gradually (by Laplace and others) the theory became highly developed;
+and, as everybody knows, in 1846 Neptune was discovered by means of it.
+
+No. 9. He recognized the comets as members of the solar system, obedient
+to the same law of gravity and moving in very elongated ellipses; so
+their return could be predicted.
+
+It was a long time before Newton recognized the comets as real members
+of the solar system, and subject to gravity like the rest. He at first
+thought they moved in straight lines. It was only in the second edition
+of the _Principia_ that the theory of comets was introduced.
+
+Halley observed a fine comet in 1682, and calculated its orbit on
+Newtonian principles. He also calculated when it ought to have been seen
+in past times; and he found the year 1607, when one was seen by Kepler;
+also the year 1531, when one was seen by Appian; again, he reckoned
+1456, 1380, 1305. All these appearances were the same comet, in all
+probability, returning every seventy-five or seventy-six years. The
+period was easily allowed to be not exact, because of perturbing
+planets. He then predicted its return for 1758, or perhaps 1759, a date
+he could not himself hope to see. He lived to a great age, but he died
+sixteen years before this date.
+
+As the time drew nigh, three-quarters of a century afterwards,
+astronomers were greatly interested in this first cometary prediction,
+and kept an eager look-out for "Halley's comet." Clairaut, a most
+eminent mathematician and student of Newton, proceeded to calculate out
+more exactly the perturbing influence of Jupiter, near which it had
+passed. After immense labour (for the difficulty of the calculation was
+extreme, and the mass of mere figures something portentous), he
+predicted its return on the 13th of April, 1759, but he considered that
+he might have made a possible error of a month. It returned on the 13th
+of March, 1759, and established beyond all doubt the rule of the
+Newtonian theory over comets.
+
+[Illustration: FIG. 71.--Well-known model exhibiting the oblate
+spheroidal form as a consequence of spinning about a central axis. The
+brass strip _a_ looks like a transparent globe when whirled, and bulges
+out equatorially.]
+
+No. 10. Applying the idea of centrifugal force to the earth considered
+as a rotating body, he perceived that it could not be a true sphere, and
+calculated its oblateness, obtaining 28 miles greater equatorial than
+polar diameter.
+
+Here we return to one of the more simple deductions. A spinning body of
+any kind tends to swell at its circumference (or equator), and shrink
+along its axis (or poles). If the body is of yielding material, its
+shape must alter under the influence of centrifugal force; and if a
+globe of yielding substance subject to known forces rotates at a
+definite pace, its shape can be calculated. Thus a plastic sphere the
+size of the earth, held together by its own gravity, and rotating once a
+day, can be shown to have its equatorial diameter twenty-eight miles
+greater than its polar diameter: the two diameters being 8,000 and 8,028
+respectively. Now we have no guarantee that the earth is of yielding
+material: for all Newton could tell it might be extremely rigid. As a
+matter of fact it is now very nearly rigid. But he argued thus. The
+water on it is certainly yielding, and although the solid earth might
+decline to bulge at the equator in deference to the diurnal rotation,
+that would not prevent the ocean from flowing from the poles to the
+equator and piling itself up as an equatorial ocean fourteen miles deep,
+leaving dry land everywhere near either pole. Nothing of this sort is
+observed: the distribution of land and water is not thus regulated.
+Hence, whatever the earth may be now, it must once have been plastic
+enough to accommodate itself perfectly to the centrifugal forces, and to
+take the shape appropriate to a perfectly plastic body. In all
+probability it was once molten, and for long afterwards pasty.
+
+Thus, then, the shape of the earth can be calculated from the length of
+its day and the intensity of its gravity. The calculation is not
+difficult: it consists in imagining a couple of holes bored to the
+centre of the earth, one from a pole and one from the equator; filling
+these both with water, and calculating how much higher the water will
+stand in one leg of the gigantic V tube so formed than in the other. The
+answer comes out about fourteen miles.
+
+The shape of the earth can now be observed geodetically, and it accords
+with calculation, but the observations are extremely delicate; in
+Newton's time the _size_ was only barely known, the _shape_ was not
+observed till long after; but on the principles of mechanics, combined
+with a little common-sense reasoning, it could be calculated with
+certainty and accuracy.
+
+No. 11. From the observed shape of Jupiter or any planet the length of
+its day could be estimated.
+
+Jupiter is much more oblate than the earth. Its two diameters are to one
+another as 17 is to 16; the ellipticity of its disk is manifest to
+simple inspection. Hence we perceive that its whirling action must be
+more violent--it must rotate quicker. As a matter of fact its day is ten
+
+[Illustration: FIG. 72.--Jupiter.]
+
+hours long--five hours daylight and five hours night. The times of
+rotation of other bodies in the solar system are recorded in a table
+above.
+
+No. 12. The so-calculated shape of the earth, in combination with
+centrifugal force, causes the weight of bodies to vary with latitude;
+and Newton calculated the amount of this variation. 194 lbs. at pole
+balance 195 lbs. at equator.
+
+But following from the calculated shape of the earth follow several
+interesting consequences. First of all, the intensity of gravity will
+not be the same everywhere; for at the equator a stone is further from
+the average bulk of the earth (say the centre) than it is at the poles,
+and owing to this fact a mass of 590 pounds at the pole; would suffice
+to balance 591 pounds at the equator, if the two could be placed in the
+pans of a gigantic balance whose beam straddled along an earth's
+quadrant. This is a _true_ variation of gravity due to the shape of the
+earth. But besides this there is a still larger _apparent_ variation due
+to centrifugal force, which affects all bodies at the equator but not
+those at the poles. From this cause, even if the earth were a true
+sphere, yet if it were spinning at its actual pace, 288 pounds at the
+pole could balance 289 pounds at the equator; because at the equator the
+true weight of the mass would not be fully appreciated, centrifugal
+force would virtually diminish it by 1/289th of its amount.
+
+In actual fact both causes co-exist, and accordingly the total variation
+of gravity observed is compounded of the real and the apparent effects;
+the result is that 194 pounds at a pole weighs as much as 195 pounds at
+the equator.
+
+No. 13. A homogeneous sphere attracts as if its mass were concentrated
+at its centre. For any other figure, such as an oblate spheroid, this is
+not exactly true. A hollow concentric spherical shell exerts no force on
+small bodies inside it.
+
+A sphere composed of uniform material, or of materials arranged in
+concentric strata, can be shown to attract external bodies as if its
+mass were concentrated at its centre. A hollow sphere, similarly
+composed, does the same, but on internal bodies it exerts no force at
+all.
+
+Hence, at all distances above the surface of the earth, gravity
+decreases in inverse proportion as the square of the distance from the
+centre of the earth increases; but, if you descend a mine, gravity
+decreases in this case also as you leave the surface, though not at the
+same rate as when you went up. For as you penetrate the crust you get
+inside a concentric shell, which is thus powerless to act upon you, and
+the earth you are now outside is a smaller one. At what rate the force
+decreases depends on the distribution of density; if the density were
+uniform all through, the law of variation would be the direct distance,
+otherwise it would be more complicated. Anyhow, the intensity of gravity
+is a maximum at the surface of the earth, and decreases as you travel
+from the surface either up or down.
+
+No. 14. The earth's equatorial protuberance, being acted on by the
+attraction of the sun and moon, must disturb its axis of rotation in a
+calculated manner; and thus is produced the precession of the equinoxes.
+
+Here we come to a truly awful piece of reasoning. A sphere attracts as
+if its mass were concentrated at its centre (No. 12), but a spheroid
+does not. The earth is a spheroid, and hence it pulls and is pulled by
+the moon with a slightly uncentric attraction. In other words, the line
+of pull does not pass through its precise centre. Now when we have a
+spinning body, say a top, overloaded on one side so that gravity acts on
+it unsymmetrically, what happens? The axis of rotation begins to rotate
+cone-wise, at a pace which depends on the rate of spin, and on the shape
+and mass of the top, as well as on the amount and leverage of the
+overloading.
+
+Newton calculated out the rapidity of this conical motion of the axis of
+the earth, produced by the slightly unsymmetrical pull of the moon, and
+found that it would complete a revolution in 26,000 years--precisely
+what was wanted to explain the precession of the equinoxes. In fact he
+had discovered the physical cause of that precession.
+
+Observe that there were three stages in this discovery of precession:--
+
+First, the observation by Hipparchus, that the nodes, or intersections
+of the earth's orbit (the sun's apparent orbit) with the plane of the
+equator, were not stationary, but slowly moved.
+
+Second, the description of this motion by Copernicus, by the statement
+that it was due to a conical motion of the earth's axis of rotation
+about its centre as a fixed point.
+
+Third, the explanation of this motion by Newton as due to the pull of
+the moon on the equatorial protuberance of the earth.
+
+The explanation _could_ not have been previously suspected, for the
+shape of the earth, on which the whole theory depends, was entirely
+unknown till Newton calculated it.
+
+Another and smaller motion of a somewhat similar kind has been worked
+out since: it is due to the unsymmetrical attraction of the other
+planets for this same equatorial protuberance. It shows itself as a
+periodic change in the obliquity of the ecliptic, or so-called recession
+of the apses, rather than as a motion of the nodes.[21]
+
+No. 15. The waters of the ocean are attracted towards the sun and moon
+on one side, and whirled a little farther away than the solid earth on
+the other side: hence Newton explained all the main phenomena of the
+tides.
+
+And now comes another tremendous generalization. The tides had long been
+an utter mystery. Kepler likens the earth to an animal, and the tides to
+his breathings and inbreathings, and says they follow the moon.
+
+Galileo chaffs him for this, and says that it is mere superstition to
+connect the moon with the tides.
+
+Descartes said the moon pressed down upon the waters by the centrifugal
+force of its vortex, and so produced a low tide under it.
+
+Everything was fog and darkness on the subject. The legend goes that an
+astronomer threw himself into the sea in despair of ever being able to
+explain the flux and reflux of its waters.
+
+Newton now with consummate skill applied his theory to the effect of
+the moon upon the ocean, and all the main details of tidal action
+gradually revealed themselves to him.
+
+He treated the water, rotating with the earth once a day, somewhat as if
+it were a satellite acted on by perturbing forces. The moon as it
+revolves round the earth is perturbed by the sun. The ocean as it
+revolves round the earth (being held on by gravitation just as the moon
+is) is perturbed by both sun and moon.
+
+The perturbing effect of a body varies directly as its mass, and
+inversely as the cube of its distance. (The simple law of inverse square
+does not apply, because a perturbation is a differential effect: the
+satellite or ocean when nearer to the perturbing body than the rest of
+the earth, is attracted more, and when further off it is attracted less
+than is the main body of the earth; and it is these differences alone
+which constitute the perturbation.) The moon is the more powerful of the
+two perturbing bodies, hence the main tides are due to the moon; and its
+chief action is to cause a pair of low waves or oceanic humps, of
+gigantic area, to travel round the earth once in a lunar day, _i.e._ in
+about 24 hours and 50 minutes. The sun makes a similar but still lower
+pair of low elevations to travel round once in a solar day of 24 hours.
+And the combination of the two pairs of humps, thus periodically
+overtaking each other, accounts for the well-known spring and neap
+tides,--spring tides when their maxima agree, neap tides when the
+maximum of one coincides with the minimum of the other: each of which
+events happens regularly once a fortnight.
+
+These are the main effects, but besides these there are the effects of
+varying distances and obliquity to be taken into account; and so we have
+a whole series of minor disturbances, very like those discussed in No.
+7, under the lunar theory, but more complex still, because there are two
+perturbing bodies instead of only one.
+
+The subject of the tides is, therefore, very recondite; and though one
+may give some elementary account of its main features, it will be best
+to defer this to a separate lecture (Lecture XVII).
+
+I had better, however, here say that Newton did not limit himself to the
+consideration of the primary oceanic humps: he pursued the subject into
+geographical detail. He pointed out that, although the rise and fall of
+the tide at mid-ocean islands would be but small, yet on stretches of
+coast the wave would fling itself, and by its momentum would propel the
+waters, to a much greater height--for instance, 20 or 30 feet;
+especially in some funnel-shaped openings like the Bristol Channel and
+the Bay of Fundy, where the concentrated impetus of the water is
+enormous.
+
+He also showed how the tidal waves reached different stations in
+successive regular order each day; and how some places might be fed with
+tide by two distinct channels; and that if the time of these channels
+happened to differ by six hours, a high tide might be arriving by one
+channel and a low tide by the other, so that the place would only feel
+the difference, and so have a very small observed rise and fall;
+instancing a port in China (in the Gulf of Tonquin) where that
+approximately occurs.
+
+In fact, although his theory was not, as we now know, complete or final,
+yet it satisfactorily explained a mass of intricate detail as well as
+the main features of the tides.
+
+No. 16. The sun's mass being known, he calculated the height of the
+solar tide.
+
+No. 17. From the observed heights of spring and neap tides he determined
+the lunar tide, and thence made an estimate of the mass of the moon.
+
+Knowing the sun's mass and distance, it was not difficult for Newton to
+calculate the height of the protuberance caused by it in a pasty ocean
+covering the whole earth. I say pasty, because, if there was any
+tendency for impulses to accumulate, as timely pushes given to a
+pendulum accumulate, the amount of disturbance might become excessive,
+and its calculation would involve a multitude of data. The Newtonian
+tide ignored this, thus practically treating the motion as either
+dead-beat, or else the impulses as very inadequately timed. With this
+reservation the mid-ocean tide due to the action of the sun alone comes
+out about one foot, or let us say one foot for simplicity. Now the
+actual tide observed in mid-Atlantic is at the springs about four feet,
+at the neaps about two. The spring tide is lunar plus solar; the neap
+tide is lunar minus solar. Hence it appears that the tide caused by the
+moon alone must be about three feet, when unaffected by momentum. From
+this datum Newton made the first attempt to approximately estimate the
+mass of the moon. I said that the masses of satellites must be
+estimated, if at all, by the perturbation they are able to cause. The
+lunar tide is a perturbation in the diurnal motion of the sea, and its
+amount is therefore a legitimate mode of calculating the moon's mass.
+The available data were not at all good, however; nor are they even now
+very perfect; and so the estimate was a good way out. It is now
+considered that the mass of the moon is about one-eightieth that of the
+earth.
+
+*      *      *      *      *
+
+Such are some of the gems extracted from their setting in the
+_Principia_, and presented as clearly as I am able before you.
+
+Do you realize the tremendous stride in knowledge--not a stride, as
+Whewell says, nor yet a leap, but a flight--which has occurred between
+the dim gropings of Kepler, the elementary truths of Galileo, the
+fascinating but wild speculations of Descartes, and this magnificent and
+comprehensive system of ordered knowledge. To some his genius seemed
+almost divine. "Does Mr. Newton eat, drink, sleep, like other men?" said
+the Marquis de l'Hopital, a French mathematician of no mean eminence; "I
+picture him to myself as a celestial genius, entirely removed from the
+restrictions of ordinary matter." To many it seemed as if there was
+nothing more to be discovered, as if the universe were now explored, and
+only a few fragments of truth remained for the gleaner. This is the
+attitude of mind expressed in Pope's famous epigram:--
+
+  "Nature and Nature's laws lay hid in Night,
+  God said, Let Newton be, and all was light."
+
+This feeling of hopelessness and impotence was very natural after the
+advent of so overpowering a genius, and it prevailed in England for
+fully a century. It was very natural, but it was very mischievous; for,
+as a consequence, nothing of great moment was done by England in
+science, and no Englishman of the first magnitude appeared, till some
+who are either living now or who have lived within the present century.
+
+It appeared to his contemporaries as if he had almost exhausted the
+possibility of discovery; but did it so appear to Newton? Did it seem to
+him as if he had seen far and deep into the truths of this great and
+infinite universe? It did not. When quite an old man, full of honour and
+renown, venerated, almost worshipped, by his contemporaries, these were
+his words:--
+
+"I know not what the world will think of my labours, but to myself it
+seems that I have been but as a child playing on the sea-shore; now
+finding some pebble rather more polished, and now some shell rather more
+agreeably variegated than another, while the immense ocean of truth
+extended itself unexplored before me."
+
+And so it must ever seem to the wisest and greatest of men when brought
+into contact with the great things of God--that which they know is as
+nothing, and less than nothing, to the infinitude of which they are
+ignorant.
+
+Newton's words sound like a simple and pleasing echo of the words of
+that great unknown poet, the writer of the book of Job:--
+
+  "Lo, these are parts of His ways,
+  But how little a portion is heard of Him;
+  The thunder of His power, who can understand?"
+
+END OF PART I.
+
+
+
+
+PART II
+
+_A COUPLE OF CENTURIES' PROGRESS._
+
+
+
+
+NOTES TO LECTURE X
+
+_Science during the century after Newton_
+
+The _Principia_ published, 1687
+
+  Roemer                1644-1710
+  James Bradley         1692-1762
+  Clairaut              1713-1765
+  Euler                 1707-1783
+  D'Alembert            1717-1783
+  Lagrange              1736-1813
+  Laplace               1749-1827
+  William Herschel      1738-1822
+
+
+_Olaus Roemer_ was born in Jutland, and studied at Copenhagen. Assisted
+Picard in 1671 to determine the exact position of Tycho's observatory on
+Huen. Accompanied Picard to Paris, and in 1675 read before the Academy
+his paper "On Successive Propagation of Light as revealed by a certain
+inequality in the motion of Jupiter's First Satellite." In 1681 he
+returned to Copenhagen as Professor of Mathematics and Astronomy, and
+died in 1710. He invented the transit instrument, mural circle,
+equatorial mounting for telescopes, and most of the other principal
+instruments now in use in observatories. He made as many observations as
+Tycho Brahe, but the records of all but the work of three days were
+destroyed by a great fire in 1728.
+
+_Bradley_, Professor of Astronomy at Oxford, discovered the aberration
+of light in 1729, while examining stars for parallax, and the nutation
+of the earth's axis in 1748. Was appointed Astronomer-Royal in 1742.
+
+
+
+
+LECTURE X
+
+ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT
+
+
+At Newton's death England stood pre-eminent among the nations of Europe
+in the sphere of science. But the pre-eminence did not last long. Two
+great discoveries were made very soon after his decease, both by
+Professor Bradley, of Oxford, and then there came a gap. A moderately
+great man often leaves behind him a school of disciples able to work
+according to their master's methods, and with a healthy spirit of
+rivalry which stimulates and encourages them. Newton left, indeed, a
+school of disciples, but his methods of work were largely unknown to
+them, and such as were known were too ponderous to be used by ordinary
+men. Only one fresh result, and that a small one, has ever been attained
+by other men working according to the methods of the _Principia_. The
+methods were studied and commented on in England to the exclusion of all
+others for nigh a century, and as a consequence no really important work
+was done.
+
+On the Continent, however, no such system of slavish imitation
+prevailed. Those methods of Newton's which had been simultaneously
+discovered by Leibnitz were more thoroughly grasped, modified, extended,
+and improved. There arose a great school of French and German
+mathematicians, and the laurels of scientific discovery passed to France
+and Germany--more especially, perhaps, at this time to France. England
+has never wholly recovered them. During the present century this country
+has been favoured with some giants who, as they become distant enough
+for their true magnitude to be perceived, may possibly stand out as
+great as any who have ever lived; but for the mass and bulk of
+scientific work at the present day we have to look to Germany, with its
+enlightened Government and extensive intellectual development. England,
+however, is waking up, and what its Government does not do, private
+enterprise is beginning to accomplish. The establishment of centres of
+scientific and literary activity in the great towns of England, though
+at present they are partially encumbered with the supply of education of
+an exceedingly rudimentary type, is a movement that in the course of
+another century or so will be seen to be one of the most important and
+fruitful steps ever taken by this country. On the Continent such centres
+have long existed; almost every large town is the seat of a University,
+and they are now liberally endowed. The University of Bologna (where,
+you may remember, Copernicus learnt mathematics) has recently celebrated
+its 800th anniversary.
+
+The scientific history of the century after Newton, summarized in the
+above table of dates, embraces the labours of the great mathematicians
+Clairaut, Euler, D'Alembert, and especially of Lagrange and Laplace.
+
+But the main work of all these men was hardly pioneering work. It was
+rather the surveying, and mapping out, and bringing into cultivation, of
+lands already discovered. Probably Herschel may be justly regarded as
+the next true pioneer. We shall not, however, properly appreciate the
+stages through which astronomy has passed, nor shall we be prepared
+adequately to welcome the discoveries of modern times unless we pay some
+attention to the intervening age. Moreover, during this era several
+facts of great moment gradually came into recognition; and the
+importance of the discovery we have now to speak of can hardly be
+over-estimated.
+
+Our whole direct knowledge of the planetary and stellar universe, from
+the early observations of the ancients down to the magnificent
+discoveries of a Herschel, depends entirely upon our happening to
+possess a sense of sight. To no other of our senses do any other worlds
+than our own in the slightest degree appeal. We touch them or hear them
+never. Consequently, if the human race had happened to be blind, no
+other world but the one it groped its way upon could ever have been
+known or imagined by it. The outside universe would have existed, but
+man would have been entirely and hopelessly ignorant of it. The bare
+idea of an outside universe beyond the world would have been
+inconceivable, and might have been scouted as absurd. We do possess the
+sense of sight; but is it to be supposed that we possess every sense
+that can be possessed by finite beings? There is not the least ground
+for such an assumption. It is easy to imagine a deaf race or a blind
+race: it is not so easy to imagine a race more highly endowed with
+senses than our own; and yet the sense of smell in animals may give us
+some aid in thinking of powers of perception which transcend our own in
+particular directions. If there were a race with higher or other senses
+than our own, or if the human race should ever in the process of
+development acquire such extra sense-organs, a whole universe of
+existent fact might become for the first time perceived by us, and we
+should look back upon our past state as upon a blind chrysalid form of
+existence in which we had been unconscious of all this new wealth of
+perception.
+
+It cannot be too clearly and strongly insisted on and brought home to
+every mind, that the mode in which the universe strikes us, our view of
+the universe, our whole idea of matter, and force, and other worlds, and
+even of consciousness, depends upon the particular set of sense-organs
+with which we, as men, happen to be endowed. The senses of force, of
+motion, of sound, of light, of touch, of heat, of taste, and of
+smell--these we have, and these are the things we primarily know. All
+else is inference founded upon these sensations. So the world appears to
+us. But given other sense-organs, and it might appear quite otherwise.
+What it is actually and truly like, therefore, is quite and for ever
+beyond us--so long as we are finite beings.
+
+Without eyes, astronomy would be non-existent. Light it is which conveys
+all the information we possess, or, as it would seem, ever can possess,
+concerning the outer and greater universe in which this small world
+forms a speck. Light is the channel, the messenger of information; our
+eyes, aided by telescopes, spectroscopes, and many other "scopes" that
+may yet be invented, are the means by which we read the information that
+light brings.
+
+Light travels from the stars to our eyes: does it come instantaneously?
+or does it loiter by the way? for if it lingers it is not bringing us
+information properly up to date--it is only telling us what the state of
+affairs was when it started on its long journey.
+
+Now, it is evidently a matter of interest to us whether we see the sun
+as he is now, or only as he was some three hundred years ago. If the
+information came by express train it would be three hundred years behind
+date, and the sun might have gone out in the reign of Queen Anne without
+our being as yet any the wiser. The question, therefore, "At what rate
+does our messenger travel?" is evidently one of great interest for
+astronomers, and many have been the attempts made to solve it. Very
+likely the ancient Greeks pondered over this question, but the earliest
+writer known to me who seriously discussed the question is Galileo. He
+suggests a rough experimental means of attacking it. First of all, it
+plainly comes quicker than sound. This can be perceived by merely
+watching distant hammering, or by noticing that the flash of a pistol is
+seen before its report is heard, or by listening to the noise of a
+flash of lightning. Sound takes five seconds to travel a mile--it has
+about the same speed as a rifle bullet; but light is much quicker than
+that.
+
+The rude experiment suggested by Galileo was to send two men with
+lanterns and screens to two distant watch-towers or neighbouring
+mountain tops, and to arrange that each was to watch alternate displays
+and obscurations of the light made by the other, and to imitate them as
+promptly as possible. Either man, therefore, on obscuring or showing his
+own light would see the distant glimmer do the same, and would be able
+to judge if there was any appreciable interval between his own action
+and the response of the distant light. The experiment was actually tried
+by the Florentine Academicians,[22] with the result that, as practice
+improved, the interval became shorter and shorter, so that there was no
+reason to suppose that there was any real interval at all. Light, in
+fact, seemed to travel instantaneously.
+
+Well might they have arrived at this result. Even if they had made far
+more perfect arrangements--for instance, by arranging a looking-glass at
+one of the stations in which a distant observer might see the reflection
+of his own lantern, and watch the obscurations and flashings made by
+himself, without having to depend on the response of human
+mechanism--even then no interval whatever could have been detected.
+
+If, by some impossibly perfect optical arrangement, a lighthouse here
+were made visible to us after reflection in a mirror erected at New
+York, so that the light would have to travel across the Atlantic and
+back before it could be seen, even then the appearance of the light on
+removing a shutter, or the eclipse on interposing it, would seem to
+happen quite instantaneously. There would certainly be an interval: the
+interval would be the fiftieth part of a second (the time a stone takes
+to drop 1/13th of an inch), but that is too short to be securely
+detected without mechanism. With mechanism the thing might be managed,
+for a series of shutters might be arranged like the teeth of a large
+wheel; so that, when the wheel rotates, eclipses follow one another very
+rapidly; if then an eye looked through the same opening as that by which
+the light goes on its way to the distant mirror, a tooth might have
+moved sufficiently to cover up this space by the time the light
+returned; in which case the whole would appear dark, for the light would
+be stopped by a tooth, either at starting or at returning, continually.
+At higher speeds of rotation some light would reappear, and at lower
+speeds it would also reappear; by noticing, therefore, the precise speed
+at which there was constant eclipse the velocity of light could be
+determined.
+
+[Illustration: FIG. 73.--Diagram of eye looking at a light reflected in
+a distant mirror through the teeth of a revolving wheel.]
+
+This experiment has now been made in a highly refined form by Fizeau,
+and repeated by M. Cornu with prodigious care and accuracy. But with
+these recent matters we have no concern at present. It may be
+instructive to say, however, that if the light had to travel two miles
+altogether, the wheel would have to possess 450 teeth and to spin 100
+times a second (at the risk of flying to pieces) in order that the ray
+starting through any one of the gaps might be stopped on returning by
+the adjacent tooth.
+
+Well might the velocity of light be called instantaneous by the early
+observers. An ordinary experiment seemed (and was) hopeless, and light
+was supposed to travel at an infinite speed. But a phenomenon was
+noticed in the heavens by a quick-witted and ingenious Danish
+astronomer, which was not susceptible of any ordinary explanation, and
+which he perceived could at once be explained if light had a certain
+rate of travel--great, indeed, but something short of infinite. This
+phenomenon was connected with the satellites of Jupiter, and the
+astronomer's name was Roemer. I will speak first of the observation and
+then of the man.
+
+[Illustration: FIG. 74.--Fizeau's wheel, shewing the appearance of
+distant image seen through its teeth. 1st, when stationary, next when
+revolving at a moderate speed, last when revolving at the high speed
+just sufficient to cause eclipse.]
+
+Jupiter's satellites are visible, precisely as our own moon is, by
+reason of the shimmer of sunlight which they reflect. But as they
+revolve round their great planet they plunge into his shadow at one part
+of their course, and so become eclipsed from sunshine and invisible to
+us. The moment of disappearance can be sharply observed.
+
+Take the first satellite as an example. The interval between successive
+eclipses ought to be its period of revolution round Jupiter. Observe
+this period. It was not uniform. On the average it was 42 hours 47
+minutes, but it seemed to depend on the time of year. When Roemer
+observed in spring it was less, and in autumn it was more than usual.
+This was evidently a puzzling fact: what on earth can our year have to
+do with the motion of a moon of Jupiter's? It was probably, therefore,
+only an apparent change, caused either by our greater or less distance
+from Jupiter, or else by our greater or less speed of travelling to or
+from him. Considering it thus, he was led to see that, when the time of
+revolution seemed longest, we were receding fastest from Jupiter, and
+when shortest, approaching fastest.
+
+_If_, then, light took time on its journey, _if_ it travelled
+progressively, the whole anomaly would be explained.
+
+In a second the earth goes nineteen miles; therefore in 42-3/4 hours
+(the time of revolution of Jupiter's first satellite) it goes 2.9
+million (say three million) miles. The eclipse happens punctually, but
+we do not see it till the light conveying the information has travelled
+the extra three million miles and caught up the earth. Evidently,
+therefore, by observing how much the apparent time of revolution is
+lengthened in one part of the earth's orbit and shortened in another,
+getting all the data accurately, and assuming the truth of our
+hypothetical explanation, we can calculate the velocity of light. This
+is what Roemer did.
+
+Now the maximum amount of retardation is just about fifteen seconds.
+Hence light takes this time to travel three million miles; therefore its
+velocity is three million divided by fifteen, say 200,000, or, as we now
+know more exactly, 186,000 miles every second. Note that the delay does
+not depend on our _distance_, but on our _speed_. One can tell this by
+common-sense as soon as we grasp the general idea of the explanation. A
+velocity cannot possibly depend on a distance only.
+
+[Illustration: FIG. 75.--Eclipses of one of Jupiter's satellites. A
+diagram intended to illustrate the dependence of its apparent time of
+revolution (from eclipse to eclipse) on the motion of the earth; but not
+illustrating the matter at all well. TT' T'' are successive positions of
+the earth, while JJ' J'' are corresponding positions of Jupiter.]
+
+Roemer's explanation of the anomaly was not accepted by astronomers. It
+excited some attention, and was discussed, but it was found not
+obviously applicable to any of the satellites except the first, and not
+very simply and satisfactorily even to that. I have, of course, given
+you the theory in its most elementary and simple form. In actual fact a
+host of disturbing and complicated considerations come in--not so
+violently disturbing for the first satellite as for the others, because
+it moves so quickly, but still complicated enough.
+
+The fact is, the real motion of Jupiter's satellites is a most difficult
+problem. The motion even of our own moon (the lunar theory) is difficult
+enough: perturbed as its motion is by the sun. You know that Newton said
+it cost him more labour than all the rest of the _Principia_. But the
+motion of Jupiter's satellites is far worse. No one, in fact, has yet
+worked their theory completely out. They are perturbed by the sun, of
+course, but they also perturb each other, and Jupiter is far from
+spherical. The shape of Jupiter, and their mutual attractions, combine
+to make their motions most peculiar and distracting.
+
+Hence an error in the time of revolution of a satellite was not
+_certainly_ due to the cause Roemer suggested, unless one could be sure
+that the inequality was not a real one, unless it could be shown that
+the theory of gravitation was insufficient to account for it. This had
+not then been done; so the half-made discovery was shelved, and properly
+shelved, as a brilliant but unverified speculation. It remained on the
+shelf for half a century, and was no doubt almost forgotten.
+
+[Illustration: FIG. 76.--A Transit-instrument for the British
+astronomical expedition, 1874. Shewing in its essential features the
+simplest form of such an instrument.]
+
+Now a word or two about the man. He was a Dane, educated at Copenhagen,
+and learned in the mathematics. We first hear of him as appointed to
+assist Picard, the eminent French geodetic surveyor (whose admirable
+work in determining the length of a degree you remember in connection
+with Newton), who had come over to Denmark with the object of fixing the
+exact site of the old and extinct Tychonic observatory in the island of
+Huen. For of course the knowledge of the exact latitude and longitude of
+every place whence numerous observations have been taken must be an
+essential to the full interpretation of those observations. The
+measurements being finished, young Roemer accompanied Picard to Paris,
+and here it was, a few years after, that he read his famous paper
+concerning "An Inequality in the Motion of Jupiter's First Satellite,"
+and its explanation by means of an hypothesis of "the successive
+propagation of light."
+
+The later years of his life he spent in Copenhagen as a professor in the
+University and an enthusiastic observer of the heavens,--not a
+descriptive observer like Herschel, but a measuring observer like Sir
+George Airy or Tycho Brahe. He was, in fact, a worthy follower of Tycho,
+and the main work of his life is the development and devising of new and
+more accurate astronomical instruments. Many of the large and accurate
+instruments with which a modern observatory is furnished are the
+invention of this Dane. One of the finest observatories in the world is
+the Russian one at Pulkowa, and a list of the instruments there reads
+like an extended catalogue of Roemer's inventions.
+
+He not only _invented_ the instruments, he had them made, being allowed
+money for the purpose; and he used them vigorously, so that at his death
+he left great piles of manuscript stored in the national observatory.
+
+Unfortunately this observatory was in the heart of the city, and was
+thus exposed to a danger from which such places ought to be as far as
+possible exempt.
+
+Some eighteen years after Roemer's death a great conflagration broke out
+in Copenhagen, and ruined large portions of the city. The successor to
+Roemer, Horrebow by name, fled from his house, with such valuables as he
+possessed, to the observatory, and there went on with his work. But
+before long the wind shifted, and to his horror he saw the flames
+coming his way. He packed up his own and his predecessor's manuscript
+observations in two cases, and prepared to escape with them, but the
+neighbours had resorted to the observatory as a place of safety, and so
+choked up the staircase with their property that he was barely able to
+escape himself, let alone the luggage, and everything was lost.
+
+[Illustration: FIG. 77.--Diagram of equatorially mounted telescope; CE
+is the polar axis parallel to the axis of the earth; AB the declination
+axis. The diurnal motion is compensated by motion about the polar axis
+only, the other being clamped.]
+
+Of all the observations, only three days' work remains, and these were
+carefully discussed by Dr. Galle, of Berlin, in 1845, and their
+nutriment extracted. These ancient observations are of great use for
+purposes of comparison with the present state of the heavens, and throw
+light upon possible changes that are going on. Of course nowadays such a
+series of observations would be printed and distributed in many
+libraries, and so made practically indestructible.
+
+Sad as the disaster was to the posthumous fame of the great observer, a
+considerable compensation was preparing. The very year that the fire
+occurred in Denmark a quiet philosopher in England was speculating and
+brooding on a remarkable observation that he had made concerning the
+apparent motion of certain stars, and he was led thereby to a discovery
+of the first magnitude concerning the speed of light--a discovery which
+resuscitated the old theory of Roemer about Jupiter's satellites, and
+made both it and him immortal.
+
+James Bradley lived a quiet, uneventful, studious life, mainly at Oxford
+but afterwards at the National Observatory at Greenwich, of which he was
+third Astronomer-Royal, Flamsteed and Halley having preceded him in that
+office. He had taken orders, and lectured at Oxford as Savilian
+Professor. It is said that he pondered his great discovery while pacing
+the Long Walk at Magdalen College--and a beautiful place it is to
+meditate in.
+
+Bradley was engaged in making observations to determine if possible the
+parallax of some of the fixed stars. Parallax means the apparent
+relative shift of bodies due to a change in the observer's position. It
+is parallax which we observe when travelling by rail and looking out of
+window at the distant landscape. Things at different distances are left
+behind at different apparent rates, and accordingly they seem to move
+relatively to each other. The most distant objects are least affected;
+and anything enormously distant, like the moon, is not subject to this
+effect, but would retain its position however far we travelled, unless
+we had some extraordinarily precise means of observation.
+
+So with the fixed stars: they were being observed from a moving
+carriage--viz. the earth--and one moving at the rate of nineteen miles a
+second. Unless they were infinitely distant, or unless they were all at
+the same distance, they must show relative apparent motions among
+themselves. Seen from one point of the earth's orbit, and then in six
+months from an opposite point, nearly 184 million miles away, surely
+they must show some difference of aspect.
+
+Remember that the old Copernican difficulty had never been removed. If
+the earth revolved round the sun, how came it that the fixed stars
+showed no parallax? The fact still remained a surprise, and the question
+a challenge. Picard, like other astronomers, supposed that it was only
+because the methods of observation had not been delicate enough; but now
+that, since the invention of the telescope and the founding of National
+Observatories, accuracy hitherto undreamt of was possible, why not
+attack the problem anew? This, then, he did, watching the stars with
+great care to see if in six months they showed any change in absolute
+position with reference to the pole of the heavens; any known secular
+motion of the pole, such as precession, being allowed for. Already he
+thought he detected a slight parallax for several stars near the pole,
+and the subject was exciting much interest.
+
+Bradley determined to attempt the same investigation. He was not
+destined to succeed in it. Not till the present century was success in
+that most difficult observation achieved; and even now it cannot be done
+by the absolute methods then attempted; but, as so often happens,
+Bradley, in attempting one thing, hit upon another, and, as it happened,
+one of still greater brilliance and importance. Let us trace the stages
+of his discovery.
+
+Atmospheric refraction made horizon observations useless for the
+delicacy of his purpose, so he chose stars near the zenith, particularly
+one--[gamma] Draconis. This he observed very carefully at different
+seasons of the year by means of an instrument specially adapted for
+zenith observations, viz. a zenith sector. The observations were made in
+conjunction with a friend of his, an amateur astronomer named Molyneux,
+and they were made at Kew. Molyneux was shortly made First Lord of the
+Admiralty, or something important of that sort, and gave up frivolous
+pursuits. So Bradley observed alone. They observed the star accurately
+early in the month of December, and then intended to wait six months.
+But from curiosity Bradley observed it again only about a week later. To
+his surprise, he found that it had already changed its position. He
+recorded his observation on the back of an old envelope: it was his wont
+thus to use up odd scraps of paper--he was not, I regret to say, a tidy
+or methodical person--and this odd piece of paper turned up long
+afterwards among his manuscripts. It has been photographed and preserved
+as an historical relic.
+
+Again and again he repeated the observation of the star, and continually
+found it moving still a little further and further south, an excessively
+small motion, but still an appreciable one--not to be set down to errors
+of observation. So it went on till March. It then waited, and after a
+bit longer began to return, until June. By September it was displaced as
+much to the north as it had been to the south, and by December it had
+got back to its original position. It had described, in fact, a small
+oscillation in the course of the year. The motion affected neighbouring
+stars in a similar way, and was called an "aberration," or wandering
+from their true place.
+
+For a long time Bradley pondered over this observation, and over others
+like them which he also made. He found one group of stars describing
+small circles, while others at a distance from them were oscillating in
+straight lines, and all the others were describing ellipses. Unless this
+state of things were cleared up, accurate astronomy was impossible. The
+fixed stars!--they were not fixed a bit. To refined and accurate
+observation, such as was now possible, they were all careering about in
+little orbits having a reference to the earth's year, besides any proper
+motion which they might really have of their own, though no such motion
+was at present known. Not till Herschel was that discovered; not till
+this extraordinary aberration was allowed for could it be discovered.
+The effect observed by Bradley and Molyneux must manifestly be only an
+apparent motion: it was absurd to suppose a real stellar motion
+regulating itself according to the position of the earth. Parallax could
+not do it, for that would displace stars relatively among each other--it
+would not move similarly a set of neighbouring stars.
+
+At length, four years after the observation, the explanation struck him,
+while in a boat upon the Thames. He noticed the apparent direction of
+the wind changed whenever the boat started. The wind veered when the
+boat's motion changed. Of course the cause of this was obvious
+enough--the speed of the wind and the speed of the boat were compounded,
+and gave an apparent direction of the wind other than the true
+direction. But this immediately suggested a cause for what he had
+observed in the heavens. He had been observing an apparent direction of
+the stars other than the true direction, because he was observing from a
+moving vehicle. The real direction was doubtless fixed: the apparent
+direction veered about with the motion of the earth. It must be that
+light did not travel instantaneously, but gradually, as Roemer had
+surmised fifty years ago; and that the motion of the light was
+compounded with the motion of the earth.
+
+Think of a stream of light or anything else falling on a moving
+carriage. The carriage will run athwart the stream, the occupants of the
+carriage will mistake its true direction. A rifle fired through the
+windows of a railway carriage by a man at rest outside would make its
+perforations not in the true line of fire unless the train is
+stationary. If the train is moving, the line joining the holes will
+point to a place in advance of where the rifle is really located.
+
+So it is with the two glasses of a telescope, the object-glass and
+eye-piece, which are pierced by the light; an astronomer, applying his
+eye to the tube and looking for the origin of the disturbance, sees it
+apparently, but not in its real position--its apparent direction is
+displaced in the direction of the telescope's motion; by an amount
+depending on the ratio of the velocity of the earth to the velocity of
+light, and on the angle between those two directions.
+
+[Illustration: FIG. 78.--Aberration diagram. The light-ray L penetrates
+the object-glass of the moving telescope at O, but does not reach the
+eye-piece until the telescope has travelled to the second position.
+Consequently a moving telescope does not point out the true direction of
+the light, but aims at a point a little in advance.]
+
+But how minute is the displacement! The greatest effect is obtained when
+the two motions are at right angles to each other, _i.e._ when the star
+seen is at right angles to the direction of the earth's motion, but even
+then it is only 20", or 1/180th part of a degree; one-ninetieth of the
+moon's apparent diameter. It could not be detected without a cross-wire
+in the telescope, and would only appear as a slight displacement from
+the centre of the field, supposing the telescope accurately pointed to
+the true direction.
+
+But if this explanation be true, it at once gives a method of
+determining the velocity of light. The maximum angle of deviation,
+represented as a ratio of arc / radius, amounts to
+
+       1                     1
+  ------------  -  .0001 = ------
+  180 x 57-1/3             10,000
+
+(a gradient of 1 foot in two miles). In other words, the velocity of
+light must be 10,000 times as great as the velocity of the earth in its
+orbit. This amounts to a speed of 190,000 miles a second--not so very
+different from what Roemer had reckoned it in order to explain the
+anomalies of Jupiter's first satellite.
+
+Stars in the direction in which the earth was moving would not be thus
+affected; there would be nothing in mere approach or recession to alter
+direction or to make itself in any way visible. Stars at right angles to
+the earth's line of motion would be most affected, and these would be
+all displaced by the full amount of 20 seconds of arc. Stars in
+intermediate directions would be displaced by intermediate amounts.
+
+But the line of the earth's motion is approximately a circle round the
+sun, hence the direction of its advance is constantly though slowly
+changing, and in one year it goes through all the points of the compass.
+The stars, being displaced always in the line of advance, must similarly
+appear to describe little closed curves, always a quadrant in advance of
+the earth, completing their orbits once a year. Those near the pole of
+the ecliptic will describe circles, being always at right angles to the
+motion. Those in the plane of the ecliptic (near the zodiac) will be
+sometimes at right angles to the motion, but at other times will be
+approached or receded from; hence these will oscillate like pendulums
+once a year; and intermediate stars will have intermediate motions--that
+is to say, will describe ellipses of varying excentricity, but all
+completed in a year, and all with the major axis 20". This agreed very
+closely with what was observed.
+
+The main details were thus clearly and simply explained by the
+hypothesis of a finite velocity for light, "the successive propagation
+of light in time." This time there was no room for hesitation, and
+astronomers hailed the discovery with enthusiasm.
+
+Not yet, however, did Bradley rest. The finite velocity of light
+explained the major part of the irregularities he had observed, but not
+the whole. The more carefully he measured the amount of the deviation,
+the less completely accurate became its explanation.
+
+There clearly was a small outstanding error or discrepancy; the stars
+were still subject to an unexplained displacement--not, indeed, a
+displacement that repeated itself every year, but one that went through
+a cycle of changes in a longer period.
+
+The displacement was only about half that of aberration, and having a
+longer period was rather more difficult to detect securely. But the
+major difficulty was the fact that the two sorts of disturbances were
+co-existent, and the skill of disentangling them, and exhibiting the
+true and complete cause of each inequality, was very brilliant.
+
+For nineteen years did Bradley observe this minor displacement, and in
+that time he saw it go through a complete cycle. Its cause was now clear
+to him; the nineteen-year period suggested the explanation. It is the
+period in which the moon goes through all her changes--a period known to
+the ancients as the lunar cycle, or Metonic cycle, and used by them to
+predict eclipses. It is still used for the first rough approximation to
+the prediction of eclipses, and to calculate Easter. The "Golden Number"
+of the Prayer-book is the number of the year in this cycle.
+
+The cause of the second inequality, or apparent periodic motion of the
+stars, Bradley made out to be a nodding motion of the earth's axis.
+
+The axis of the earth describes its precessional orbit or conical
+motion every 26,000 years, as had long been known; but superposed upon
+this great movement have now been detected minute nods, each with a
+period of nineteen years.
+
+The cause of the nodding is completely accounted for by the theory of
+gravitation, just as the precession of the equinoxes was. Both
+disturbances result from the attraction of the moon on the non-spherical
+earth--on its protuberant equator.
+
+"Nutation" is, in fact, a small perturbation of precession. The motion
+may be observed in a non-sleeping top. The slow conical motion of the
+top's slanting axis represents the course of precession. Sometimes this
+path is loopy, and its little nods correspond to nutation.
+
+The probable existence of some such perturbation had not escaped the
+sagacity of Newton, and he mentions something about it in the
+_Principia_, but thinks it too small to be detected by observation. He
+was thinking, however, of a solar disturbance rather than a lunar one,
+and this is certainly very small, though it, too, has now been observed.
+
+Newton was dead before Bradley made these great discoveries, else he
+would have been greatly pleased to hear of them.
+
+These discoveries of aberration and nutation, says Delambre, the great
+French historian of science, secure to their author a distinguished
+place after Hipparchus and Kepler among the astronomers of all ages and
+all countries.
+
+
+
+
+NOTES TO LECTURE XI
+
+
+_Lagrange_ and _Laplace_, both tremendous mathematicians, worked very
+much in alliance, and completed Newton's work. The _Mecanique Celeste_
+contains the higher intricacies of astronomy mathematically worked out
+according to the theory of gravitation. They proved the solar system to
+be stable; all its inequalities being periodic, not cumulative. And
+Laplace suggested the "nebular hypothesis" concerning the origin of sun
+and planets: a hypothesis previously suggested, and to some extent,
+elaborated, by Kant.
+
+A list of some of the principal astronomical researches of Lagrange and
+Laplace:--Libration of the moon. Long inequality of Jupiter and Saturn.
+Perturbations of Jupiter's satellites. Perturbations of comets.
+Acceleration of the moon's mean motion. Improved lunar theory.
+Improvements in the theory of the tides. Periodic changes in the form
+and obliquity of the earth's orbit. Stability of the solar system
+considered as an assemblage of rigid bodies subject to gravity.
+
+The two equations which establish the stability of the solar system
+are:--
+
+  _Sum (me^2[square root]d) = constant,_
+
+  and
+
+  _Sum (m tan^2[theta][square root]d) = constant;_
+
+where _m_ is the mass of each planet, _d_ its mean distance from the
+sun, _e_ the excentricity of its orbit, and [theta] the inclination
+of its plane. However the expressions above formulated may change for
+individual planets, the sum of them for all the planets remains
+invariable.
+
+The period of the variations in excentricity of the earth's orbit is
+86,000 years; the period of conical revolution of the earth's axis is
+25,800 years. About 18,000 years ago the excentricity was at a maximum.
+
+
+
+
+LECTURE XI
+
+LAGRANGE AND LAPLACE--THE STABILITY OF THE SOLAR SYSTEM, AND THE NEBULAR
+HYPOTHESIS
+
+
+Laplace was the son of a small farmer or peasant of Normandy. His
+extraordinary ability was noticed by some wealthy neighbours, and by
+them he was sent to a good school. From that time his career was one
+brilliant success, until in the later years of his life his prominence
+brought him tangibly into contact with the deteriorating influence of
+politics. Perhaps one ought rather to say trying than deteriorating; for
+they seem trying to a strong character, deteriorating to a weak one--and
+unfortunately, Laplace must be classed in this latter category.
+
+It has always been the custom in France for its high scientific men to
+be conspicuous also in politics. It seems to be now becoming the fashion
+in this country also, I regret to say.
+
+The _life_ of Laplace is not specially interesting, and I shall not go
+into it. His brilliant mathematical genius is unquestionable, and almost
+unrivalled. He is, in fact, generally considered to come in this respect
+next after Newton. His talents were of a more popular order than those
+of Lagrange, and accordingly he acquired fame and rank, and rose to the
+highest dignities. Nevertheless, as a man and a politician he hardly
+commands our respect, and in time-serving adjustability he is comparable
+to the redoubtable Vicar of Bray. His scientific insight and genius
+were however unquestionably of the very highest order, and his work has
+been invaluable to astronomy.
+
+I will give a short sketch of some of his investigations, so far as they
+can be made intelligible without overmuch labour. He worked very much in
+conjunction with Lagrange, a more solid though a less brilliant man, and
+it is both impossible and unnecessary for us to attempt to apportion
+respective shares of credit between these two scientific giants, the
+greatest scientific men that France ever produced.
+
+First comes a research into the libration of the moon. This was
+discovered by Galileo in his old age at Arcetri, just before his
+blindness. The moon, as every one knows, keeps the same face to the
+earth as it revolves round it. In other words, it does not rotate with
+reference to the earth, though it does rotate with respect to outside
+bodies. Its libration consists in a sort of oscillation, whereby it
+shows us now a little more on one side, now a little more on the other,
+so that altogether we are cognizant of more than one-half of its
+surface--in fact, altogether of about three-fifths. It is a simple and
+unimportant matter, easily explained.
+
+     The motion of the moon may be analyzed into a rotation about its
+     own axis combined with a revolution about the earth. The speed of
+     the rotation is quite uniform, the speed of the revolution is not
+     quite uniform, because the orbit is not circular but elliptical,
+     and the moon has to travel faster in perigee than in apogee (in
+     accordance with Kepler's second law). The consequence of this is
+     that we see a little too far round the body of the moon, first on
+     one side, then on the other. Hence it _appears_ to oscillate
+     slightly, like a lop-sided fly-wheel whose revolutions have been
+     allowed to die away so that they end in oscillations of small
+     amplitude.[23] Its axis of rotation, too, is not precisely
+     perpendicular to its plane of revolution, and therefore we
+     sometimes see a few hundred miles beyond its north pole, sometimes
+     a similar amount beyond its south. Lastly, there is a sort of
+     parallax effect, owing to the fact that we see the rising moon from
+     one point of view, and the setting moon from a point 8,000 miles
+     distant; and this base-line of the earth's diameter gives us again
+     some extra glimpses. This diurnal or parallactic libration is
+     really more effective than the other two in extending our vision
+     into the space-facing hemisphere of the moon.
+
+     These simple matters may as well be understood, but there is
+     nothing in them to dwell upon. The far side of the moon is probably
+     but little worth seeing. Its features are likely to be more blurred
+     with accumulations of meteoric dust than are those of our side, but
+     otherwise they are likely to be of the same general character.
+
+The thing of real interest is the fact that the moon does turn the same
+face towards us; _i.e._ has ceased to rotate with respect to the earth
+(if ever it did so). The stability of this state of things was shown by
+Lagrange to depend on the shape of the moon. It must be slightly
+egg-shape, or prolate--extended in the direction of the earth; its
+earth-pointing diameter being a few hundred feet longer than its visible
+diameter; a cause slight enough, but nevertheless sufficient to maintain
+stability, except under the action of a distinct disturbing cause. The
+prolate or lemon-like shape is caused by the gravitative pull of the
+earth, balanced by the centrifugal whirl. The two forces balance each
+other as regards motion, but between them they have strained the moon a
+trifle out of shape. The moon has yielded as if it were perfectly
+plastic; in all probability it once was so.
+
+It may be interesting to note for a moment the correlative effect of
+this aspect of the moon, if we transfer ourselves to its surface in
+imagination, and look at the earth (cf. Fig. 41). The earth would be
+like a gigantic moon of four times our moon's diameter, and would go
+through its phases in regular order. But it would not rise or set: it
+would be fixed in the sky, and subject only to a minute oscillation to
+and fro once a month, by reason of the "libration" we have been speaking
+of. Its aspect, as seen by markings on its surface, would rapidly
+change, going through a cycle in twenty-four hours; but its permanent
+features would be usually masked by lawless accumulations of cloud,
+mainly aggregated in rude belts parallel to the equator. And these
+cloudy patches would be the most luminous, the whitest portions; for of
+course it would be their silver lining that we would then be looking
+on.[24]
+
+Next among the investigations of Lagrange and Laplace we will mention
+the long inequality of Jupiter and Saturn. Halley had found that Jupiter
+was continually lagging behind its true place as given by the theory of
+gravitation; and, on the other hand, that Saturn was being accelerated.
+The lag on the part of Jupiter amounted to about 34-1/2 minutes in a
+century. Overhauling ancient observations, however, Halley found signs
+of the opposite state of things, for when he got far enough back Jupiter
+was accelerated and Saturn was being retarded.
+
+Here was evidently a case of planetary perturbation, and Laplace and
+Lagrange undertook the working of it out. They attacked it as a case of
+the problem of three bodies, viz. the sun, Jupiter, and Saturn; which
+are so enormously the biggest of the known bodies in the system that
+insignificant masses like the Earth, Mars, and the rest, may be wholly
+neglected. They succeeded brilliantly, after a long and complex
+investigation: succeeded, not in solving the problem of the three
+bodies, but, by considering their mutual action as perturbations
+superposed on each other, in explaining the most conspicuous of the
+observed anomalies of their motion, and in laying the foundation of a
+general planetary theory.
+
+[Illustration: FIG. 79.--Shewing the three conjunction places in the
+orbits of Jupiter and Saturn. The two planets are represented as leaving
+one of the conjunctions where Jupiter was being pulled back and Saturn
+being pulled forward by their mutual attraction.]
+
+     One of the facts that plays a large part in the result was known to
+     the old astrologers, viz. that Jupiter and Saturn come into
+     conjunction with a certain triangular symmetry; the whole scheme
+     being called a trigon, and being mentioned several times by Kepler.
+     It happens that five of Jupiter's years very nearly equal two of
+     Saturn's,[25] so that they get very nearly into conjunction three
+     times in every five Jupiter years, but not exactly. The result of
+     this close approach is that periodically one pulls the other on and
+     is itself pulled back; but since the three points progress, it is
+     not always the same planet which gets pulled back. The complete
+     theory shows that in the year 1560 there was no marked
+     perturbation: before that it was in one direction, while afterwards
+     it was in the other direction, and the period of the whole cycle of
+     disturbances is 929 of our years. The solution of this long
+     outstanding puzzle by the theory of gravitation was hailed with the
+     greatest enthusiasm by astronomers, and it established the fame of
+     the two French mathematicians.
+
+Next they attacked the complicated problem of the motions of Jupiter's
+satellites. They succeeded in obtaining a theory of their motions which
+represented fact very nearly indeed, and they detected the following
+curious relationship between the satellites:--The speed of the first
+satellite + twice the speed of the second is equal to the speed of the
+third.
+
+They found this, not empirically, after the manner of Kepler, but as a
+deduction from the law of gravitation; for they go on to show that even
+if the satellites had not started with this relation they would sooner
+or later, by mutual perturbation, get themselves into it. One singular
+consequence of this, and of another quite similar connection between
+their positions, is that all three satellites can never be eclipsed at
+once.
+
+The motion of the fourth satellite is less tractable; it does not so
+readily form an easy system with the others.
+
+After these great successes the two astronomers naturally proceeded to
+study the mutual perturbations of all other bodies in the solar system.
+And one very remarkable discovery they made concerning the earth and
+moon, an account of which will be interesting, though the details and
+processes of calculation are quite beyond us in a course like this.
+
+Astronomical theory had become so nearly perfect by this time, and
+observations so accurate, that it was possible to calculate many
+astronomical events forwards or backwards, over even a thousand years or
+more, with admirable precision.
+
+Now, Halley had studied some records of ancient eclipses, and had
+calculated back by means of the lunar theory to see whether the
+calculation of the time they ought to occur would agree with the record
+of the time they did occur. To his surprise he found a discrepancy, not
+a large one, but still one quite noticeable. To state it as we know it
+now:--An eclipse a century ago happened twelve seconds later than it
+ought to have happened by theory; two centuries back the error amounted
+to forty-eight seconds, in three centuries it would be 108 seconds, and
+so on; the lag depending on the square of the time. By research, and
+help from scholars, he succeeded in obtaining the records of some very
+ancient eclipses indeed. One in Egypt towards the end of the tenth
+century A.D.; another in 201 A.D.; another a little before Christ; and
+one, the oldest of all of which any authentic record has been preserved,
+observed by the Chaldaean astronomers in Babylon in the reign of
+Hezekiah.
+
+Calculating back to this splendid old record of a solar eclipse, over
+the intervening 2,400 years, the calculated and the observed times were
+found to disagree by nearly two hours. Pondering over an explanation of
+the discrepancy, Halley guessed that it must be because the moon's
+motion was not uniform, it must be going quicker and quicker, gaining
+twelve seconds each century on its previous gain--a discovery announced
+by him as "the acceleration of the moon's mean motion." The month was
+constantly getting shorter.
+
+What was the physical cause of this acceleration according to the theory
+of gravitation? Many attacked the question, but all failed. This was the
+problem Laplace set himself to work out. A singular and beautiful result
+rewarded his efforts.
+
+You know that the earth describes an elliptic orbit round the sun: and
+that an ellipse is a circle with a certain amount of flattening or
+"excentricity."[26] Well, Laplace found that the excentricity of the
+earth's orbit must be changing, getting slightly less; and that this
+change of excentricity would have an effect upon the length of the
+month. It would make the moon go quicker.
+
+One can almost see how it comes about. A decrease in excentricity means
+an increase in mean distance of the earth from the sun. This means to
+the moon a less solar perturbation. Now one effect of the solar
+perturbation is to keep the moon's orbit extra large: if the size of its
+orbit diminishes, its velocity must increase, according to Kepler's
+third law.
+
+Laplace calculated the amount of acceleration so resulting, and found it
+ten seconds a century; very nearly what observation required; for,
+though I have quoted observation as demanding twelve seconds per
+century, the facts were not then so distinctly and definitely
+ascertained.
+
+This calculation for a long time seemed thoroughly satisfactory, but it
+is not the last word on the subject. Quite lately an error has been
+found in the working, which diminishes the theoretical
+gravitation-acceleration to six seconds a century instead of ten, thus
+making it insufficient to agree exactly with fact. The theory of
+gravitation leaves an outstanding error. (The point is now almost
+thoroughly understood, and we shall return to it in Lecture XVIII).
+
+But another question arises out of this discussion. I have spoken of the
+excentricity of the earth's orbit as decreasing. Was it always
+decreasing? and if so, how far back was it so excentric that at
+perihelion the earth passed quite near the sun? If it ever did thus pass
+near the sun, the inference is manifest--the earth must at one time have
+been thrown off, or been separated off, from the sun.
+
+If a projectile could be fired so fast that it described an orbit round
+the earth--and the speed of fire to attain this lies between five and
+seven miles a second (not less than the one, nor more than the
+other)--it would ever afterwards pass through its point of projection
+as one point of its elliptic orbit; and its periodic return through that
+point would be the sign of its origin. Similarly, if a satellite does
+_not_ come near its central orb, and can be shown never to have been
+near it, the natural inference is that it has _not_ been born from it,
+but has originated in some other way.
+
+The question which presented itself in connexion with the variable
+ellipticity of the earth's orbit was the following:--Had it always been
+decreasing, so that once it was excentric enough just to graze the sun
+at perihelion as a projected body would do?
+
+Into the problem thus presented Lagrange threw himself, and he succeeded
+in showing that no such explanation of the origin of the earth is
+possible. The excentricity of the orbit, though now decreasing, was not
+always decreasing; ages ago it was increasing: it passes through
+periodic changes. Eighteen thousand years ago its excentricity was a
+maximum; since then it has been diminishing, and will continue to
+diminish for 25,000 years more, when it will be an almost perfect
+circle; it will then begin to increase again, and so on. The obliquity
+of the ecliptic is also changing periodically, but not greatly: the
+change is less than three degrees.
+
+This research has, or ought to have, the most transcendent interest for
+geologists and geographers. You know that geologists find traces of
+extraordinary variations of temperature on the surface of the earth.
+England was at one time tropical, at another time glacial. Far away
+north, in Spitzbergen, evidence of the luxuriant vegetation of past ages
+has been found; and the explanation of these great climatic changes has
+long been a puzzle. Does not the secular variation in excentricity of
+the earth's orbit, combined with the precession of the equinoxes, afford
+a key? And if a key at all, it will be an accurate key, and enable us to
+calculate back with some precision to the date of the glacial epoch;
+and again to the time when a tropical flora flourished in what is now
+northern Europe, _i.e._ to the date of the Carboniferous era.
+
+This aspect of the subject has recently been taught with vigour and
+success by Dr. Croll in his book "Climate and Time."
+
+     A brief and partial explanation of the matter may be given, because
+     it is a point of some interest and is also one of fair simplicity.
+
+     Every one knows that the climatic conditions of winter and summer
+     are inverted in the two hemispheres, and that at present the sun is
+     nearest to us in our (northern) winter. In other words, the earth's
+     axis is inclined so as to tilt its north pole away from the sun at
+     perihelion, or when the earth is at the part of its elliptic orbit
+     nearest the sun's focus; and to tilt it towards the sun at
+     aphelion. The result of this present state of things is to diminish
+     the intensity of the average northern winter and of the average
+     northern summer, and on the other hand to aggravate the extremes of
+     temperature in the southern hemisphere; all other things being
+     equal. Of course other things are not equal, and the distribution
+     of land and sea is a still more powerful climatic agent than is the
+     three million miles or so extra nearness of the sun. But it is
+     supposed that the Antarctic ice-cap is larger than the northern,
+     and increased summer radiation with increased winter cold would
+     account for this.
+
+     But the present state of things did not always obtain. The conical
+     movement of the earth's axis (now known by a curious perversion of
+     phrase as "precession") will in the course of 13,000 years or so
+     cause the tilt to be precisely opposite, and then we shall have the
+     more extreme winters and summers instead of the southern
+     hemisphere.
+
+     If the change were to occur now, it might not be overpowering,
+     because now the excentricity is moderate. But if it happened some
+     time back, when the excentricity was much greater, a decidedly
+     different arrangement of climate may have resulted. There is no
+     need to say _if_ it happened some time back: it did happen, and
+     accordingly an agent for affecting the distribution of mean
+     temperature on the earth is to hand; though whether it is
+     sufficient to achieve all that has been observed by geologists is a
+     matter of opinion.
+
+     Once more, the whole diversity of the seasons depends on the tilt
+     of the earth's axis, the 23 deg. by which it is inclined to a
+     perpendicular to the orbital plane; and this obliquity or tilt is
+     subject to slow fluctuations. Hence there will come eras when all
+     causes combine to produce a maximum extremity of seasons in the
+     northern hemisphere, and other eras when it is the southern
+     hemisphere which is subject to extremes.
+
+But a grander problem still awaited solution--nothing less than the fate
+of the whole solar system. Here are a number of bodies of various sizes
+circulating at various rates round one central body, all attracted by
+it, and all attracting each other, the whole abandoned to the free play
+of the force of gravitation: what will be the end of it all? Will they
+ultimately approach and fall into the sun, or will they recede further
+and further from him, into the cold of space? There is a third possible
+alternative: may they not alternately approach and recede from him, so
+as on the whole to maintain a fair approximation to their present
+distances, without great and violent extremes of temperature either way?
+
+If any one planet of the system were to fall into the sun, more
+especially if it were a big one like Jupiter or Saturn, the heat
+produced would be so terrific that life on this earth would be
+destroyed, even at its present distance; so that we are personally
+interested in the behaviour of the other planets as well as in the
+behaviour of our own.
+
+The result of the portentously difficult and profoundly interesting
+investigation, here sketched in barest outline, is that the solar system
+is stable: that is to say, that if disturbed a little it will oscillate
+and return to its old state; whereas if it were unstable the slightest
+disturbance would tend to accumulate, and would sooner or later bring
+about a catastrophe. A hanging pendulum is stable, and oscillates about
+a mean position; its motion is periodic. A top-heavy load balanced on a
+point is unstable. All the changes of the solar system are periodic,
+_i.e._ they repeat themselves at regular intervals, and they never
+exceed a certain moderate amount.
+
+The period is something enormous. They will not have gone through all
+their changes until a period of 2,000,000 years has elapsed. This is
+the period of the planetary oscillation: "a great pendulum of eternity
+which beats ages as our pendulums beat seconds." Enormous it seems; and
+yet we have reason to believe that the earth has existed through many
+such periods.
+
+     The two laws of stability discovered and stated by Lagrange and
+     Laplace I can state, though they may be difficult to understand:--
+
+     Represent the masses of the several planets by m_1, m_2, &c.; their
+     mean distances from the sun (or radii vectores) by r_1, r_2, &c.;
+     the excentricities of their orbits by e_1, e_2, &c.; and the
+     obliquity of the planes of these orbits, reckoned from a single
+     plane of reference or "invariable plane," by [theta]_1, [theta]_2,
+     &c.; then all these quantities (except m) are liable to
+     fluctuate; but, however much they change, an increase for one
+     planet will be accompanied by a decrease for some others; so that,
+     taking all the planets into account, the sum of a set of terms like
+     these, m_1e_1^2 [square root]r_1 + m_2e_2^2 [square root]r_2
+     + &c., will remain always the same. This is summed up briefly in
+     the following statement:
+
+  [Sigma](me^2 [square root]r) = constant.
+
+     That is one law, and the other is like it, but with inclination of
+     orbit instead of excentricity, viz.:
+
+  [Sigma](m[theta]^2 [square root]r) = constant.
+
+     The value of each of these two constants can at any time be
+     calculated. At present their values are small. Hence they always
+     were and always will be small; being, in fact, invariable. Hence
+     neither _e_ nor _r_ nor [theta] can ever become infinite, nor can
+     their average value for the system ever become zero.
+
+The planets may share the given amount of total excentricity and
+obliquity in various proportions between themselves; but even if it were
+all piled on to one planet it would not be very excessive, unless the
+planet were so small a one as Mercury; and it would be most improbable
+that one planet should ever have all the excentricity of the solar
+system heaped upon itself. The earth, therefore, never has been, nor
+ever will be, enormously nearer the sun than it is at present: nor can
+it ever get very much further off. Its changes are small and are
+periodic--an increase is followed by a decrease, like the swing of a
+pendulum.
+
+The above two laws have been called the Magna Charta of the solar
+system, and were long supposed to guarantee its absolute permanence. So
+far as the theory of gravitation carries us, they do guarantee its
+permanence; but something more remains to be said on the subject in a
+future lecture (XVIII).
+
+And now, finally, we come to a sublime speculation, thrown out by
+Laplace, not as the result of profound calculation, like the results
+hitherto mentioned, not following certainly from the theory of
+gravitation, or from any other known theory, and therefore not to be
+accepted as more than a brilliant hypothesis, to be confirmed or
+rejected as our knowledge extends. This speculation is the "Nebular
+hypothesis." Since the time of Laplace the nebular hypothesis has had
+ups and downs of credence, sometimes being largely believed in,
+sometimes being almost ignored. At the present time it holds the field
+with perhaps greater probability of ultimate triumph than has ever
+before seemed to belong to it--far greater than belonged to it when
+first propounded.
+
+It had been previously stated clearly and well by the philosopher Kant,
+who was intensely interested in "the starry heavens" as well as in the
+"mind of man," and who shewed in connexion with astronomy also a most
+surprising genius. The hypothesis ought by rights perhaps to be known
+rather by his name than by that of Laplace.
+
+The data on which it was founded are these:--Every motion in the solar
+system known at that time took place in one direction, and in one
+direction only. Thus the planets revolve round the sun, all going the
+same way round; moons revolve round the planets, still maintaining the
+same direction of rotation, and all the bodies that were known to rotate
+on their own axis did so with still the same kind of spin. Moreover,
+all these motions take place in or near a single plane. The ancients
+knew that sun moon and planets all keep near to the ecliptic, within a
+belt known as the zodiac: none strays away into other parts of the sky.
+Satellites also, and rings, are arranged in or near the same plane; and
+the plane of diurnal spin, or equator of the different bodies, is but
+slightly tilted.
+
+Now all this could not be the result of chance. What could have caused
+it? Is there any connection or common ancestry possible, to account for
+this strange family likeness? There is no connection now, but there may
+have been once. Must have been, we may almost say. It is as though they
+had once been parts of one great mass rotating as a whole; for if such a
+rotating mass broke up, its parts would retain its direction of
+rotation. But such a mass, filling all space as far as or beyond Saturn,
+although containing the materials of the whole solar system in itself,
+must have been of very rare consistency. Occupying so much bulk it could
+not have been solid, nor yet liquid, but it might have been gaseous.
+
+Are there any such gigantic rotating masses of gas in the heaven now?
+Certainly there are; there are the nebulae. Some of the nebulae are now
+known to be gaseous, and some of them at least are in a state of
+rotation. Laplace could not have known this for certain, but he
+suspected it. The first distinctly spiral nebula was discovered by the
+telescope of Lord Rosse; and quite recently a splendid photograph of the
+great Andromeda nebula, by our townsman, Mr. Isaac Roberts, reveals what
+was quite unsuspected--and makes it clear that this prodigious mass also
+is in a state of extensive and majestic whirl.
+
+Very well, then, put this problem:--A vast mass of rotating gas is left
+to itself to cool for ages and to condense as it cools: how will it
+behave? A difficult mathematical problem, worthy of being attacked
+to-day; not yet at all adequately treated. There are those who believe
+that by the complete treatment of such a problem all the history of the
+solar system could be evolved.
+
+[Illustration: FIG. 80.--Lord Rosse's drawing of the spiral nebula in
+Canes Venatici, with the stub marks of the draughtsman unduly emphasised
+into features by the engraver.]
+
+Laplace pictured to himself this mass shrinking and thereby whirling
+more and more rapidly. A spinning body shrinking in size and retaining
+its original amount of rotation, as it will unless a brake is applied,
+must spin more and more rapidly as it shrinks. It has what
+mathematicians call a constant moment of momentum; and what it loses in
+leverage, as it shrinks, it gains in speed. The mass is held together by
+gravitation, every particle attracting every other particle; but since
+all the particles are describing curved paths, they will tend to fly off
+tangentially, and only a small excess of the gravitation force over the
+centrifugal is left to pull the particles in, and slowly to concentrate
+the nebula. The mutual gravitation of the parts is opposed by the
+centrifugal force of the whirl. At length a point is reached where the
+two forces balance. A portion outside a certain line will be in
+equilibrium; it will be left behind, and the rest must contract without
+it. A ring is formed, and away goes the inner nucleus contracting
+further and further towards a centre. After a time another ring will be
+left behind in the same way, and so on. What happens to these rings?
+They rotate with the motion they possess when thrown or shrunk off; but
+will they remain rings? If perfectly regular they may; if there be any
+irregularity they are liable to break up. They will break into one or
+two or more large masses, which are ultimately very likely to collide
+and become one. The revolving body so formed is still a rotating gaseous
+mass; and it will go on shrinking and cooling and throwing off rings,
+like the larger nucleus by which it has been abandoned. As any nucleus
+gets smaller, its rate of rotation increases, and so the rings last
+thrown off will be spinning faster than those thrown off earliest. The
+final nucleus or residual central body will be rotating fastest of all.
+
+The nucleus of the whole original mass we now see shrunk up into what we
+call the sun, which is spinning on its axis once every twenty-five days.
+The rings successively thrown off by it are now the planets--some large,
+some small--those last thrown off rotating round him comparatively
+quickly, those outside much more slowly. The rings thrown off by the
+planetary gaseous masses as they contracted have now become satellites;
+except one ring which has remained without breaking up, and is to be
+seen rotating round Saturn still.
+
+One other similar ring, an abortive attempt at a planet, is also left
+round the sun (the zone of asteroids).
+
+Such, crudely and baldly, is the famous nebular hypothesis of Laplace.
+It was first stated, as has been said above, by the philosopher Kant,
+but it was elaborated into much fuller detail by the greatest of French
+mathematicians and astronomers.
+
+The contracting masses will condense and generate great quantities of
+heat by their own shrinkage; they will at a certain stage condense to
+liquid, and after a time will begin to cool and congeal with a
+superficial crust, which will get thicker and thicker; but for ages they
+will remain hot, even after they have become thoroughly solid. The small
+ones will cool fastest; the big ones will retain their heat for an
+immense time. Bullets cool quickly, cannon-balls take hours or days to
+cool, planets take millions of years. Our moon may be nearly cold, but
+the earth is still warm--indeed, very hot inside. Jupiter is believed by
+some observers still to glow with a dull red heat; and the high
+temperature of the much larger and still liquid mass of the sun is
+apparent to everybody. Not till it begins to scum over will it be
+perceptibly cooler.
+
+[Illustration: FIG. 81.--Saturn.]
+
+Many things are now known concerning heat which were not known to
+Laplace (in the above paragraph they are only hinted at), and these
+confirm and strengthen the general features of his hypothesis in a
+striking way; so do the most recent telescopic discoveries. But fresh
+possibilities have now occurred to us, tidal phenomena are seen to have
+an influence then wholly unsuspected, and it will be in a modified and
+amplified form that the philosopher of next century will still hold to
+the main features of this famous old Nebular Hypothesis respecting the
+origin of the sun and planets--the Evolution of the solar system.
+
+
+
+
+NOTES TO LECTURE XII
+
+
+The subject of stellar astronomy was first opened up by Sir William
+Herschel, the greatest observing astronomer.
+
+_Frederick William Herschel_ was born in Hanover in 1738, and brought up
+as a musician. Came to England in 1756. First saw a telescope in 1773.
+Made a great many himself, and began a survey of the heavens. His sister
+Caroline, born in 1750, came to England in 1772, and became his devoted
+assistant to the end of his life. Uranus discovered in 1781. Music
+finally abandoned next year, and the 40-foot telescope begun. Discovered
+two moons of Saturn and two of Uranus. Reviewed, described, and gauged
+all the visible heavens. Discovered and catalogued 2,500 nebulae and 806
+double stars. Speculated concerning the Milky Way, the nebulosity of
+stars, the origin and growth of solar systems. Discovered that the stars
+were in motion, not fixed, and that the sun as one of them was
+journeying towards a point in the constellation Hercules. Died in 1822,
+eighty-four years old. Caroline Herschel discovered eight comets, and
+lived on to the age of ninety-eight.
+
+
+
+
+LECTURE XII
+
+HERSCHEL AND THE MOTION OF THE FIXED STARS
+
+
+We may admit, I think, that, with a few notable exceptions, the work of
+the great men we have been recently considering was rather to complete
+and round off the work of Newton, than to strike out new and original
+lines.
+
+This was the whole tendency of eighteenth century astronomy. It appeared
+to be getting into an adult and uninteresting stage, wherein everything
+could be calculated and predicted. Labour and ingenuity, and a severe
+mathematical training, were necessary to work out the remote
+consequences of known laws, but nothing fresh seemed likely to turn up.
+Consequently men's minds began turning in other directions, and we find
+chemistry and optics largely studied by some of the greatest minds,
+instead of astronomy.
+
+But before the century closed there was destined to arise one remarkable
+exception--a man who was comparatively ignorant of that which had been
+done before--a man unversed in mathematics and the intricacies of
+science, but who possessed such a real and genuine enthusiasm and love
+of Nature that he overcame the force of adverse circumstances, and
+entering the territory of astronomy by a by-path, struck out a new line
+for himself, and infused into the science a healthy spirit of fresh life
+and activity.
+
+This man was William Herschel.
+
+"The rise of Herschel," says Miss Clerke, "is the one conspicuous
+anomaly in the otherwise somewhat quiet and prosy eighteenth century. It
+proved decisive of the course of events in the nineteenth. It was
+unexplained by anything that had gone before, yet all that came after
+hinged upon it. It gave a new direction to effort; it lent a fresh
+impulse to thought. It opened a channel for the widespread public
+interest which was gathering towards astronomical subjects to flow in."
+
+Herschel was born at Hanover in 1738, the son of an oboe player in a
+military regiment. The father was a good musician, and a cultivated man.
+The mother was a German _Frau_ of the period, a strong, active,
+business-like woman, of strong character and profound ignorance. Herself
+unable to write, she set her face against learning and all new-fangled
+notions. The education of the sons she could not altogether control,
+though she lamented over it, but the education of her two daughters she
+strictly limited to cooking, sewing, and household management. These,
+however, she taught them well.
+
+It was a large family, and William was the fourth child. We need only
+remember the names of his younger brother Alexander, and of his much
+younger sister Caroline.
+
+They were all very musical--the youngest boy was once raised upon a
+table to play the violin at a public performance. The girls were
+forbidden to learn music by their mother, but their father sometimes
+taught them a little on the sly. Alexander was besides an ingenious
+mechanician.
+
+At the age of seventeen, William became oboist to the Hanoverian Guards,
+shortly before the regiment was ordered to England. Two years later he
+removed himself from the regiment, with the approval of his parents,
+though probably without the approbation or consent of the commanding
+officer, by whom such removal would be regarded as simple desertion,
+which indeed it was; and George III. long afterwards handed him an
+official pardon for it.
+
+At the age of nineteen, he was thus launched in England with an outfit
+of some French, Latin, and English, picked up by himself; some skill in
+playing the hautboy, the violin, and the organ, as taught by his father;
+and some good linen and clothing, and an immense stock of energy,
+provided by his mother.
+
+He lived as musical instructor to one or two militia bands in Yorkshire,
+and for three years we hear no more than this of him. But, at the end of
+that time, a noted organist, Dr. Miller, of Durham, who had heard his
+playing, proposed that he should come and live with him and play at
+concerts, which he was very glad to do. He next obtained the post of
+organist at Halifax; and some four or five years later he was invited to
+become organist at the Octagon Chapel in Bath, and soon led the musical
+life of that then very fashionable place.
+
+About this time he went on a short visit to his family at Hanover, by
+all of whom he was very much beloved, especially by his young sister
+Caroline, who always regarded him as specially her own brother. It is
+rather pitiful, however, to find that her domestic occupations still
+unfairly repressed and blighted her life. She says:--
+
+     "Of the joys and pleasures which all felt at this long-wished-for
+     meeting with my--let me say my dearest--brother, but a small
+     portion could fall to my share; for with my constant attendance at
+     church and school, besides the time I was employed in doing the
+     drudgery of the scullery, it was but seldom I could make one in the
+     group when the family were assembled together."
+
+While at Bath he wrote many musical pieces--glees, anthems, chants,
+pieces for the harp, and an orchestral symphony. He taught a large
+number of pupils, and lived a hard and successful life. After fourteen
+hours or so spent in teaching and playing, he would retire at night to
+instruct his mind with a study of mathematics, optics, Italian, or
+Greek, in all of which he managed to make some progress. He also about
+this time fell in with some book on astronomy.
+
+In 1763 his father was struck with paralysis, and two years later he
+died.
+
+William then proposed that Alexander should come over from Hanover and
+join him at Bath, which was done. Next they wanted to rescue their
+sister Caroline from her humdrum existence, but this was a more
+difficult matter. Caroline's journal gives an account of her life at
+this time that is instructive. Here are a few extracts from it:--
+
+     "My father wished to give me something like a polished education,
+     but my mother was particularly determined that it should be a
+     rough, but at the same time a useful one; and nothing further she
+     thought was necessary but to send me two or three months to a
+     sempstress to be taught to make household linen....
+
+     "My mother would not consent to my being taught French, ... so all
+     my father could do for me was to indulge me (and please himself)
+     sometimes with a short lesson on the violin, when my mother was
+     either in good humour or out of the way.... She had cause for
+     wishing me not to know more than was necessary for being useful in
+     the family; for it was her certain belief that my brother William
+     would have returned to his country, and my eldest brother not have
+     looked so high, if they had had a little less learning."
+
+However, seven years after the death of their father, William went over
+to Germany and returned to England in triumph, bringing Caroline with
+him: she being then twenty-two.
+
+So now began a busy life in Bath. For Caroline the work must have been
+tremendous. For, besides having to learn singing, she had to learn
+English. She had, moreover, to keep accounts and do the marketing.
+
+When the season at Bath was over, she hoped to get rather more of her
+brother William's society; but he was deep in optics and astronomy, used
+to sleep with the books under his pillow, read them during meals, and
+scarcely ever thought of anything else.
+
+He was determined to see for himself all the astronomical wonders; and
+there being a small Gregorian reflector in one of the shops, he hired
+it. But he was not satisfied with this, and contemplated making a
+telescope 20 feet long. He wrote to opticians inquiring the price of a
+mirror suitable, but found there were none so large, and that even the
+smaller ones were beyond his means. Nothing daunted, he determined to
+make some for himself. Alexander entered into his plans: tools, hones,
+polishers, and all sorts of rubbish were imported into the house, to the
+sister's dismay, who says:--
+
+[Illustration: FIG. 82.--Principle of Newtonian reflector.]
+
+     "And then, to my sorrow, I saw almost every room turned into a
+     workshop. A cabinet-maker making a tube and stands of all
+     descriptions in a handsomely furnished drawing-room; Alex. putting
+     up a huge turning-machine (which he had brought in the autumn from
+     Bristol, where he used to spend the summer) in a bed-room, for
+     turning patterns, grinding glasses, and turning eye-pieces, &c. At
+     the same time music durst not lie entirely dormant during the
+     summer, and my brother had frequent rehearsals at home."
+
+Finally, in 1774, at the age of thirty-six, he had made himself a
+5-1/2-foot telescope, and began to view the heavens. So attached was he
+to the instrument that he would run from the concert-room between the
+parts, and take a look at the stars.
+
+He soon began another telescope, and then another. He must have made
+some dozen different telescopes, always trying to get them bigger and
+bigger; at last he got a 7-foot and then a 10-foot instrument, and began
+a systematic survey of the heavens; he also began to communicate his
+results to the Royal Society.
+
+He now took a larger house, with more room for workshops, and a grass
+plot for a 20-foot telescope, and still he went on grinding
+mirrors--literally hundreds of them.
+
+I read another extract from the diary of his sister, who waited on him
+and obeyed him like a spaniel:--
+
+     "My time was taken up with copying music and practising, besides
+     attendance on my brother when polishing, since by way of keeping
+     him alive I was constantly obliged to feed him by putting the
+     victuals by bits into his mouth. This was once the case when, in
+     order to finish a 7-foot mirror, he had not taken his hands from it
+     for sixteen hours together. In general he was never unemployed at
+     meals, but was always at those times contriving or making drawings
+     of whatever came in his mind. Generally I was obliged to read to
+     him whilst he was at the turning-lathe, or polishing mirrors--_Don
+     Quixote_, _Arabian Nights' Entertainments_, the novels of Sterne,
+     Fielding, &c.; serving tea and supper without interrupting the work
+     with which he was engaged, ... and sometimes lending a hand. I
+     became, in time, as useful a member of the workshop as a boy might
+     be to his master in the first year of his apprenticeship.... But as
+     I was to take a part the next year in the oratorios, I had, for a
+     whole twelvemonth, two lessons per week from Miss Fleming, the
+     celebrated dancing-mistress, to drill me for a gentlewoman (God
+     knows how she succeeded). So we lived on without interruption. My
+     brother Alex. was absent from Bath for some months every summer,
+     but when at home he took much pleasure in executing some turning or
+     clockmaker's work for his brother."
+
+The music, and the astronomy, and the making of telescopes, all went on
+together, each at high pressure, and enough done in each to satisfy any
+ordinary activity. But the Herschels knew no rest. Grinding mirrors by
+day, concerts and oratorios in the evening, star-gazing at night. It is
+strange his health could stand it.
+
+The star-gazing, moreover, was no _dilettante_ work; it was based on a
+serious system--a well thought out plan of observation. It was nothing
+less than this--to pass the whole heavens steadily and in order through
+the telescope, noting and describing and recording every object that
+should be visible, whether previously known or unknown. The operation is
+called sweeping; but it is not a rapid passage from one object to
+another, as the term might suggest; it is a most tedious business, and
+consists in following with the telescope a certain field of view for
+some minutes, so as to be sure that nothing is missed, then shifting it
+to the next overlapping field, and watching again. And whatever object
+appears must be scrutinized anxiously to see what there is peculiar
+about it. If a star, it may be double, or it may be coloured, or it may
+be nebulous; or again it may be variable, and so its brightness must be
+estimated in order to compare with a subsequent observation.
+
+Four distinct times in his life did Herschel thus pass the whole visible
+heavens under review; and each survey occupied him several years. He
+discovered double stars, variable stars, nebulae, and comets; and Mr.
+William Herschel, of Bath, the amateur astronomer, was gradually
+emerging from his obscurity, and becoming a known man.
+
+Tuesday, the 13th of March, 1781, is a date memorable in the annals of
+astronomy. "On this night," he writes to the Royal Society, "in
+examining the small stars near _[eta]_ Geminorum, I perceived one
+visibly larger than the rest. Struck with its uncommon appearance, I
+compared it to _[eta]_ Geminorum and another star, and finding it so
+much larger than either, I suspected it to be a comet."
+
+The "comet" was immediately observed by professional astronomers, and
+its orbit was computed by some of them. It was thus found to move in
+nearly a circle instead of an elongated ellipse, and to be nearly twice
+as far from the sun as Saturn. It was no comet, it was a new planet;
+more than 100 times as big as the earth, and nearly twice as far away as
+Saturn. It was presently christened "Uranus."
+
+This was a most striking discovery, and the news sped over Europe. To
+understand the interest it excited we must remember that such a
+discovery was unique. Since the most ancient times of which men had any
+knowledge, the planets Mercury, Venus, Mars, Jupiter, Saturn, had been
+known, and there had been no addition to their number. Galileo and
+others had discovered satellites indeed, but a new primary planet was an
+entire and utterly unsuspected novelty.
+
+One of the most immediate consequences of the event was the discovery of
+Herschel himself. The Royal Society made him a Fellow the same year. The
+University of Oxford dubbed him a doctor; and the King sent for him to
+bring his telescope and show it at Court. So to London and Windsor he
+went, taking with him his best telescope. Maskelyne, the then
+Astronomer-Royal, compared it with the National one at Greenwich, and
+found Herschel's home-made instrument far the better of the two. He had
+a stand made after Herschel's pattern, but was so disgusted with his own
+instrument now that he scarcely thought it worthy of the stand when it
+was made. At Windsor, George III. was very civil, and Mr. Herschel was
+in great request to show the ladies of the Court Saturn and other
+objects of interest. Mr. Herschel exhibited a piece of worldly wisdom
+under these circumstances, that recalls faintly the behaviour of Tycho
+Brahe under similar circumstances. The evening when the exhibition was
+to take place threatened to become cloudy and wet, so Herschel rigged up
+an artificial Saturn, constructed of card and tissue paper, with a lamp
+behind it, in the distant wall of a garden; and, when the time came, his
+new titled friends were regaled with a view of this imitation Saturn
+through the telescope--the real one not being visible. They went away
+much pleased.
+
+He stayed hovering between Windsor and Greenwich, and uncertain what was
+to be the outcome of all this regal patronizing. He writes to his sister
+that he would much rather be back grinding mirrors at Bath. And she
+writes begging him to come, for his musical pupils were getting
+impatient. They had to get the better of their impatience, however, for
+the King ultimately appointed him astronomer or rather telescope-maker
+to himself, and so Caroline and the whole household were sent for, and
+established in a small house at Datchet.
+
+From being a star-gazing musician, Herschel thus became a practical
+astronomer. Henceforth he lived in his observatory; only on wet and
+moonlight nights could he be torn away from it. The day-time he devoted
+to making his long-contemplated 20-foot telescope.
+
+Not yet, however, were all their difficulties removed. The house at
+Datchet was a tumble-down barn of a place, chosen rather as a workshop
+and observatory than as a dwelling-house. And the salary allowed him by
+George III. was scarcely a princely one. It was, as a matter of fact,
+L200 a year. The idea was that he would earn his living by making
+telescopes, and so indeed he did. He made altogether some hundreds.
+Among others, four for the King. But this eternal making of telescopes
+for other people to use or play with was a weariness to the flesh. What
+he wanted was to observe, observe, observe.
+
+Sir William Watson, an old friend of his, and of some influence at
+Court, expressed his mind pretty plainly concerning Herschel's position;
+and as soon as the King got to understand that there was anything the
+matter, he immediately offered L2,000 for a gigantic telescope to be
+made for Herschel's own use. Nothing better did he want in life. The
+whole army of carpenters and craftsmen resident in Datchet were pressed
+into the service. Furnaces for the speculum metal were built, stands
+erected, and the 40-foot telescope fairly begun. It cost L4,000 before
+it was finished, but the King paid the whole.
+
+[Illustration: FIG. 83.--Herschel's 40-foot telescope.]
+
+With it he discovered two more satellites to Saturn (five hitherto had
+been known), and two moons to his own planet Uranus. These two are now
+known as Oberon and Titania. They were not seen again till some forty
+years after, when his son, Sir John Herschel, reobserved them. And in
+1847, Mr. Lassell, at his house, "Starfield," near Liverpool, discovered
+two more, called Ariel and Umbriel, making the number four, as now
+known. Mr. Lassell also discovered, with a telescope of his own making,
+an eighth satellite of Saturn--Hyperion--and a satellite to Neptune.
+
+A letter from a foreign astronomer about this period describes Herschel
+and his sister's method of work:--
+
+     "I spent the night of the 6th of January at Herschel's, in Datchet,
+     near Windsor, and had the good luck to hit on a fine evening. He
+     has his 20-foot Newtonian telescope in the open air, and mounted in
+     his garden very simply and conveniently. It is moved by an
+     assistant, who stands below it.... Near the instrument is a clock
+     regulated to sidereal time.... In the room near it sits Herschel's
+     sister, and she has Flamsteed's atlas open before her. As he gives
+     her the word, she writes down the declination and right ascension,
+     and the other circumstances of the observation. In this way
+     Herschel examines the whole sky without omitting the least part. He
+     commonly observes with a magnifying power of one hundred and fifty,
+     and is sure that after four or five years he will have passed in
+     review every object above our horizon. He showed me the book in
+     which his observations up to this time are written, and I am
+     astonished at the great number of them. Each sweep covers 2 deg. 15' in
+     declination, and he lets each star pass at least three times
+     through the field of his telescope, so that it is impossible that
+     anything can escape him. He has already found about 900 double
+     stars, and almost as many nebulae. I went to bed about one o'clock,
+     and up to that time he had found that night four or five new
+     nebulae. The thermometer in the garden stood at 13 deg. Fahrenheit; but,
+     in spite of this, Herschel observes the whole night through, except
+     that he stops every three or four hours and goes into the room for
+     a few moments. For some years Herschel has observed the heavens
+     every hour when the weather is clear, and this always in the open
+     air, because he says that the telescope only performs well when it
+     is at the same temperature as the air. He protects himself against
+     the weather by putting on more clothing. He has an excellent
+     constitution, and thinks about nothing else in the world but the
+     celestial bodies. He has promised me in the most cordial way,
+     entirely in the service of astronomy, and without thinking of his
+     own interest, to see to the telescopes I have ordered for European
+     observatories, and he will himself attend to the preparation of the
+     mirrors."
+
+[Illustration: _Painted by Abbott._
+
+_Engraved by Ryder._
+
+FIG. 84.--WILLIAM HERSCHEL.
+
+_From an Original Picture in the Possession of_ WM. WATSON, M.D.,
+F.R.S.]
+
+In 1783, Herschel married an estimable lady who sympathized with his
+pursuits. She was the only daughter of a City magnate, so his pecuniary
+difficulties, such as they were (they were never very troublesome to
+him), came to an end. They moved now into a more commodious house at
+Slough. Their one son, afterwards the famous Sir John Herschel, was
+born some nine years later. But the marriage was rather a blow to his
+devoted sister: henceforth she lived in lodgings, and went over at
+night-time to help him observe. For it must be remarked that this family
+literally turned night into day. Whatever sleep they got was in the
+day-time. Every fine night without exception was spent in observing: and
+the quite incredible fierceness of the pursuit is illustrated, as
+strongly as it can be, by the following sentence out of Caroline's
+diary, at the time of the move from Datchet to Slough: "The last night
+at Datchet was spent in sweeping till daylight, and by the next evening
+the telescope stood ready for observation at Slough."
+
+Caroline was now often allowed to sweep with a small telescope on her
+own account. In this way she picked up a good many nebulae in the course
+of her life, and eight comets, four of which were quite new, and one of
+which, known since as Encke's comet, has become very famous.
+
+The work they got through between them is something astonishing. He made
+with his own hands 430 parabolic mirrors for reflecting telescopes,
+besides a great number of complete instruments. He was forty-two when he
+began contributing to the Royal Society; yet before he died he had sent
+them sixty-nine long and elaborate treatises. One of these memoirs is a
+catalogue of 1000 nebulae. Fifteen years after he sends in another 1000;
+and some years later another 500. He also discovered 806 double stars,
+which he proved were really corrected from the fact that they revolved
+round each other (p. 309). He lived to see some of them perform half a
+revolution. For him the stars were not fixed: they moved slowly among
+themselves. He detected their proper motions. He passed the whole
+northern firmament in review four distinct times; counted the stars in
+3,400 gauge-fields, and estimated the brightness of hundreds of stars.
+He also measured as accurately as he could their proper motions,
+devising for this purpose the method which still to this day remains in
+use.
+
+And what is the outcome of it all? It is not Uranus, nor the satellites,
+nor even the double stars and the nebulae considered as mere objects: it
+is the beginning of a science of the stars.
+
+[Illustration: FIG. 85.--CAROLINE HERSCHEL.
+
+_From a Drawing from Life, by_ GEORGE MUeLLER, 1847.]
+
+Hitherto the stars had only been observed for nautical and practical
+purposes. Their times of rising and southing and setting had been noted;
+they had been treated as a clock or piece of dead mechanism, and as
+fixed points of reference. All the energies of astronomers had gone out
+towards the solar system. It was the planets that had been observed.
+Tycho had observed and tabulated their positions. Kepler had found out
+some laws of their motion. Galileo had discovered their peculiarities
+and attendants. Newton and Laplace had perceived every detail of their
+laws.
+
+But for the stars--the old Ptolemaic system might still have been true.
+They might still be mere dots in a vast crystalline sphere, all set at
+about one distance, and subservient to the uses of the earth.
+
+Herschel changed all this. Instead of sameness, he found variety;
+instead of uniformity of distance, limitless and utterly limitless
+fields and boundless distances; instead of rest and quiescence, motion
+and activity; instead of stagnation, life.
+
+[Illustration: FIG. 86.--The double-double star [epsilon] Lyrae as seen
+under three different powers.]
+
+Yes, that is what Herschel discovered--the life and activity of the
+whole visible universe. No longer was our little solar system to be the
+one object of regard, no longer were its phenomena to be alone
+interesting to man. With Herschel every star was a solar system. And
+more than that: he found suns revolving round suns, at distances such as
+the mind reels at, still obeying the same law of gravitation as pulls an
+apple from a tree. He tried hard to estimate the distance of the stars
+from the earth, but there he failed: it was too hopeless a problem. It
+was solved some time after his death by Bessel, and the distances of
+many stars are now known but these distances are awful and unspeakable.
+Our distance from the sun shrinks up into a mere speck--the whole solar
+system into a mere unit of measurement, to be repeated hundreds of
+thousands of times before we reach the stars.
+
+Yet their motion is visible--yes, to very accurate measurement quite
+plain. One star, known as 61 Cygni, was then and is now rushing along at
+the rate of 100 miles every second. Not that you must imagine that this
+makes any obvious and apparent change in its position. No, for all
+ordinary and practical purposes they are still fixed stars; thousands of
+years will show us no obvious change; "Adam" saw precisely the same
+constellations as we do: it is only by refined micrometric measurement
+with high magnifying power that their flight can be detected.
+
+But the sun is one of the stars--not by any means a specially large or
+bright one; Sirius we now know to be twenty times as big as the sun. The
+sun is one of the stars: then is it at rest? Herschel asked this
+question and endeavoured to answer it. He succeeded in the most
+astonishing manner. It is, perhaps, his most remarkable discovery, and
+savours of intuition. This is how it happened. With imperfect optical
+means and his own eyesight to guide him, he considered and pondered over
+the proper motion of the stars as he had observed it, till he discovered
+a kind of uniformity running through it all. Mixed up with
+irregularities and individualities, he found that in a certain part of
+the heavens the stars were on the whole opening out--separating slowly
+from each other; on the opposite side of the heavens they were on the
+average closing up--getting slightly nearer to each other; while in
+directions at right angles to this they were fairly preserving their
+customary distances asunder.
+
+Now, what is the moral to be drawn from such uniformity of behaviour
+among unconnected bodies? Surely that this part of their motion is only
+apparent--that it is we who are moving. Travelling over a prairie
+bounded by a belt of trees, we should see the trees in our line of
+advance opening out, and those behind closing up; we should see in fact
+the same kind of apparent motion as Herschel was able to detect among
+the stars: the opening out being most marked near the constellation
+Hercules. The conclusion is obvious: the sun, with all its planets, must
+be steadily moving towards a point in the constellation Hercules. The
+most accurate modern research has been hardly able to improve upon this
+statement of Herschel's. Possibly the solar system may ultimately be
+found to revolve round some other body, but what that is no one knows.
+All one can tell is the present direction of the majestic motion: since
+it was discovered it has continued unchanged, and will probably so
+continue for thousands of years.
+
+[Illustration: FIG. 87.--Old drawing of the cluster in Hercules.]
+
+And, finally, concerning the nebulae. These mysterious objects exercised
+a strong fascination for Herschel, and many are the speculations he
+indulges in concerning them. At one time he regards them all as clusters
+of stars, and the Milky Way as our cluster; the others he regards as
+other universes almost infinitely distant; and he proceeds to gauge and
+estimate the shape of our own universe or galaxy of suns, the Milky Way.
+
+Later on, however, he pictures to himself the nebulae as nascent suns:
+solar systems before they are formed. Some he thinks have begun to
+aggregate, while some are still glowing gas.
+
+[Illustration: FIG. 88.--Old drawing of the Andromeda nebula.]
+
+He likens the heavens to a garden in which there are plants growing in
+all manner of different stages: some shooting, some in leaf, some in
+flower, some bearing seed, some decaying; and thus at one inspection we
+have before us the whole life-history of the plant.
+
+Just so he thinks the heavens contain worlds, some old, some dead, some
+young and vigorous, and some in the act of being formed. The nebulae are
+these latter, and the nebulous stars are a further stage in the
+condensation towards a sun.
+
+And thus, by simple observation, he is led towards something very like
+the nebular hypothesis of Laplace; and his position, whether it be true
+or false, is substantially the same as is held to-day.
+
+[Illustration: FIG. 89.--The great nebula in Orion.]
+
+We _know_ now that many of the nebulae consist of innumerable isolated
+particles and may be spoken of as gas. We know that some are in a state
+of whirling motion. We know also that such gas left to itself will
+slowly as it cools condense and shrink, so as to form a central solid
+nucleus; and also, if it were in whirling motion, that it would send off
+rings from itself, and that these rings could break up into planets. In
+two familiar cases the ring has not yet thus aggregated into planet or
+satellite--the zone of asteroids, and Saturn's ring.
+
+The whole of this could not have been asserted in Herschel's time: for
+further information the world had to wait.
+
+These are the problems of modern astronomy--these and many others, which
+are the growth of this century, aye, and the growth of the last thirty
+or forty, and indeed of the last ten years. Even as I write, new and
+very confirmatory discoveries are being announced. The Milky Way _does_
+seem to have some affinity with our sun. And the chief stars of the
+constellation of Orion constitute another family, and are enveloped in
+the great nebula, now by photography perceived to be far greater than
+had ever been imagined.
+
+What is to be the outcome of it all I know not; but sure I am of this,
+that the largest views of the universe that we are able to frame, and
+the grandest manner of its construction that we can conceive, are
+certain to pale and shrink and become inadequate when confronted with
+the truth.
+
+
+
+
+NOTES TO LECTURE XIII
+
+
+BODE'S LAW.--Write down the series 0, 3, 6, 12, 24, 48, &c.; add 4 to
+each, and divide by 10; you get the series:
+
+    .4       .7    1.0    1.6    2.8    5.2     10.0    19.6   38.8
+  Mercury  Venus  Earth   Mars  ----  Jupiter  Saturn  Uranus  ----
+
+numbers which very fairly represent the distances of the then known
+planets from the sun in the order specified.
+
+Ceres was discovered on the 1st of January, 1801, by Piazzi; Pallas in
+March, 1802, by Olbers; Juno in 1804, by Harding; and Vesta in 1807, by
+Olbers. No more asteroids were discovered till 1845, but there are now
+several hundreds known. Their diameters range from 500 to 20 miles.
+
+Neptune was discovered from the perturbations of Uranus by sheer
+calculation, carried on simultaneously and independently by Leverrier in
+Paris, and Adams in Cambridge. It was first knowingly seen by Galle, of
+Berlin, on the 23rd of September, 1846.
+
+
+
+
+LECTURE XIII
+
+THE DISCOVERY OF THE ASTEROIDS
+
+
+Up to the time of Herschel, astronomical interest centred on the solar
+system. Since that time it has been divided, and a great part of our
+attention has been given to the more distant celestial bodies. The solar
+system has by no means lost its interest--it has indeed gained in
+interest continually, as we gain in knowledge concerning it; but in
+order to follow the course of science it will be necessary for us to
+oscillate to and fro, sometimes attending to the solar system--the
+planets and their satellites--sometimes extending our vision to the
+enormously more distant stellar spaces.
+
+Those who have read the third lecture in Part I. will remember the
+speculation in which Kepler indulged respecting the arrangements of the
+planets, the order in which they succeeded one another in space, and the
+law of their respective distances from the sun; and his fanciful guess
+about the five regular solids inscribed and circumscribed about their
+orbits.
+
+The rude coincidences were, however, accidental, and he failed to
+discover any true law. No thoroughly satisfactory law is known at the
+present day. And yet, if the nebular hypothesis or anything like it be
+true, there must be some law to be discovered hereafter, though it may
+be a very complicated one.
+
+An empirical relation is, however, known: it was suggested by Tatius,
+and published by Bode, of Berlin, in 1772. It is always known as Bode's
+law.
+
+     Bode's law asserts that the distance of each planet is
+     approximately double the distance of the inner adjacent planet from
+     the sun, but that the rate of increase is distinctly slower than
+     this for the inner ones; consequently a better approximation will
+     be obtained by adding a constant to each term of an appropriate
+     geometrical progression. Thus, form a doubling series like this,
+     1-1/2, 3, 6, 12, 24, &c. doubling each time; then add 4 to each,
+     and you get a series which expresses very fairly the relative
+     distances of the successive planets from the sun, except that the
+     number for Mercury is rather erroneous, and we now know that at the
+     other extreme the number for Neptune is erroneous too.
+
+     I have stated it in the notes above in a form calculated to give
+     the law every chance, and a form that was probably fashionable
+     after the discovery of Uranus; but to call the first term of the
+     doubling series 0 is evidently not quite fair, though it puts
+     Mercury's distance right. Neptune's distance, however, turns out to
+     be more nearly 30 times the earth's distance than 38.8. The others
+     are very nearly right: compare column D of the table preceding
+     Lecture III. on p. 57, with the numbers in the notes on p. 294.
+
+The discovery of Uranus a few years afterwards, in 1781, at 19.2 times
+the earth's distance from the sun, lent great _eclat_ to the law, and
+seemed to establish its right to be regarded as at least a close
+approximation to the truth.
+
+The gap between Mars and Jupiter, which had often been noticed, and
+which Kepler filled with a hypothetical planet too small to see, comes
+into great prominence by this law of Bode. So much so, that towards the
+end of last century an enthusiastic German, von Zach, after some search
+himself for the expected planet, arranged a committee of observing
+astronomers, or, as he termed it, a body of astronomical detective
+police, to begin a systematic search for this missing subject of the
+sun.
+
+[Illustration: FIG. 90.--Planetary orbits to scale; showing the
+Asteroidal region between Jupiter and Mars. (The orbits of satellites
+are exaggerated.)]
+
+In 1800 the preliminaries were settled: the heavens near the zodiac
+were divided into twenty-four regions, each of which was intrusted to
+one observer to be swept. Meanwhile, however, quite independently of
+these arrangements in Germany, and entirely unknown to this committee, a
+quiet astronomer in Sicily, Piazzi, was engaged in making a catalogue of
+the stars. His attention was directed to a certain region in Taurus by
+an error in a previous catalogue, which contained a star really
+non-existent.
+
+In the course of his scrutiny, on the 1st of January, 1801, he noticed a
+small star which next evening appeared to have shifted. He watched it
+anxiously for successive evenings, and by the 24th of January he was
+quite sure he had got hold of some moving body, not a star: probably, he
+thought, a comet. It was very small, only of the eighth magnitude; and
+he wrote to two astronomers (one of them Bode himself) saying what he
+had observed. He continued to observe till the 11th of February, when he
+was attacked by illness and compelled to cease.
+
+His letters did not reach their destination till the end of March.
+Directly Bode opened his letter he jumped to the conclusion that this
+must be the missing planet. But unfortunately he was unable to verify
+the guess, for the object, whatever it was, had now got too near the sun
+to be seen. It would not be likely to be out again before September, and
+by that time it would be hopelessly lost again, and have just as much to
+be rediscovered as if it had never been seen.
+
+Mathematical astronomers tried to calculate a possible orbit for the
+body from the observations of Piazzi, but the observed places were so
+desperately few and close together. It was like having to determine a
+curve from three points close together. Three observations ought to
+serve,[27] but if they are taken with insufficient interval between
+them it is extremely difficult to construct the whole circumstances of
+the orbit from them. All the calculations gave different results, and
+none were of the slightest use.
+
+The difficulty as it turned out was most fortunate. It resulted in the
+discovery of one of the greatest mathematicians, perhaps the greatest,
+that Germany has ever produced--Gauss. He was then a young man of
+twenty-five, eking out a living by tuition. He had invented but not
+published several powerful mathematical methods (one of them now known
+as "the method of least squares"), and he applied them to Piazzi's
+observations. He was thus able to calculate an orbit, and to predict a
+place where, by the end of the year, the planet should be visible. On
+the 31st of December of that same year, very near the place predicted by
+Gauss, von Zach rediscovered it, and Olbers discovered it also the next
+evening. Piazzi called it Ceres, after the tutelary goddess of Sicily.
+
+Its distance from the sun as determined by Gauss was 2.767 times the
+earth's distance. Bode's law made it 2.8. It was undoubtedly the missing
+planet. But it was only one hundred and fifty or two hundred miles in
+diameter--the smallest heavenly body known at the time of its discovery.
+It revolves the same way as other planets, but the plane of its orbit is
+tilted 10 deg. to the plane of the ecliptic, which was an exceptionally
+large amount.
+
+Very soon, a more surprising discovery followed. Olbers, while searching
+for Ceres, had carefully mapped the part of the heavens where it was
+expected; and in March, 1802, he saw in this place a star he had not
+previously noticed. In two hours he detected its motion, and in a month
+he sent his observations to Gauss, who returned as answer the calculated
+orbit. It was distant 2.67, like Ceres, and was a little smaller, but it
+had a very excentric orbit: its plane being tilted 34-1/2 deg., an
+extraordinary inclination. This was called Pallas.
+
+Olbers at once surmised that these two planets were fragments of a
+larger one, and kept an eager look out for other fragments.
+
+In two years another was seen, in the course of charting the region of
+the heavens traversed by Ceres and Pallas. It was smaller than either,
+and was called Juno.
+
+In 1807 the persevering search of Olbers resulted in the discovery of
+another, with a very oblique orbit, which Gauss named Vesta. Vesta is
+bigger than any of the others, being five hundred miles in diameter, and
+shines like a star of the sixth magnitude. Gauss by this time had become
+so practised in the difficult computations that he worked out the
+complete orbit of Vesta within ten hours of receiving the observational
+data from Olbers.
+
+For many weary years Olbers kept up a patient and unremitting search for
+more of these small bodies, or fragments of the large planet as he
+thought them; but his patience went unrewarded, and he died in 1840
+without seeing or knowing of any more. In 1845 another was found,
+however, in Germany, and a few weeks later two others by Mr. Hind in
+England. Since then there seems no end to them; numbers have been
+discovered in America, where Professors Peters and Watson have made a
+specialty of them, and have themselves found something like a hundred.
+
+Vesta is the largest--its area being about the same as that of Central
+Europe, without Russia or Spain--and the smallest known is about twenty
+miles in diameter, or with a surface about the size of Kent. The whole
+of them together do not nearly equal the earth in bulk.
+
+The main interest of these bodies to us lies in the question, What is
+their history? Can they have been once a single planet broken up? or are
+they rather an abortive attempt at a planet never yet formed into one?
+
+The question is not _entirely_ settled, but I can tell you which way
+opinion strongly tends at the present time.
+
+Imagine a shell travelling in an elliptic orbit round the earth to
+suddenly explode: the centre of gravity of all its fragments would
+continue moving along precisely the same path as had been traversed by
+the centre of the shell before explosion, and would complete its orbit
+quite undisturbed. Each fragment would describe an orbit of its own,
+because it would be affected by a different initial velocity; but every
+orbit would be a simple ellipse, and consequently every piece would in
+time return through its starting-point--viz. the place at which the
+explosion occurred. If the zone of asteroids had a common point through
+which they all successively passed, they could be unhesitatingly
+asserted to be the remains of an exploded planet. But they have nothing
+of the kind; their orbits are scattered within a certain broad zone--a
+zone everywhere as broad as the earth's distance from the sun,
+92,000,000 miles--with no sort of law indicating an origin of this kind.
+
+It must be admitted, however, that the fragments of our supposed shell
+might in the course of ages, if left to themselves, mutually perturb
+each other into a different arrangement of orbits from that with which
+they began. But their perturbations would be very minute, and moreover,
+on Laplace's theory, would only result in periodic changes, provided
+each mass were rigid. It is probable that the asteroids were at one time
+not rigid, and hence it is difficult to say what may have happened to
+them; but there is not the least reason to believe that their present
+arrangement is derivable in any way from an explosion, and it is certain
+that an enormous time must have elapsed since such an event if it ever
+occurred.
+
+It is far more probable that they never constituted one body at all, but
+are the remains of a cloudy ring thrown off by the solar system in
+shrinking past that point: a small ring after the immense effort which
+produced Jupiter and his satellites: a ring which has aggregated into a
+multitude of little lumps instead of a few big ones. Such an event is
+not unique in the solar system; there is a similar ring round Saturn.
+At first sight, and to ordinary careful inspection, this differs from
+the zone of asteroids in being a solid lump of matter, like a quoit. But
+it is easy to show from the theory of gravitation, that a solid ring
+could not possibly be stable, but would before long get precipitated
+excentrically upon the body of the planet. Devices have been invented,
+such as artfully distributed irregularities calculated to act as
+satellites and maintain stability; but none of these things really work.
+Nor will it do to imagine the rings fluid; they too would destroy each
+other. The mechanical behaviour of a system of rings, on different
+hypotheses as to their constitution, has been worked out with consummate
+skill by Clerk Maxwell; who finds that the only possible constitution
+for Saturn's assemblage of rings is a multitude of discrete particles
+each pursuing its independent orbit. Saturn's ring is, in fact, a very
+concentrated zone of minor asteroids, and there is every reason to
+conclude that the origin of the solar asteroids cannot be very unlike
+the origin of the Saturnian ones. The nebular hypothesis lends itself
+readily to both.
+
+The interlockings and motions of the particles in Saturn's rings are
+most beautiful, and have been worked out and stated by Maxwell with
+marvellous completeness. His paper constituted what is called "The Adams
+Prize Essay" for 1856. Sir George Airy, one of the adjudicators
+(recently Astronomer-Royal), characterized it as "one of the most
+remarkable applications of mathematics to physics that I have ever
+seen."
+
+There are several distinct constituent rings in the entire Saturnian
+zone, and each perturbs the other, with the result that they ripple and
+pulse in concord. The waves thus formed absorb the effect of the mutual
+perturbations, and prevent an accumulation which would be dangerous to
+the persistence of the whole.
+
+The only effect of gravitational perturbation and of collisions is
+gradually to broaden out the whole ring, enlarging its outer and
+diminishing its inner diameter. But if there were any frictional
+resistance in the medium through which the rings spin, then other
+effects would slowly occur, which ought to be looked for with interest.
+So complete and intimate is the way Maxwell works out and describes the
+whole circumstances of the motion of such an assemblage of particles,
+and so cogent his argument as to the necessity that they must move
+precisely so, and no otherwise, else the rings would not be stable, that
+it was a Cambridge joke concerning him that he paid a visit to Saturn
+one evening, and made his observations on the spot.
+
+
+
+
+NOTES TO LECTURE XIV
+
+
+The total number of stars in the heavens visible to a good eye is about
+5,000. The total number at present seen by telescope is about
+50,000,000. The number able to impress a photographic plate has not yet
+been estimated; but it is enormously greater still. Of those which we
+can see in these latitudes, about 14 are of the first magnitude, 48 of
+the second, 152 of the third, 313 of the fourth, 854 of the fifth, and
+2,010 of the sixth; total, 3,391.
+
+The quickest-moving stars known are a double star of the sixth
+magnitude, called 61 Cygni, and one of the seventh magnitude, called
+Groombridge 1830. The velocity of the latter is 200 miles a second. The
+nearest known stars are 61 Cygni and [alpha] Centauri. The distance
+of these from us is about 400,000 times the distance of the sun. Their
+parallax is accordingly half a second of arc. Sirius is more than a
+million times further from us than our sun is, and twenty times as big;
+many of the brightest stars are at more than double this distance. The
+distance of Arcturus is too great to measure even now. Stellar parallax
+was first securely detected in 1838, by Bessel, for 61 Cygni. Bessel was
+born in 1784, and died in 1846, shortly before the discovery of Neptune.
+
+The stars are suns, and are most likely surrounded by planets. One
+planet belonging to Sirius has been discovered. It was predicted by
+Bessel, its position calculated by Peters, and seen by Alvan Clark in
+1862. Another predicted one, belonging to Procyon, has not yet been
+seen.
+
+A velocity of 5 miles a second could carry a projectile right round the
+earth. A velocity of 7 miles a second would carry it away from the
+earth, and round the sun. A velocity of 27 miles a second would carry a
+projectile right out of the solar system never to return.
+
+
+
+
+LECTURE XIV
+
+BESSEL--THE DISTANCES OF THE STARS, AND THE DISCOVERY OF STELLAR PLANETS
+
+
+We will now leave the solar system for a time, and hastily sketch the
+history of stellar astronomy from the time of Sir William Herschel.
+
+You remember how greatly Herschel had changed the aspect of the heavens
+for man,--how he had found that none of the stars were really fixed, but
+were moving in all manner of ways: some of this motion only apparent,
+much of it real. Nevertheless, so enormously distant are they, that if
+we could be transported back to the days of the old Chaldaean
+astronomers, or to the days of Noah, we should still see the heavens
+with precisely the same aspect as they wear now. Only by refined
+apparatus could any change be discoverable in all those centuries. For
+all practical purposes, therefore, the stars may still be well called
+fixed.
+
+Another thing one may notice, as showing their enormous distances, is
+that from every planet of the solar system the aspect of the heavens
+will be precisely the same. Inhabitants of Mars, or Jupiter, or Saturn,
+or Uranus, will see exactly the same constellations as we do. The whole
+dimensions of the solar system shrink up into a speck when so
+contemplated. And from the stars none of the planetary orbs of our
+system are visible at all; nothing but the sun is visible, and that
+merely as a twinkling star, brighter than some, but fainter than many
+others.
+
+The sun and the stars are one. Try to realize this distinctly, and keep
+it in mind. I find it often difficult to drive this idea home. After
+some talk on the subject a friendly auditor will report, "the lecturer
+then described the stars, including that greatest and most magnificent
+of all stars, the sun." It would be difficult more completely to
+misapprehend the entire statement. When I say the sun is one of the
+stars, I mean one among the others; we are a long way from them, they
+are a long way from each other. They need be no more closely packed
+among each other than we are closely packed among them; except that some
+of them are double or multiple, and we are not double.
+
+     It is highly desirable to acquire an intimate knowledge of the
+     constellations and a nodding acquaintance with their principal
+     stars. A description of their peculiarities is dull and
+     uninteresting unless they are at least familiar by name. A little
+     _viva voce_ help to begin with, supplemented by patient night
+     scrutiny with a celestial globe or star maps under a tent or shed,
+     is perhaps the easiest way: a very convenient instrument for the
+     purpose of learning the constellations is the form of map called a
+     "planisphere," because it can be made to show all the
+     constellations visible at a given time at a given date, and no
+     others. The Greek alphabet also is a thing that should be learnt by
+     everybody. The increased difficulty in teaching science owing to
+     the modern ignorance of even a smattering of Greek is becoming
+     grotesque. The stars are named from their ancient grouping into
+     constellations, and by the prefix of a Greek letter to the larger
+     ones, and of numerals to the smaller ones. The biggest of all have
+     special Arabic names as well. The brightest stars are called of
+     "the first magnitude," the next are of "the second magnitude," and
+     so on. But this arrangement into magnitudes has become technical
+     and precise, and intermediate or fractional magnitudes are
+     inserted. Those brighter than the ordinary first magnitude are
+     therefore now spoken of as of magnitude 1/2, for instance, or .6,
+     which is rather confusing. Small telescopic stars are often only
+     named by their numbers in some specified catalogue--a dull but
+     sufficient method.
+
+     Here is a list of the stars visible from these latitudes, which are
+     popularly considered as of the first magnitude. All of them should
+     be familiarly recognized in the heavens, whenever seen.
+
+  Star.                Constellation.
+
+  Sirius               Canis major
+  Procyon              Canis minor
+  Rigel                Orion
+  Betelgeux            Orion
+  Castor               Gemini
+  Pollux               Gemini
+  Aldebaran            Taurus
+  Arcturus             Booetes
+  Vega                 Lyra
+  Capella              Auriga
+  Regulus              Leo
+  Altair               Aquila
+  Fomalhaut            Southern Fish
+  Spica                Virgo
+
+     [alpha] Cygni is a little below the first magnitude. So,
+     perhaps, is Castor. In the southern heavens, Canopus and [alpha]
+     Centauri rank next after Sirius in brightness.
+
+[Illustration: FIG. 91.--Diagram illustrating Parallax.]
+
+The distances of the fixed stars had, we know, been a perennial problem,
+and many had been the attempts to solve it. All the methods of any
+precision have depended on the Copernican fact that the earth in June
+was 184 million miles away from its position in December, and that
+accordingly the grouping and aspect of the heavens should be somewhat
+different when seen from so different a point of view. An apparent
+change of this sort is called generally parallax; _the_ parallax of a
+star being technically defined as the angle subtended at the star by the
+radius of the earth's orbit: that is to say, the angle E[sigma]S;
+where E is the earth, S the sun, and [sigma] a star (Fig. 91).
+
+Plainly, the further off [sigma] is, the more nearly parallel will
+the two lines to it become. And the difficulty of determining the
+parallax was just this, that the more accurately the observations were
+made, the more nearly parallel did those lines become. The angle was, in
+fact, just as likely to turn out negative as positive--an absurd result,
+of course, to be attributed to unavoidable very minute inaccuracies.
+
+For a long time absolute methods of determining parallax were attempted;
+for instance, by observing the position of the star with respect to the
+zenith at different seasons of the year. And many of these
+determinations appeared to result in success. Hooke fancied he had
+measured a parallax for Vega in this way, amounting to 30" of arc.
+Flamsteed obtained 40" for [gamma] Draconis. Roemer made a serious
+attempt by comparing observations of Vega and Sirius, stars almost the
+antipodes of each other in the celestial vault; hoping to detect some
+effect due to the size of the earth's orbit, which should apparently
+displace them with the season of the year. All these fancied results
+however, were shown to be spurious, and their real cause assigned, by
+the great discovery of the aberration of light by Bradley.
+
+After this discovery it was possible to watch for still outstanding very
+minute discrepancies; and so the problem of stellar parallax was
+attacked with fresh vigour by Piazzi, by Brinkley, and by Struve. But
+when results were obtained, they were traced after long discussion to
+age and gradual wear of the instrument, or to some other minute
+inaccuracy. The more carefully the observation was made, the more nearly
+zero became the parallax--the more nearly infinite the distance of the
+stars. The brightest stars were the ones commonly chosen for the
+investigation, and Vega was a favourite, because, going near the zenith,
+it was far removed from the fluctuating and tiresome disturbances of
+atmospheric refraction. The reason bright stars were chosen was because
+they were presumably nearer than the others; and indeed a rough guess at
+their probable distance was made by supposing them to be of the same
+size as the sun, and estimating their light in comparison with sunlight.
+By this confessedly unsatisfactory method it had been estimated that
+Sirius must be 140,000 times further away than the sun is, if he be
+equally big. We now know that Sirius is much further off than this; and
+accordingly that he is much brighter, perhaps sixty times as bright,
+though not necessarily sixty times as big, as our sun. But even
+supposing him of the same light-giving power as the sun, his parallax
+was estimated as 1".8, a quantity very difficult to be sure of in any
+absolute determination.
+
+Relative methods were, however, also employed, and the advantages of one
+of these (which seems to have been suggested by Galileo) so impressed
+themselves upon William Herschel that he made a serious attempt to
+compass the problem by its means. The method was to take two stars in
+the same telescopic field and carefully to estimate their apparent
+angular distance from each other at different seasons of the year. All
+such disturbances as precession, aberration, nutation, refraction, and
+the like, would affect them both equally, and could thus be eliminated.
+If they were at the same distance from the solar system, relative
+parallax would, indeed, also be eliminated; but if, as was probable,
+they were at different distances, then they would apparently shift
+relatively to one another, and the amount of shift, if it could be
+observed, would measure, not indeed the distance of either from the
+earth, but their distance from each other. And this at any rate would be
+a step. It might be completed by similarly treating other stars in the
+same field, taking them in pairs together. A bright and a faint star
+would naturally be suitable, because their distances were likely to be
+unequal; and so Herschel fixed upon a number of doublets which he knew
+of, containing one bright and one faint component. For up to that time
+it had been supposed that such grouping in occasional pairs or triplets
+was chance coincidence, the two being optically foreshortened together,
+but having no real connection or proximity. Herschel failed in what he
+was looking for, but instead of that he discovered the real connection
+of a number of these doublets, for he found that they were slowly
+revolving round each other. There are a certain number of merely optical
+or accidental doublets, but the majority of them are real pairs of suns
+revolving round each other.
+
+This relative method of mapping micrometrically a field of neighbouring
+stars, and comparing their configuration now and six months hence, was,
+however, the method ultimately destined to succeed; and it is, I
+believe, the only method which has succeeded down to the present day.
+Certainly it is the method regularly employed, at Dunsink, at the Cape
+of Good Hope, and everywhere else where stellar parallax is part of the
+work.
+
+Between 1830 and 1840 the question was ripe for settlement, and, as
+frequently happens with a long-matured difficulty, it gave way in three
+places at once. Bessel, Henderson, and Struve almost simultaneously
+announced a stellar parallax which could reasonably be accepted. Bessel
+was a little the earliest, and by far the most accurate. His, indeed,
+was the result which commanded confidence, and to him the palm must be
+awarded.
+
+He was largely a self-taught student, having begun life in a
+counting-house, and having abandoned business for astronomy. But
+notwithstanding these disadvantages, he became a highly competent
+mathematician as well as a skilful practical astronomer. He was
+appointed to superintend the construction of Germany's first great
+astronomical observatory, that of Koenigsberg, which, by his system,
+zeal, and genius, he rapidly made a place of the first importance.
+
+Struve at Dorpat, Bessel at Koenigsberg, and Henderson at the Cape of
+Good Hope--all of them at newly-equipped observatories--were severally
+engaged at the same problem.
+
+But the Russian and German observers had the advantage of the work of
+one of the most brilliant opticians--I suppose the most brilliant--that
+has yet appeared: Fraunhofer, of Munich. An orphan lad, apprenticed to a
+maker of looking-glasses, and subject to hard struggles and privations
+in early life, he struggled upwards, and ultimately became head of the
+optical department of a Munich firm of telescope-makers. Here he
+constructed the famous "Dorpat refractor" for Struve, which is still at
+work; and designed the "Koenigsberg heliometer" for Bessel. He also made
+a long and most skilful research into the solar spectrum, which has
+immortalized his name. But his health was broken by early trials, and he
+died at the age of thirty-nine, while planning new and still more
+important optical achievements.
+
+A heliometer is the most accurate astronomical instrument for relative
+measurements of position, as a transit circle is the most accurate for
+absolute determinations. It consists of an equatorial telescope with
+object-glass cut right across, and each half movable by a sliding
+movement one past the other, the amount by which the two halves are
+dislocated being read off by a refined method, and the whole instrument
+having a multitude of appendages conducive to convenience and accuracy.
+Its use is to act as a micrometer or measurer of small distances.[28]
+Each half of the object-glass gives a distinct image, which may be
+allowed to coincide or may be separated as occasion requires. If it be
+the components of a double star that are being examined, each component
+will in general be seen double, so that four images will be seen
+altogether; but by careful adjustment it will be possible to arrange
+that one image of each pair shall be superposed on or coincide with each
+other, in which case only three images are visible; the amount of
+dislocation of the halves of the object-glass necessary to accomplish
+this is what is read off. The adjustment is one that can be performed
+with extreme accuracy, and by performing it again and again with all
+possible modifications, an extremely accurate determination of the
+angular distance between the two components is obtained.
+
+[Illustration: FIG. 92.--Heliometer.]
+
+Bessel determined to apply this beautiful instrument to the problem of
+stellar parallax; and he began by considering carefully the kind of star
+for which success was most likely. Hitherto the brightest had been most
+attended to, but Bessel thought that quickness of proper motion would be
+a still better test of nearness. Not that either criterion is conclusive
+as to distance, but there was a presumption in favour of either a very
+bright or an obviously moving star being nearer than a faint or a
+stationary one; and as the "bright" criterion had already been often
+applied without result, he decided to try the other. He had already
+called attention to a record by Piazzi in 1792 of a double star in
+Cygnus whose proper motion was five seconds of arc every year--a motion
+which caused this telescopic object, 61 Cygni, to be known as "the
+flying star." Its motion is not really very perceptible, for it will
+only have traversed one-third of a lunar diameter in the course of a
+century; still it was the quickest moving star then known. The position
+of this interesting double he compared with two other stars which were
+seen simultaneously in the field of the heliometer, by the method I have
+described, throughout the whole year 1838; and in the last month of that
+year he was able to announce with confidence a distinct though very
+small parallax; substantiating it with a mass of detailed evidence which
+commanded the assent of astronomers. The amount of it he gave as
+one-third of a second. We know now that he was very nearly right, though
+modern research makes it more like half a second.[29]
+
+Soon afterwards, Struve announced a quarter of a second as the parallax
+of Vega, but that is distinctly too great; and Henderson announced for
+[alpha] Centauri (then thought to be a double) a parallax of one
+second, which, if correct, would make it quite the nearest of all the
+stars, but the result is now believed to be about twice too big.
+
+Knowing the distance of 61 Cygni, we can at once tell its real rate of
+travel--at least, its rate across our line of sight: it is rather over
+three million miles a day.
+
+Now just consider the smallness of the half second of arc, thus
+triumphantly though only approximately measured. It is the angle
+subtended by twenty-six feet at a distance of 2,000 miles. If a
+telescope planted at New York could be directed to a house in England,
+and be then turned so as to set its cross-wire first on one end of an
+ordinary room and then on the other end of the same room, it would have
+turned through half a second, the angle of greatest stellar parallax.
+Or, putting it another way. If the star were as near us as New York is,
+the sun, on the same scale, would be nine paces off. As twenty-six feet
+is to the distance of New York, so is ninety-two million miles to the
+distance of the nearest fixed star.
+
+Suppose you could arrange some sort of telegraphic vehicle able to carry
+you from here to New York in the tenth part of a second--_i.e._ in the
+time required to drop two inches--such a vehicle would carry you to the
+moon in twelve seconds, to the sun in an hour and a quarter. Travelling
+thus continually, in twenty-four hours you would leave the last member
+of the solar system behind you, and begin your plunge into the depths of
+space. How long would it be before you encountered another object? A
+month, should you guess? Twenty years you must journey with that
+prodigious speed before you reach the nearest star, and then another
+twenty years before you reach another. At these awful distances from one
+another the stars are scattered in space, and were they not brilliantly
+self-luminous and glowing like our sun, they would be hopelessly
+invisible.
+
+I have spoken of 61 Cygni as a flying star, but there is another which
+goes still quicker, a faint star, 1830 in Groombridge's Catalogue. Its
+distance is far greater than that of 61 Cygni, and yet it is seen to
+move almost as quickly. Its actual speed is about 200 miles a
+second--greater than the whole visible firmament of fifty million stars
+can control; and unless the universe is immensely larger than anything
+we can see with the most powerful telescopes, or unless there are crowds
+of invisible non-luminous stars mixed up with the others, it can only be
+a temporary visitor to this frame of things; it is rushing from an
+infinite distance to an infinite distance; it is passing through our
+visible universe for the first and only time--it will never return. But
+so gigantic is the extent of visible space, that even with its amazing
+speed of 200 miles every second, this star will take two or three
+million years to get out of sight of our present telescopes, and several
+thousand years before it gets perceptibly fainter than it is now.
+
+Have we any reason for supposing that the stars we see are all there
+are? In other words, have we any reason for supposing all celestial
+objects to be sufficiently luminous to be visible? We have every ground
+for believing the contrary. Every body in the solar system is dull and
+dark except the sun, though probably Jupiter is still red-hot. Why may
+not some of the stars be dark too? The genius of Bessel surmised this,
+and consistently upheld the doctrine that the astronomy of the future
+would have to concern itself with dark and invisible bodies; he preached
+"an astronomy of the invisible." Moreover he predicted the presence of
+two such dark bodies--one a companion of Sirius, the other of Procyon.
+He noticed certain irregularities in the motions of these stars which he
+asserted must be caused by their revolving round other bodies in a
+period of half a century. He announced in 1844 that both Sirius and
+Procyon were double stars, but that their companions, though large, were
+dark, and therefore invisible.
+
+No one accepted this view, till Peters, in America, found in 1851 that
+the hypothesis accurately explained the anomalous motion of Sirius, and,
+in fact, indicated an exact place where the companion ought to be. The
+obscure companion of Sirius became now a recognized celestial object,
+although it had never been seen, and it was held to revolve round Sirius
+in fifty years, and to be about half as big.
+
+In 1862, the firm of Alvan Clark and Sons, of New York, were completing
+a magnificent 18-inch refractor, and the younger Clark was trying it on
+Sirius, when he said: "Why, father, the star has a companion!" The elder
+Clark also looked, and sure enough there was a faint companion due east
+of the bright star, and in just the position required by theory. Not
+that the Clarks knew anything about the theory. They were keen-sighted
+and most skilful instrument-makers, and they made the discovery by
+accident. After it had once been seen, it was found that several of the
+large telescopes of the world were able to show it. It is half as big,
+but it only gives 1/10000th part of the light that Sirius gives. No
+doubt it shines partly with a borrowed light and partly with a dull heat
+of its own. It is a real planet, but as yet too hot to live on. It will
+cool down in time, as our earth has cooled and as Jupiter is cooling,
+and no doubt become habitable enough. It does revolve round Sirius in a
+period of 49.4 years--almost exactly what Bessel assigned to it.
+
+But Bessel also assigned a dark companion to Procyon. It and its
+luminous neighbour are considered to revolve round each other in a
+period of forty years, and astronomers feel perfectly assured of its
+existence, though at present it has not been seen by man.
+
+
+
+
+LECTURE XV
+
+THE DISCOVERY OF NEPTUNE
+
+
+We approach to-night perhaps the greatest, certainly the most
+conspicuous, triumphs of the theory of gravitation. The explanation by
+Newton of the observed facts of the motion of the moon, the way he
+accounted for precession and nutation and for the tides, the way in
+which Laplace explained every detail of the planetary motions--these
+achievements may seem to the professional astronomer equally, if not
+more, striking and wonderful; but of the facts to be explained in these
+cases the general public are necessarily more or less ignorant, and so
+no beauty or thoroughness of treatment appeals to them, nor can excite
+their imaginations. But to predict in the solitude of the study, with no
+weapons other than pen, ink, and paper, an unknown and enormously
+distant world, to calculate its orbit when as yet it had never been
+seen, and to be able to say to a practical astronomer, "Point your
+telescope in such a direction at such a time, and you will see a new
+planet hitherto unknown to man"--this must always appeal to the
+imagination with dramatic intensity, and must awaken some interest in
+almost the dullest.
+
+Prediction is no novelty in science; and in astronomy least of all is it
+a novelty. Thousands of years ago, Thales, and others whose very names
+we have forgotten, could predict eclipses with some certainty, though
+with only rough accuracy. And many other phenomena were capable of
+prediction by accumulated experience. We have seen, for instance (coming
+to later times), how a gap between Mars and Jupiter caused a missing
+planet to be suspected and looked for, and to be found in a hundred
+pieces. We have seen, also, how the abnormal proper-motion of Sirius
+suggested to Bessel the existence of an unseen companion. And these last
+instances seem to approach very near the same class of prediction as
+that of the discovery of Neptune. Wherein, then, lies the difference?
+How comes it that some classes of prediction--such as that if you put
+your finger in fire it will get burnt--are childishly easy and
+commonplace, while others excite in the keenest intellects the highest
+feelings of admiration? Mainly, the difference lies, first, in the
+grounds on which the prediction is based; second, on the difficulty of
+the investigation whereby it is accomplished; third, in the completeness
+and the accuracy with which it can be verified. In all these points, the
+discovery of Neptune stands out pre-eminently among the verified
+predictions of science, and the circumstances surrounding it are of
+singular interest.
+
+*      *      *      *      *
+
+In 1781, Sir William Herschel discovered the planet Uranus. Now you know
+that three distinct observations suffice to determine the orbit of a
+planet completely, and that it is well to have the three observations as
+far apart as possible so as to minimize the effects of minute but
+necessary errors of observation. (See p. 298.) Directly Uranus was
+found, therefore, old records of stellar observations were ransacked,
+with the object of discovering whether it had ever been unwittingly seen
+before. If seen, it had been thought of course to be a star (for it
+shines like a star of the sixth magnitude, and can therefore be just
+seen without a telescope if one knows precisely where to look for it,
+and if one has good sight), but if it had been seen and catalogued as a
+star it would have moved from its place, and the catalogue would by that
+entry be wrong. The thing to detect, therefore, was errors in the
+catalogues: to examine all entries, and see if the stars entered
+actually existed, or were any of them missing. If a wrong entry were
+discovered, it might of course have been due to some clerical error,
+though that is hardly probable considering the care taken over these
+things, or it might have been some tailless comet or other, or it might
+have been the newly found planet.
+
+So the next thing was to calculate backwards, and see if by any
+possibility the planet could have been in that place at that time.
+Examined in this way the tabulated observations of Flamsteed showed that
+he had unwittingly observed Uranus five distinct times, the first time
+in 1690, nearly a century before Herschel discovered its true nature.
+But more remarkable still, Le Monnier, of Paris, had observed it eight
+times in one month, cataloguing it each time as a different star. If
+only he had reduced and compared his observations, he would have
+anticipated Herschel by twelve years. As it was, he missed it
+altogether. It was seen once by Bradley also. Altogether it had been
+seen twenty times.
+
+These old observations of Flamsteed and those of Le Monnier, combined
+with those made after Herschel's discovery, were very useful in
+determining an exact orbit for the new planet, and its motion was
+considered thoroughly known. It was not an _exact_ ellipse, of course:
+none of the planets describe _exact_ ellipses--each perturbs all the
+rest, and these small perturbations must be taken into account, those of
+Jupiter and Saturn being by far the most important.
+
+For a time Uranus seemed to travel regularly and as expected, in the
+orbit which had been calculated for it; but early in the present century
+it began to be slightly refractory, and by 1820 its actual place showed
+quite a distinct discrepancy from its position as calculated with the
+aid of the old observations. It was at first thought that this
+discrepancy must be due to inaccuracies in the older observations, and
+they were accordingly rejected, and tables prepared for the planet based
+on the newer and more accurate observations only. But by 1830 it became
+apparent that it would not accurately obey even these. The error
+amounted to some 20". By 1840 it was as much as 90', or a minute and a
+half. This discrepancy is quite distinct, but still it is very small,
+and had two objects been in the heavens at once, the actual Uranus and
+the theoretical Uranus, no unaided eye could possibly have distinguished
+them or detected that they were other than a single star.
+
+[Illustration: FIG. 93.--Perturbations of Uranus.
+
+The chance observations by Flamsteed, by Le Monnier, and others, are
+plotted in this diagram, as well as the modern determinations made after
+Herschel had discovered the nature of the planet. The decades are laid
+off horizontally. Vertical distance represents the difference between
+observed and subsequently calculated longitudes--in other words, the
+principal perturbations caused by Neptune. To show the scale, a number
+of standard things are represented too by lengths measured upwards from
+the line of time, viz: the smallest quantity perceptible to the naked
+eye,--the maximum angle of aberration, of nutation, and of stellar
+parallax; though this last is too small to be properly indicated. The
+perturbations are much bigger than these; but compared with what can be
+seen without a telescope they are small--the distance between the
+component pairs of [epsilon] Lyrae (210") (see fig. 86, page 288), which
+a few keen-eyed persons can see as a simple double star, being about
+twice the greatest perturbation.]
+
+The diagram shows all the irregularities plotted in the light of our
+present knowledge; and, to compare with their amounts, a few standard
+things are placed on the same scale, such as the smallest interval
+capable of being detected with the unaided eye, the distance of the
+component stars in [epsilon] Lyrae, the constants of aberration, of
+nutation, and of stellar parallax.
+
+The errors of Uranus therefore, though small, were enormously greater
+than things which had certainly been observed; there was an unmistakable
+discrepancy between theory and observation. Some cause was evidently at
+work on this distant planet, causing it to disagree with its motion as
+calculated according to the law of gravitation. Some thought that the
+exact law of gravitation did not apply to so distant a body. Others
+surmised the presence of some foreign and unknown body, some comet, or
+some still more distant planet perhaps, whose gravitative attraction for
+Uranus was the cause of the whole difficulty--some perturbations, in
+fact, which had not been taken into account because of our ignorance of
+the existence of the body which caused them.
+
+But though such an idea was mentioned among astronomers, it was not
+regarded with any special favour, and was considered merely as one among
+a number of hypotheses which could be suggested as fairly probable.
+
+It is perfectly right not to attach much importance to unelaborated
+guesses. Not until the consequences of an hypothesis have been
+laboriously worked out--not until it can be shown capable of producing
+the effect quantitatively as well as qualitatively--does its statement
+rise above the level of a guess, and attain the dignity of a theory. A
+later stage still occurs when the theory has been actually and
+completely verified by agreement with observation.
+
+     Now the errors in the motion of Uranus, _i.e._ the discrepancy
+     between its observed and calculated longitudes--all known
+     disturbing causes, such as Jupiter and Saturn, being allowed
+     for--are as follows (as quoted by Dr. Haughton) in seconds of
+     arc:--
+
+  ANCIENT OBSERVATIONS (casually made, as of a star).
+
+  Flamsteed         1690    +61.2
+      "             1712    +92.7
+      "             1715    +73.8
+  Le Monnier        1750    -47.6
+  Bradley           1753    -39.5
+  Mayer             1756    -45.7
+  Le Monnier        1764    -34.9
+     "              1769    -19.3
+     "              1771     -2.3
+
+  MODERN OBSERVATIONS.
+
+  1780                 +3.46
+  1783                 +8.45
+  1786                +12.36
+  1789                +19.02
+  1801                +22.21
+  1810                +23.16
+  1822                +20.97
+  1825                +18.16
+  1828                +10.82
+  1831                 -3.98
+  1834                -20.80
+  1837                -42.66
+  1840                -66.64
+
+     These are the numbers plotted in the above diagram (Fig. 92), where
+     H marks the discovery of the planet and the beginning of its
+     regular observation.
+
+Something was evidently the matter with the planet. If the law of
+gravitation held exactly at so great a distance from the sun, there must
+be some perturbing force acting on it besides all those known ones which
+had been fully taken into account. Could it be an outer planet? The
+question occurred to several, and one or two tried if they could solve
+the problem, but were soon stopped by the tremendous difficulties of
+calculation.
+
+The ordinary problem of perturbation is difficult enough: Given a
+disturbing planet in such and such a position, to find the perturbations
+it produces. This problem it was that Laplace worked out in the
+_Mecanique Celeste_.
+
+But the inverse problem: Given the perturbations, to find the planet
+which causes them--such a problem had never yet been attacked, and by
+only a few had its possibility been conceived. Bessel made preparations
+for trying what he could do at it in 1840, but he was prevented by fatal
+illness.
+
+In 1841 the difficulties of the problem presented by these residual
+perturbations of Uranus excited the imagination of a young student, an
+undergraduate of St. John's College, Cambridge--John Couch Adams by
+name--and he determined to have a try at it as soon as he was through
+his Tripos. In January, 1843, he graduated as Senior Wrangler, and
+shortly afterwards he set to work. In less than two years he reached a
+definite conclusion; and in October, 1845, he wrote to the
+Astronomer-Royal, at Greenwich, Professor Airy, saying that the
+perturbations of Uranus would be explained by assuming the existence of
+an outer planet, which he reckoned was now situated in a specified
+latitude and longitude.
+
+We know now that had the Astronomer-Royal put sufficient faith in this
+result to point his big telescope to the spot indicated and commence
+sweeping for a planet, he would have detected it within 1-3/4 deg. of the
+place assigned to it by Mr. Adams. But any one in the position of the
+Astronomer-Royal knows that almost every post brings an absurd letter
+from some ambitious correspondent or other, some of them having just
+discovered perpetual motion, or squared the circle, or proved the earth
+flat, or discovered the constitution of the moon, or of ether, or of
+electricity; and out of this mass of rubbish it requires great skill and
+patience to detect such gems of value as there may be.
+
+Now this letter of Mr. Adams's was indeed a jewel of the first water,
+and no doubt bore on its face a very different appearance from the
+chaff of which I have spoken; but still Mr. Adams was an unknown man: he
+had graduated as Senior Wrangler it is true, but somebody must graduate
+as Senior Wrangler every year, and every year by no means produces a
+first-rate mathematician. Those behind the scenes, as Professor Airy of
+course was, having been a Senior Wrangler himself, knew perfectly well
+that the labelling of a young man on taking his degree is much more
+worthless as a testimony to his genius and ability than the general
+public are apt to suppose.
+
+Was it likely that a young and unknown man should have successfully
+solved so extremely difficult a problem? It was altogether unlikely.
+Still, he would test him: he would ask for further explanations
+concerning some of the perturbations which he himself had specially
+noticed, and see if Mr. Adams could explain these also by his
+hypothesis. If he could, there might be something in his theory. If he
+failed--well, there was an end of it. The questions were not difficult.
+They concerned the error of the radius vector. Mr. Adams could have
+answered them with perfect ease; but sad to say, though a brilliant
+mathematician, he was not a man of business. He did not answer Professor
+Airy's letter.
+
+It may to many seem a pity that the Greenwich Equatoreal was not pointed
+to the place, just to see whether any foreign object did happen to be in
+that neighbourhood; but it is no light matter to derange the work of an
+Observatory, and alter the work mapped out for the staff into a sudden
+sweep for a new planet, on the strength of a mathematical investigation
+just received by post. If observatories were conducted on these
+unsystematic and spasmodic principles, they would not be the calm,
+accurate, satisfactory places they are.
+
+Of course, if any one could have known that a new planet was to be had
+for the looking, _any_ course would have been justified; but no one
+could know this. I do not suppose that Mr. Adams himself could feel all
+that confidence in his attempted prediction. So there the matter
+dropped. Mr. Adams's communication was pigeon-holed, and remained in
+seclusion for eight or nine months.
+
+Meanwhile, and quite independently, something of the same sort was going
+on in France. A brilliant young mathematician, born in Normandy in 1811,
+had accepted the post of Astronomical Professor at the Ecole
+Polytechnique, then recently founded by Napoleon. His first published
+papers directed attention to his wonderful powers; and the official head
+of astronomy in France, the famous Arago, suggested to him the
+unexplained perturbations of Uranus as a worthy object for his fresh and
+well-armed vigour.
+
+At once he set to work in a thorough and systematic way. He first
+considered whether the discrepancies could be due to errors in the
+tables or errors in the old observations. He discussed them with minute
+care, and came to the conclusion that they were not thus to be explained
+away. This part of the work he published in November, 1845.
+
+He then set to work to consider the perturbations produced by Jupiter
+and Saturn, to see if they had been with perfect accuracy allowed for,
+or whether some minute improvements could be made sufficient to destroy
+the irregularities. He introduced several fresh terms into these
+perturbations, but none of them of sufficient magnitude to do more than
+slightly lessen the unexplained perturbations.
+
+He next examined the various hypotheses that had been suggested to
+account for them:--Was it a failure in the law of gravitation? Was it
+due to the presence of a resisting medium? Was it due to some unseen but
+large satellite? Or was it due to a collision with some comet?
+
+All these he examined and dismissed for various reasons one after the
+other. It was due to some steady continuous cause--for instance, some
+unknown planet. Could this planet be inside the orbit of Uranus? No, for
+then it would perturb Saturn and Jupiter also, and they were not
+perturbed by it. It must, therefore, be some planet outside the orbit of
+Uranus, and in all probability, according to Bode's empirical law, at
+nearly double the distance from the sun that Uranus is. Lastly he
+proceeded to examine where this planet was, and what its orbit must be
+to produce the observed disturbances.
+
+[Illustration: FIG. 94.--Uranus's and Neptune's relative positions.
+
+The above diagram, drawn to scale by Dr. Haughton, shows the paths of
+Uranus and Neptune, and their positions from 1781 to 1840, and
+illustrates the _direction_ of their mutual perturbing force. In 1822
+the planets were in conjunction, and the force would then perturb the
+radius vector (or distance from the sun), but not the longitude (or
+place in orbit). Before that date Uranus had been hurried along, and
+after that date it had been retarded, by the pull of Neptune, and thus
+the observed discrepancies from its computed place were produced. The
+problem was first to disentangle the outstanding perturbations from
+those which would be caused by Jupiter and Saturn and all other known
+causes, and then to assign the place of an outer planet able to produce
+precisely those perturbations in Uranus.]
+
+Not without failures and disheartening complications was this part of
+the process completed. This was, after all, the real tug of war. So many
+unknown quantities: its mass, its distance, its excentricity, the
+obliquity of its orbit, its position at any time--nothing known, in
+fact, about the planet except the microscopic disturbance it caused in
+Uranus, some thousand million miles away from it.
+
+Without going into further detail, suffice it to say that in June, 1846,
+he published his last paper, and in it announced to the world his
+theoretical position for the planet.
+
+Professor Airy received a copy of this paper before the end of the
+month, and was astonished to find that Leverrier's theoretical place for
+the planet was within 1 deg. of the place Mr. Adams had assigned to it eight
+months before. So striking a coincidence seemed sufficient to justify a
+Herschelian "sweep" for a week or two.
+
+But a sweep for so distant a planet would be no easy matter. When seen
+in a large telescope it would still only look like a star, and it would
+require considerable labour and watching to sift it out from the other
+stars surrounding it. We know that Uranus had been seen twenty times,
+and thought to be a star, before its true nature was by Herschel
+discovered; and Uranus is only about half as far away as Neptune is.
+
+Neither in Paris nor yet at Greenwich was any optical search undertaken;
+but Professor Airy wrote to ask M. Leverrier the same old question as he
+had fruitlessly put to Mr. Adams: Did the new theory explain the errors
+of the radius vector or not? The reply of Leverrier was both prompt and
+satisfactory--these errors were explained, as well as all the others.
+The existence of the object was then for the first time officially
+believed in.
+
+The British Association met that year at Southampton, and Sir John
+Herschel was one of its Sectional Presidents. In his inaugural address,
+on September 10th, 1846, he called attention to the researches of
+Leverrier and Adams in these memorable words:--
+
+     "The past year has given to us the new [minor] planet Astraea; it
+     has done more--it has given us the probable prospect of another.
+     We see it as Columbus saw America from the shores of Spain. Its
+     movements have been felt trembling along the far-reaching line of
+     our analysis with a certainty hardly inferior to ocular
+     demonstration."
+
+It was about time to begin to look for it. So the Astronomer-Royal
+thought on reading Leverrier's paper. But as the national telescope at
+Greenwich was otherwise occupied, he wrote to Professor Challis, at
+Cambridge, to know if he would permit a search to be made for it with
+the Northumberland Equatoreal, the large telescope of Cambridge
+University, presented to it by one of the Dukes of Northumberland.
+
+Professor Challis said he would conduct the search himself; and shortly
+commenced a leisurely and dignified series of sweeps round about the
+place assigned by theory, cataloguing all the stars which he observed,
+intending afterwards to sort out his observations, compare one with
+another, and find out whether any one star had changed its position;
+because if it had it must be the planet. He thus, without giving an
+excessive time to the business, accumulated a host of observations,
+which he intended afterwards to reduce and sift at his leisure.
+
+The wretched man thus actually saw the planet twice--on August 4th and
+August 12th, 1846--without knowing it. If only he had had a map of the
+heavens containing telescopic stars down to the tenth magnitude, and if
+he had compared his observations with this map as they were made, the
+process would have been easy, and the discovery quick. But he had no
+such map. Nevertheless one was in existence: it had just been completed
+in that country of enlightened method and industry--Germany. Dr.
+Bremiker had not, indeed, completed his great work--a chart of the whole
+zodiac down to stars of the tenth magnitude--but portions of it were
+completed, and the special region where the new planet was expected
+happened to be among the portions already just done. But in England
+this was not known.
+
+Meanwhile, Mr. Adams wrote to the Astronomer-Royal several additional
+communications, making improvements in his theory, and giving what he
+considered nearer and nearer approximations for the place of the planet.
+He also now answered quite satisfactorily, but too late, the question
+about the radius vector sent to him months before.
+
+Let us return to Leverrier. This great man was likewise engaged in
+improving his theory and in considering how best the optical search
+could be conducted. Actuated, probably, by the knowledge that in such
+matters as cataloguing and mapping Germany was then, as now, far ahead
+of all the other nations of the world, he wrote in September (the same
+September as Sir John Herschel delivered his eloquent address at
+Southampton) to Berlin. Leverrier wrote, I say, to Dr. Galle, head of
+the Observatory at Berlin, saying to him, clearly and decidedly, that
+the new planet was now in or close to such and such a position, and that
+if he would point his telescope to that part of the heavens he would see
+it; and, moreover, that he would be able to tell it from a star by its
+having a sensible magnitude, or disk, instead of being a mere point.
+
+Galle got the letter on the 23rd of September, 1846. That same evening
+he did point his telescope to the place Leverrier told him, and he saw
+the planet that very night. He recognized it first by its appearance. To
+his practised eye it did seem to have a small disk, and not quite the
+same aspect as an ordinary star. He then consulted Bremiker's great star
+chart, the part just engraved and finished, and sure enough on that
+chart there was no such star there. Undoubtedly it was the planet.
+
+The news flashed over Europe at the maximum speed with which news could
+travel at that date (which was not very fast); and by the 1st of October
+Professor Challis and Mr. Adams heard it at Cambridge, and had the
+pleasure of knowing that they were forestalled, and that England was
+out of the race.
+
+It was an unconscious race to all concerned, however. Those in France
+knew nothing of the search going on in England. Mr. Adams's papers had
+never been published; and very annoyed the French were when a claim was
+set up on his behalf to a share in this magnificent discovery.
+Controversies and recriminations, excuses and justifications, followed;
+but the discussion has now settled down. All the world honours the
+bright genius and mathematical skill of Mr. Adams, and recognizes that
+he first solved the problem by calculation. All the world, too,
+perceives clearly the no less eminent mathematical talents of M.
+Leverrier, but it recognizes in him something more than the mere
+mathematician--the man of energy, decision, and character.
+
+
+
+
+LECTURE XVI
+
+COMETS AND METEORS
+
+
+We have now considered the solar system in several aspects, and we have
+passed in review something of what is known about the stars. We have
+seen how each star is itself, in all probability, the centre of another
+and distinct solar system, the constituents of which are too dark and
+far off to be visible to us; nothing visible here but the central sun
+alone, and that only as a twinkling speck.
+
+But between our solar system and these other suns--between each of these
+suns and all the rest--there exist vast empty spaces, apparently devoid
+of matter.
+
+We have now to ask, Are these spaces really empty? Is there really
+nothing in space but the nebulae, the suns, their planets, and their
+satellites? Are all the bodies in space of this gigantic size? May there
+not be an infinitude of small bodies as well?
+
+The answer to this question is in the affirmative. There appears to be
+no special size suited to the vastness of space; we find, as a matter of
+fact, bodies of all manner of sizes, ranging by gradations from the most
+tremendous suns, like Sirius, down through ordinary suns to smaller
+ones, then to planets of all sizes, satellites still smaller, then the
+asteroids, till we come to the smallest satellite of Mars, only about
+ten miles in diameter, and weighing only some billion tons--the smallest
+of the regular bodies belonging to the solar system known.
+
+But, besides all these, there are found to occur other masses, not much
+bigger and some probably smaller, and these we call comets when we see
+them. Below these, again, we find masses varying from a few tons in
+weight down to only a few pounds or ounces, and these when we see them,
+which is not often, we call meteors or shooting-stars; and to the size
+of these meteorites there would appear to be no limit: some may be
+literal grains of dust. There seems to be a regular gradation of size,
+therefore, ranging from Sirius to dust; and apparently we must regard
+all space as full of these cosmic particles--stray fragments, as it
+were, perhaps of some older world, perhaps going to help to form a new
+one some day. As Kepler said, there are more "comets" in the sky than
+fish in the sea. Not that they are at all crowded together, else they
+would make a cosmic haze. The transparency of space shows that there
+must be an enormous proportion of clear space between each, and they are
+probably much more concentrated near one of the big bodies than they are
+in interstellar space.[30] Even during the furious hail of meteors in
+November 1866 it was estimated that their average distance apart in the
+thickest of the shower was 35 miles.
+
+Consider the nature of a meteor or shooting-star. We ordinarily see them
+as a mere streak of light; sometimes they leave a luminous tail behind
+them; occasionally they appear as an actual fire-ball, accompanied by an
+explosion; sometimes, but very seldom, they are seen to drop, and may
+subsequently be dug up as a lump of iron or rock, showing signs of rough
+treatment by excoriation and heat. These last are the meteorites, or
+siderites, or aerolites, or bolides, of our museums. They are popularly
+spoken of as thunderbolts, though they have nothing whatever to do with
+atmospheric electricity.
+
+[Illustration: FIG. 95.--Meteorite.]
+
+They appear to be travelling rocky or metallic fragments which in their
+journey through space are caught in the earth's atmosphere and
+instantaneously ignited by the friction. Far away in the depths of space
+one of these bodies felt the attracting power of the sun, and began
+moving towards him. As it approached, its speed grew gradually quicker
+and quicker continually, until by the time it has approached to within
+the distance of the earth, it whizzes past with the velocity of
+twenty-six miles a second. The earth is moving on its own account
+nineteen miles every second. If the two bodies happened to be moving in
+opposite directions, the combined speed would be terrific; and the
+faintest trace of atmosphere, miles above the earth's surface, would
+exert a furious grinding action on the stone. A stream of particles
+would be torn off; if of iron, they would burn like a shower of filings
+from a firework, thus forming a trail; and the mass itself would be
+dissipated, shattered to fragments in an instant.
+
+[Illustration: FIG. 96.--Meteor stream crossing field of telescope.]
+
+[Illustration: FIG. 97.--Diagram of direction of earth's orbital
+motion, showing that after midnight, _i.e._ between midnight and noon,
+more asteroids are likely to be swept up by any locality than between
+noon and midnight. [From Sir R.S. Ball.]]
+
+Even if the earth were moving laterally, the same thing would occur. But
+if earth and stone happened to be moving in the same direction, there
+would be only the differential velocity of seven miles a second; and
+though this is in all conscience great enough, yet there might be a
+chance for a residue of the nucleus to escape entire destruction, though
+it would be scraped, heated, and superficially molten by the friction;
+but so much of its speed would be rubbed out of it, that on striking
+the earth it might bury itself only a few feet or yards in the soil, so
+that it could be dug out. The number of those which thus reach the earth
+is comparatively infinitesimal. Nearly all get ground up and dissipated
+by the atmosphere; and fortunate it is for us that they are so. This
+bombardment of the exposed face of the moon must be something
+terrible.[31]
+
+Thus, then, every shooting-star we see, and all the myriads that we do
+not and cannot see because they occur in the day-time, all these bright
+flashes or streaks, represent the death and burial of one of these
+flying stones. It had been careering on its own account through space
+for untold ages, till it meets a planet. It cannot strike the actual
+body of the planet--the atmosphere is a sufficient screen; the
+tremendous friction reduces it to dust in an instant, and this dust then
+quietly and leisurely settles down on to the surface.
+
+Evidence of the settlement of meteoric dust is not easy to obtain in
+such a place as England, where the dust which accumulates is seldom of a
+celestial character; but on the snow-fields of Greenland or the
+Himalayas dust can be found; and by a Committee of the British
+Association distinct evidence of molten globules of iron and other
+materials appropriate to aerolites has been obtained, by the simple
+process of collecting, melting, and filtering long exposed snow.
+Volcanic ash may be mingled with it, but under the microscope the
+volcanic and the meteoric constituents have each a distinctive
+character.
+
+The quantity of meteoric material which reaches the earth as dust must
+be immensely in excess of the minute quantity which arrives in the form
+of lumps. Hundreds or thousands of tons per annum must be received; and
+the accretion must, one would think, in the course of ages be able to
+exert some influence on the period of the earth's rotation--the length
+of the day. It is too small, however, to have been yet certainly
+detected. Possibly, it is altogether negligible.
+
+It has been suggested that those stones which actually fall are not the
+true cosmic wanderers, but are merely fragments of our own earth, cast
+up by powerful volcanoes long ago when the igneous power of the earth
+was more vigorous than now--cast up with a speed of close upon seven
+miles a second; and now in these quiet times gradually being swept up by
+the earth, and so returning whence they came.
+
+I confess I am unable to draw a clear distinction between one set and
+the other. Some falling stars may have had an origin of this sort, but
+certainly others have not; and it would seem very unlikely that one set
+only should fall bodily upon the earth, while the others should always
+be rubbed to powder. Still, it is a possibility to be borne in mind.
+
+We have spoken of these cosmic visitors as wandering masses of stone or
+iron; but we should be wrong if we associated with the term "wandering"
+any ideas of lawlessness and irregularity of path. These small lumps of
+matter are as obedient to the law of gravity as any large ones can be.
+They must all, therefore, have definite orbits, and these orbits will
+have reference to the main attracting power of our system--they will, in
+fact, be nearly all careering round the sun.
+
+Each planet may, in truth, have a certain following of its own. Within
+the limited sphere of the earth's predominant attraction, for instance,
+extending some way beyond the moon, we may have a number of satellites
+that we never see, all revolving regularly in elliptic orbits round the
+earth. But, comparatively speaking, these satellite meteorites are few.
+The great bulk of them will be of a planetary character--they will be
+attendant upon the sun.
+
+It may seem strange that such minute bodies should have regular orbits
+and obey Kepler's laws, but they must. All three laws must be as
+rigorously obeyed by them as by the planets themselves. There is nothing
+in the smallness of a particle to excuse it from implicit obedience to
+law. The only consequence of their smallness is their inability to
+perturb others. They cannot appreciably perturb either the planets they
+approach or each other. The attracting power of a lump one million tons
+in weight is very minute. A pound, on the surface of such a body of the
+same density as the earth, would be only pulled to it with a force equal
+to that with which the earth pulls a grain. So the perturbing power of
+such a mass on distant bodies is imperceptible. It is a good thing it is
+so: accurate astronomy would be impossible if we had to take into
+account the perturbations caused by a crowd of invisible bodies.
+Astronomy would then approach in complexity some of the problems of
+physics.
+
+But though we may be convinced from the facts of gravitation that these
+meteoric stones, and all other bodies flying through space near our
+solar system, must be constrained by the sun to obey Kepler's laws, and
+fly round it in some regular elliptic or hyperbolic orbit, what chance
+have we of determining that orbit? At first sight, a very poor chance,
+for we never see them except for the instant when they splash into our
+atmosphere; and for them that instant is instant death. It is unlikely
+that any escape that ordeal, and even if they do, their career and orbit
+are effectually changed. Henceforward they must become attendants on the
+earth. They may drop on to its surface, or they may duck out of our
+atmosphere again, and revolve round us unseen in the clear space between
+earth and moon.
+
+Nevertheless, although the problem of determining the original orbit of
+any given set of shooting-stars before it struck us would seem nearly
+insoluble, it has been solved, and solved with some approach to
+accuracy; being done by the help of observations of certain other
+bodies. The bodies by whose help this difficult problem has been
+attacked and resolved are comets. What are comets?
+
+I must tell you that the scientific world is not entirely and completely
+decided on the structure of comets. There are many floating ideas on the
+subject, and some certain knowledge. But the subject is still, in many
+respects, an open one, and the ideas I propose to advocate you will
+accept for no more than they are worth, viz. as worthy to be compared
+with other and different views.
+
+Up to the time of Newton, the nature of comets was entirely unknown.
+They were regarded with superstitious awe as fiery portents, and were
+supposed to be connected with the death of some king, or with some
+national catastrophe.
+
+Even so late as the first edition of the _Principia_ the problem of
+comets was unsolved, and their theory is not given; but between the
+first and the second editions a large comet appeared, in 1680, and
+Newton speculated on its appearance and behaviour. It rushed down very
+close to the sun, spun half round him very quickly, and then receded
+from him again. If it were a material substance, to which the law of
+gravitation applied, it must be moving in a conic section with the sun
+in one focus, and its radius vector must sweep out equal areas in equal
+times. Examining the record of its positions made at observatories, he
+found its observed path quite accordant with theory; and the motion of
+comets was from that time understood. Up to that time no one had
+attempted to calculate an orbit for a comet. They had been thought
+irregular and lawless bodies. Now they were recognized as perfectly
+obedient to the law of gravitation, and revolving round the sun like
+everything else--as members, in fact, of our solar system, though not
+necessarily permanent members.
+
+But the orbit of a comet is very different from a planetary one. The
+excentricity of its orbit is enormous--in other words, it is either a
+very elongated ellipse or a parabola. The comet of 1680, Newton found
+to move in an orbit so nearly a parabola that the time of describing it
+must be reckoned in hundreds of years at the least. It is now thought
+possible that it may not be quite a parabola, but an ellipse so
+elongated that it will not return till 2255. Until that date arrives,
+however, uncertainty will prevail as to whether it is a periodic comet,
+or one of those that only visit our system once. If it be periodic, as
+suspected, it is the same as appeared when Julius Caesar was killed, and
+which likewise appeared in the years 531 and 1106 A.D. Should it appear
+in 2255, our posterity will probably regard it as a memorial of Newton.
+
+[Illustration: FIG. 98.--Parabolic and elliptic orbits. The _a b_
+(visible) portions are indistinguishable.]
+
+The next comet discussed in the light of the theory of gravitation was
+the famous one of Halley. You know something of the history of this.
+Its period is 75-1/2 years. Halley saw it in 1682, and predicted its
+return in 1758 or 1759--the first cometary prediction. Clairaut
+calculated its return right within a month (p. 219). It has been back
+once more, in 1835; and this time its date was correctly predicted
+within three days, because Uranus was now known. It was away at its
+furthest point in 1873. It will be back again in 1911.
+
+[Illustration: FIG. 99.--Orbit of Halley's comet.]
+
+Coming to recent times, we have the great comets of 1843 and of 1858,
+the history of neither being known. Quite possibly they arrived then for
+the first time. Possibly the second will appear again in 3808. But
+besides these great comets, there are a multitude of telescopic ones,
+which do not show these striking features, and have no gigantic tail.
+Some have no tail at all, others have at best a few insignificant
+streamers, and others show a faint haze looking like a microscopic
+nebula.
+
+All these comets are of considerable extent--some millions of miles
+thick usually, and yet stars are clearly visible through them. Hence
+they must be matter of very small density; their tails can be nothing
+more dense than a filmy mist, but their nucleus must be something more
+solid and substantial.
+
+[Illustration: FIG. 100.--Various appearances of Halley's comet when
+last seen.]
+
+I have said that comets arrive from the depths of space, rush towards
+and round the sun, whizzing past the earth with a speed of twenty-six
+miles a second, on round the sun with a far greater velocity than that,
+and then rush off again. Now, all the time they are away from the sun
+they are invisible. It is only as they get near him that they begin to
+expand and throw off tails and other appendages. The sun's heat is
+evidently evaporating them, and driving away a cloud of mist and
+volatile matter. This is when they can be seen. The comet is most
+gorgeous when it is near the sun, and as soon as it gets a reasonable
+distance away from him it is perfectly invisible.
+
+The matter evaporated from the comet by the sun's heat does not
+return--it is lost to the comet; and hence, after a few such journeys,
+its volatile matter gets appreciably diminished, and so old-established
+periodic comets have no tails to speak of. But the new visitants, coming
+from the depths of space for the first time--these have great supplies
+of volatile matter, and these are they which show the most magnificent
+tails.
+
+[Illustration: FIG. 101.--Head of Donati's comet of 1858.]
+
+The tail of a comet is always directed away from the sun as if it were
+repelled. To this rule there is no exception. It is suggested, and held
+as most probable, that the tail and sun are similarly electrified, and
+that the repulsion of the tail is electrical repulsion. Some great force
+is obviously at work to account for the enormous distance to which the
+tail is shot in a few hours. The pressure of the sun's light can do
+something, and is a force that must not be ignored when small particles
+are being dealt with. (Cf. _Modern Views of Electricity_, 2nd edition,
+p. 363.)
+
+Now just think what analogies there are between comets and meteors. Both
+are bodies travelling in orbits round the sun, and both are mostly
+invisible, but both become visible to us under certain circumstances.
+Meteors become visible when they plunge into the extreme limits of our
+atmosphere. Comets become visible when they approach the sun. Is it
+possible that comets are large meteors which dip into the solar
+atmosphere, and are thus rendered conspicuously luminous? Certainly they
+do not dip into the actual main atmosphere of the sun, else they would
+be utterly destroyed; but it is possible that the sun has a faint trace
+of atmosphere extending far beyond this, and into this perhaps these
+meteors dip, and glow with the friction. The particles thrown off might
+be, also by friction, electrified; and the vaporous tail might be thus
+accounted for.
+
+[Illustration: FIG. 102.--Halley's Comet.]
+
+Let us make this hypothesis provisionally--that comets are large
+meteors, or a compact swarm of meteors, which, coming near the sun, find
+a highly rarefied sort of atmosphere, in which they get heated and
+partly vaporized, just as ordinary meteorites do when they dip into the
+atmosphere of the earth. And let us see whether any facts bear out the
+analogy and justify the hypothesis.
+
+I must tell you now the history of three bodies, and you will see that
+some intimate connection between comets and meteors is proved. The
+three bodies are known as, first, Encke's comet; second, Biela's comet;
+third, the November swarm of meteors.
+
+Encke's comet (one of those discovered by Miss Herschel) is an
+insignificant-looking telescopic comet of small period, the orbit of
+which was well known, and which was carefully observed at each
+reappearance after Encke had calculated its orbit. It was the quickest
+of the comets, returning every 3-1/2 years.
+
+[Illustration: FIG. 103.--Encke's comet.]
+
+It was found, however, that its period was not quite constant; it kept
+on getting slightly shorter. The comet, in fact, returned to the sun
+slightly before its time. Now this effect is exactly what friction
+against a solar atmosphere would bring about. Every time it passed near
+the sun a little velocity would be rubbed out of it. But the velocity is
+that which carries it away, hence it would not go quite so far, and
+therefore would return a little sooner. Any revolving body subject to
+friction must revolve quicker and quicker, and get nearer and nearer
+its central body, until, if the process goes on long enough, it must
+drop upon its surface. This seems the kind of thing happening to Encke's
+comet. The effect is very small, and not thoroughly proved; but, so far
+as it goes, the evidence points to a greatly extended rare solar
+atmosphere, which rubs some energy out of it at every perihelion
+passage.
+
+[Illustration: FIG. 104.--Biela's comet as last seen, in two portions.]
+
+Next, Biela's comet. This also was a well known and carefully observed
+telescopic comet, with a period of six years. In one of its distant
+excursions, it was calculated that it must pass very near Jupiter, and
+much curiosity was excited as to what would happen to it in consequence
+of the perturbation it must experience. As I have said, comets are only
+visible as they approach the sun, and a watch was kept for it about its
+appointed time. It was late, but it did ultimately arrive.
+
+The singular thing about it, however, was that it was now double. It had
+apparently separated into two. This was in 1846. It was looked for again
+in 1852, and this time the components were further separated. Sometimes
+one was brighter, sometimes the other. Next time it ought to have come
+round no one could find either portion. The comet seemed to have wholly
+disappeared. It has never been seen since. It was then recorded and
+advertised as the missing comet.
+
+But now comes the interesting part of the story. The orbit of this Biela
+comet was well known, and it was found that on a certain night in 1872
+the earth would cross the orbit, and had some chance of encountering the
+comet. Not a very likely chance, because it need not be in that part of
+its orbit at the time; but it was suspected not to be far off--if still
+existent. Well, the night arrived, the earth did cross the orbit, and
+there was seen, not the comet, but a number of shooting-stars. Not one
+body, nor yet two, but a multitude of bodies--in fact, a swarm of
+meteors. Not a very great swarm, such as sometimes occurs, but still a
+quite noticeable one; and this shower of meteors is definitely
+recognized as flying along the track of Biela's comet. They are known as
+the Andromedes.
+
+This observation has been generalized. Every cometary orbit is marked by
+a ring of meteoric stones travelling round it, and whenever a number of
+shooting-stars are seen quickly one after the other, it is an evidence
+that we are crossing the track of some comet. But suppose instead of
+only crossing the track of a comet we were to pass close to the comet
+itself, we should then expect to see an extraordinary swarm--a multitude
+of shooting-stars. Such phenomena have occurred. The most famous are
+those known as the November meteors, or Leonids.
+
+This is the third of those bodies whose history I had to tell you.
+Professor H.A. Newton, of America, by examining ancient records arrived
+at the conclusion that the earth passed through a certain definite
+meteor shoal every thirty-three years. He found, in fact, that every
+thirty-three years an unusual flight of shooting-stars was witnessed in
+November, the earliest record being 599 A.D. Their last appearance had
+been in 1833, and he therefore predicted their return in 1866 or 1867.
+Sure enough, in November, 1866, they appeared; and many must remember
+seeing that glorious display. Although their hail was almost continuous,
+it is estimated that their average distance apart was thirty-five miles!
+Their radiant point was and always is in the constellation Leo, and
+hence their name Leonids.
+
+[Illustration: FIG. 105.--Radiant point perspective. The arrows
+represent a number of approximately parallel meteor-streaks
+foreshortened from a common vanishing-point.]
+
+     A parallel stream fixed in space necessarily exhibits a definite
+     aspect with reference to the fixed stars. Its aspect with respect
+     to the earth will be very changeable, because of the rotation and
+     revolution of that body, but its position with respect to
+     constellations will be steady. Hence each meteor swarm, being a
+     steady parallel stream of rushing masses, always strikes us from
+     the same point in stellar space, and by this point (or radiant) it
+     is identified and named.
+
+     The paths do not appear to us to be parallel, because of
+     perspective: they seem to radiate and spread in all directions from
+     a fixed centre like spokes, but all these diverging streaks are
+     really parallel lines optically foreshortened by different amounts
+     so as to produce the radiant impression.
+
+     The annexed diagram (Fig. 105) clearly illustrates the fact that
+     the "radiant" is the vanishing point of a number of parallel lines.
+
+[Illustration: FIG. 106.--Orbit of November meteors.]
+
+This swarm is specially interesting to us from the fact that we cross
+its orbit every year. Its orbit and the earth's intersect. Every
+November we go through it, and hence every November we see a few
+stragglers of this immense swarm. The swarm itself takes thirty-three
+years on its revolution round the sun, and hence we only encounter it
+every thirty-three years.
+
+The swarm is of immense size. In breadth it is such that the earth,
+flying nineteen miles a second, takes four or five hours to cross it,
+and this is therefore the time the display lasts. But in length it is
+far more enormous. The speed with which it travels is twenty-five miles
+a second, (for its orbit extends as far as Uranus, although by no means
+parabolic), and yet it takes more than a year to pass. Imagine a
+procession 200,000 miles broad, every individual rushing along at the
+rate of twenty-five miles every second, and the whole procession so long
+that it takes more than a year to pass. It is like a gigantic shoal of
+herrings swimming round and round the sun every thirty-three years, and
+travelling past the earth with that tremendous velocity of twenty-five
+miles a second. The earth dashes through the swarm and sweeps up
+myriads. Think of the countless numbers swept up by the whole earth in
+crossing such a shoal as that! But heaps more remain, and probably the
+millions which are destroyed every thirty-three years have not yet made
+any very important difference to the numbers still remaining.
+
+The earth never misses this swarm. Every thirty-three years it is bound
+to pass through some part of them, for the shoal is so long that if the
+head is just missed one November the tail will be encountered next
+November. This is a plain and obvious result of its enormous length. It
+may be likened to a two-foot length of sewing silk swimming round and
+round an oval sixty feet in circumference. But, you will say, although
+the numbers are so great that destroying a few millions or so every
+thirty-three years makes but little difference to them, yet, if this
+process has been going on from all eternity, they ought to be all swept
+up. Granted; and no doubt the most ancient swarms have already all or
+nearly all been swept up.
+
+[Illustration: FIG. 107.--Orbit of November meteors; showing their
+probable parabolic orbit previous to 126 A.D., and its sudden conversion
+into an elliptic orbit by the violent perturbation caused by Uranus,
+which at that date occupied the position shown.]
+
+The August meteors, or Perseids, are an example. Every August we cross
+their path, and we have a small meteoric display radiating from the
+sword-hand of Perseus, but never specially more in one August than
+another. It would seem as if the main shoal has disappeared, and nothing
+is now left but the stragglers; or perhaps it is that the shoal has
+gradually become uniformly distributed all along the path. Anyhow, these
+August meteors are reckoned much more ancient members of the solar
+system than are the November meteors. The November meteors are believed
+to have entered the solar system in the year 126 A.D.
+
+This may seem an extraordinary statement. It is not final, but it is
+based on the calculations of Leverrier--confirmed recently by Mr. Adams.
+A few moments will suffice to make the grounds of it clear. Leverrier
+calculated the orbit of the November meteors, and found them to be an
+oval extending beyond Uranus. It was perturbed by the outer planets near
+which it went, so that in past times it must have moved in a slightly
+different orbit. Calculating back to their past positions, it was found
+that in a certain year it must have gone very near to Uranus, and that
+by the perturbation of this planet its path had been completely changed.
+Originally it had in all probability been a comet, flying in a parabolic
+orbit towards the sun like many others. This one, encountering Uranus,
+was pulled to pieces as it were, and its orbit made elliptical as shown
+in Fig. 107. It was no longer free to escape and go away into the depths
+of space: it was enchained and made a member of the solar system. It
+also ceased to be a comet; it was degraded into a shoal of meteors.
+
+This is believed to be the past history of this splendid swarm. Since
+its introduction to the solar system it has made 52 revolutions: its
+next return is due in November, 1899, and I hope that it may occur in
+the English dusk, and (see Fig. 97) in a cloudless after-midnight sky,
+as it did in 1866.
+
+
+
+
+NOTES FOR LECTURE XVII
+
+
+The tide-generating force of one body on another is directly as the mass
+of the one body and inversely as the cube of the distance between them.
+Hence the moon is more effective in producing terrestrial tides than the
+sun.
+
+The tidal wave directly produced by the moon in the open ocean is about
+5 feet high, that produced by the sun is about 2 feet. Hence the average
+spring tide is to the average neap as about 7 to 3. The lunar tide
+varies between apogee and perigee from 4.3 to 5.9.
+
+The solar tide varies between aphelion and perihelion from 1.9 to 2.1.
+Hence the highest spring tide is to the lowest neap as 5.9 + 2.1 is to
+4.3 -2.1, or as 8 to 2.2.
+
+The semi-synchronous oscillation of the Southern Ocean raises the
+magnitude of oceanic tides somewhat above these directly generated
+values.
+
+Oceanic tides are true waves, not currents. Coast tides are currents.
+The momentum of the water, when the tidal wave breaks upon a continent
+and rushes up channels, raises coast tides to a much greater height--in
+some places up to 50 or 60 feet, or even more.
+
+Early observed connections between moon and tides would be these:--
+
+     1st. Spring tides at new and full moon.
+
+     2nd. Average interval between tide and tide is half a lunar, not a
+     solar, day--a lunar day being the interval between two successive
+     returns of the moon to the meridian: 24 hours and 50 minutes.
+
+     3rd. The tides of a given place at new and full moon occur always
+     at the same time of day whatever the season of the year.
+
+
+
+
+LECTURE XVII
+
+THE TIDES
+
+
+Persons accustomed to make use of the Mersey landing-stages can hardly
+fail to have been struck with two obvious phenomena. One is that the
+gangways thereto are sometimes almost level, and at other times very
+steep; another is that the water often rushes past the stage rather
+violently, sometimes south towards Garston, sometimes north towards the
+sea. They observe, in fact, that the water has two periodic motions--one
+up and down, the other to and fro--a vertical and a horizontal motion.
+They may further observe, if they take the trouble, that a complete
+swing of the water, up and down, or to and fro, takes place about every
+twelve and a half hours; moreover, that soon after high and low water
+there is no current--the water is stationary, whereas about half-way
+between high and low it is rushing with maximum speed either up or down
+the river.
+
+To both these motions of the water the name _tide_ is given, and both
+are extremely important. Sailors usually pay most attention to the
+horizontal motion, and on charts you find the tide-races marked; and the
+places where there is but a small horizontal rush of the water are
+labelled "very little tide here." Landsmen, or, at any rate, such of the
+more philosophic sort as pay any attention to the matter at all, think
+most of the vertical motion of the water--its amount of rise and fall.
+
+Dwellers in some low-lying districts in London are compelled to pay
+attention to the extra high tides of the Thames, because it is, or was,
+very liable to overflow its banks and inundate their basements.
+
+Sailors, however, on nearing a port are also greatly affected by the
+time and amount of high water there, especially when they are in a big
+ship; and we know well enough how frequently Atlantic liners, after
+having accomplished their voyage with good speed, have to hang around
+for hours waiting till there is enough water to lift them over the
+Bar--that standing obstruction, one feels inclined to say disgrace, to
+the Liverpool harbour.
+
+[Illustration: FIG. 108.--The Mersey]
+
+To us in Liverpool the tides are of supreme importance--upon them the
+very existence of the city depends--for without them Liverpool would not
+be a port. It may be familiar to many of you how this is, and yet it is
+a matter that cannot be passed over in silence. I will therefore call
+your attention to the Ordnance Survey of the estuaries of the Mersey and
+the Dee. You see first that there is a great tendency for sand-banks to
+accumulate all about this coast, from North Wales right away round to
+Southport. You see next that the port of Chester has been practically
+silted up by the deposits of sand in the wide-mouthed Dee, while the
+port of Liverpool remains open owing to the scouring action of the tide
+in its peculiarly shaped channel. Without the tides the Mersey would be
+a wretched dribble not much bigger than it is at Warrington. With them,
+this splendid basin is kept open, and a channel is cut of such depth
+that the _Great Eastern_ easily rode in it in all states of the water.
+
+The basin is filled with water every twelve hours through its narrow
+neck. The amount of water stored up in this basin at high tide I
+estimate as 600 million tons. All this quantity flows through the neck
+in six hours, and flows out again in the next six, scouring and
+cleansing and carrying mud and sand far out to sea. Just at present the
+currents set strongest on the Birkenhead side of the river, and
+accordingly a "Pluckington bank" unfortunately grows under the Liverpool
+stage. Should this tendency to silt up the gates of our docks increase,
+land can be reclaimed on the other side of the river between Tranmere
+and Rock Ferry, and an embankment made so as to deflect the water over
+Liverpool way, and give us a fairer proportion of the current. After
+passing New Brighton the water spreads out again to the left; its
+velocity forward diminishes; and after a few miles it has no power to
+cut away that sandbank known as the Bar. Should it be thought desirable
+to make it accomplish this, and sweep the Bar further out to sea into
+deeper water, it is probable that a rude training wall (say of old
+hulks, or other removable partial obstruction) on the west of Queen's
+Channel, arranged so as to check the spreading out over all this useless
+area, may be quite sufficient to retain the needed extra impetus in the
+water, perhaps even without choking up the useful old Rock Channel,
+through which smaller ships still find convenient exit.
+
+Now, although the horizontal rush of the tide is necessary to our
+existence as a port, it does not follow that the accompanying rise and
+fall of the water is an unmixed blessing. To it is due the need for all
+the expensive arrangements of docks and gates wherewith to store up the
+high-level water. Quebec and New York are cities on such magnificent
+rivers that the current required to keep open channel is supplied
+without any tidal action, although Quebec is nearly 1,000 miles from the
+open ocean; and accordingly, Atlantic liners do not hover in mid-river
+and discharge passengers by tender, but they proceed straight to the
+side of the quays lining the river, or, as at New York, they dive into
+one of the pockets belonging to the company running the ship, and there
+discharge passengers and cargo without further trouble, and with no need
+for docks or gates. However, rivers like the St. Lawrence and the Hudson
+are the natural property of a gigantic continent; and we in England may
+be well contented with the possession of such tidal estuaries as the
+Mersey, the Thames, and the Humber. That by pertinacious dredging the
+citizens of Glasgow manage to get large ships right up their small
+river, the Clyde, to the quays of the town, is a remarkable fact, and
+redounds very highly to their credit.
+
+We will now proceed to consider the connection existing between the
+horizontal rush of water and its vertical elevation, and ask, Which is
+cause and which is effect? Does the elevation of the ocean cause the
+tidal flow, or does the tidal flow cause the elevation? The answer is
+twofold: both statements are in some sense true. The prime cause of the
+tide is undoubtedly a vertical elevation of the ocean, a tidal wave or
+hump produced by the attraction of the moon. This hump as it passes the
+various channels opening into the ocean raises their level, and causes
+water to flow up them. But this simple oceanic tide, although the cause
+of all tide, is itself but a small affair. It seldom rises above six or
+seven feet, and tides on islands in mid-ocean have about this value or
+less. But the tides on our coasts are far greater than this--they rise
+twenty or thirty feet, or even fifty feet occasionally, at some places,
+as at Bristol. Why is this? The horizontal motion of the water gives it
+such an impetus or momentum that its motion far transcends that of the
+original impulse given to it, just as a push given to a pendulum may
+cause it to swing over a much greater arc than that through which the
+force acts. The inrushing water flowing up the English Channel or the
+Bristol Channel or St. George's Channel has such an impetus that it
+propels itself some twenty or thirty feet high before it has exhausted
+its momentum and begins to descend. In the Bristol Channel the gradual
+narrowing of the opening so much assists this action that the tides
+often rise forty feet, occasionally fifty feet, and rush still further
+up the Severn in a precipitous and extraordinary hill of water called
+"the bore."
+
+Some places are subject to considerable rise and fall of water with very
+little horizontal flow; others possess strong tidal races, but very
+little elevation and depression. The effect observed at any given place
+entirely depends on whether the place has the general character of a
+terminus, or whether it lies _en route_ to some great basin.
+
+You must understand, then, that all tide takes its rise in the free and
+open ocean under the action of the moon. No ordinary-sized sea like the
+North Sea, or even the Mediterranean, is big enough for more than a just
+appreciable tide to be generated in it. The Pacific, the Atlantic, and
+the Southern Oceans are the great tidal reservoirs, and in them the
+tides of the earth are generated as low flat humps of gigantic area,
+though only a few feet high, oscillating up and down in the period of
+approximately twelve hours. The tides we, and other coast-possessing
+nations, experience are the overflow or back-wash of these oceanic
+humps, and I will now show you in what manner the great Atlantic
+tide-wave reaches the British Isles twice a day.
+
+[Illustration: FIG. 109.--Co-tidal lines.]
+
+Fig. 109 shows the contour lines of the great wave as it rolls in east
+from the Atlantic, getting split by the Land's End and by Ireland into
+three portions; one of which rushes up the English Channel and through
+the Straits of Dover. Another rolls up the Irish Sea, with a minor
+offshoot up the Bristol Channel, and, curling round Anglesey, flows
+along the North Wales coast and fills Liverpool Bay and the Mersey. The
+third branch streams round the north coast of Ireland, past the Mull of
+Cantyre and Rathlin Island; part fills up the Firth of Clyde, while the
+rest flows south, and, swirling round the west side of the Isle of Man,
+helps the southern current to fill the Bay of Liverpool. The rest of the
+great wave impinges on the coast of Scotland, and, curling round it,
+fills up the North Sea right away to the Norway coast, and then flows
+down below Denmark, joining the southern and earlier arriving stream.
+The diagram I show you is a rough chart of cotidal lines, which I made
+out of the information contained in _Whitaker's Almanac_.
+
+A place may thus be fed with tide by two distinct channels, and many
+curious phenomena occur in certain places from this cause. Thus it may
+happen that one channel is six hours longer than the other, in which
+case a flow will arrive by one at the same time as an ebb arrives by the
+other; and the result will be that the place will have hardly any tide
+at all, one tide interfering with and neutralizing the other. This is
+more markedly observed at other parts of the world than in the British
+Isles. Whenever a place is reached by two channels of different length,
+its tides are sure to be peculiar, and probably small.
+
+Another cause of small tide is the way the wave surges to and fro in a
+channel. The tidal wave surging up the English Channel, for instance,
+gets largely reflected by the constriction at Dover, and so a crest
+surges back again, as we may see waves reflected in a long trough or
+tilted bath. The result is that Southampton has two high tides rapidly
+succeeding one another, and for three hours the high-water level varies
+but slightly--a fact of evident convenience to the port.
+
+Places on a nodal line, so to speak, about the middle of the length of
+the channel, have a minimum of rise and fall, though the water rushes
+past them first violently up towards Dover, where the rise is
+considerable, and then back again towards the ocean. At Portland, for
+instance, the total rise and fall is very small: it is practically on a
+node. Yarmouth, again, is near a less marked node in the North Sea,
+where stationary waves likewise surge to and fro, and accordingly the
+tidal rise and fall at Yarmouth is only about five feet (varying from
+four and a half to six), whereas at London it is twenty or thirty feet,
+and at Flamborough Head or Leith it is from twelve to sixteen feet.
+
+It is generally supposed that water never flows up-hill, but in these
+cases of oscillation it flows up-hill for three hours together. The
+water is rushing up the English Channel towards Dover long after it is
+highest at the Dover end; it goes on piling itself up, until its
+momentum is checked by the pressure, and then it surges back. It
+behaves, in fact, very like the bob of a pendulum, which rises against
+gravity at every quarter swing.
+
+To get a very large tide, the place ought to be directly accessible by a
+long sweep of a channel to the open ocean, and if it is situate on a
+gradually converging opening, the ebb and flow may be enormous. The
+Severn is the best example of this on the British Isles; but the largest
+tides in the world are found, I believe, in the Bay of Fundy, on the
+coast of North America, where they sometimes rise one hundred and twenty
+feet. Excessive or extra tides may be produced occasionally in any place
+by the propelling force of a high wind driving the water towards the
+shore; also by a low barometer, _i.e._ by a local decrease in the
+pressure of the air.
+
+Well, now, leaving these topographical details concerning tides, which
+we see to be due to great oceanic humps (great in area that is, though
+small in height), let us proceed to ask what causes these humps; and if
+it be the moon that does it, how does it do it?
+
+The statement that the moon causes the tides sounds at first rather an
+absurdity, and a mere popular superstition. Galileo chaffed Kepler for
+believing it. Who it was that discovered the connection between moon and
+tides we know not--probably it is a thing which has been several times
+rediscovered by observant sailors or coast-dwellers--and it is certainly
+a very ancient piece of information.
+
+Probably the first connection observed was that about full moon and
+about new moon the tides are extra high, being called spring tides,
+whereas about half-moon the tides are much less, and are called neap
+tides. The word spring in this connection has no reference to the season
+of the year; except that both words probably represent the same idea of
+energetic uprising or upspringing, while the word neap comes from nip,
+and means pinched, scanty, nipped tide.
+
+The next connection likely to be observed would be that the interval
+between two day tides was not exactly a solar day of twenty-four hours,
+but a lunar day of fifty minutes longer. For by reason of the moon's
+monthly motion it lags behind the sun about fifty minutes a day, and the
+tides do the same, and so perpetually occur later and later, about fifty
+minutes a day later, or 12 hours and 25 minutes on the average between
+tide and tide.
+
+A third and still more striking connection was also discovered by some
+of the ancient great navigators and philosophers--viz. that the time of
+high water at a given place at full moon is always the same, or very
+nearly so. In other words, the highest or spring tides always occur
+nearly at the same time of day at a given place. For instance, at
+Liverpool this time is noon and midnight. London is about two hours and
+a half later. Each port has its own time for receiving a given tide, and
+the time is called the "establishment" of the port. Look out a day when
+the moon is full, and you will find the Liverpool high tide occurs at
+half-past eleven, or close upon it. The same happens when the moon is
+new. A day after full or new moon the spring tides rise to their
+highest, and these extra high tides always occur in Liverpool at noon
+and at midnight, whatever the season of the year. About the equinoxes
+they are liable to be extraordinarily high. The extra low tides here are
+therefore at 6 a.m. and 6 p.m., and the 6 p.m. low tide is a nuisance to
+the river steamers. The spring tides at London are highest about
+half-past two.
+
+*      *      *      *      *
+
+It is, therefore, quite clear that the moon has to do with the tides. It
+and the sun together are, in fact, the whole cause of them; and the mode
+in which these bodies act by gravitative attraction was first made out
+and explained in remarkably full detail by Sir Isaac Newton. You will
+find his account of the tides in the second and third books of the
+_Principia_; and though the theory does not occupy more than a few pages
+of that immortal work, he succeeds not only in explaining the local
+tidal peculiarities, much as I have done to-night, but also in
+calculating the approximate height of mid-ocean solar tide; and from the
+observed lunar tide he shows how to determine the then quite unknown
+mass of the moon. This was a quite extraordinary achievement, the
+difficulty of which it is not easy for a person unused to similar
+discussions fully to appreciate. It is, indeed, but a small part of what
+Newton accomplished, but by itself it is sufficient to confer
+immortality upon any ordinary philosopher, and to place him in a front
+rank.
+
+[Illustration: FIG. 110.--Whirling earth model.]
+
+To make intelligible Newton's theory of the tides, I must not attempt to
+go into too great detail. I will consider only the salient points.
+First, you know that every mass of matter attracts every other piece of
+matter; second, that the moon revolves round the earth, or rather that
+the earth and moon revolve round their common centre of gravity once a
+month; third, that the earth spins on its own axis once a day; fourth,
+that when a thing is whirled round, it tends to fly out from the centre
+and requires a force to hold it in. These are the principles involved.
+You can whirl a bucket full of water vertically round without spilling
+it. Make an elastic globe rotate, and it bulges out into an oblate or
+orange shape; as illustrated by the model shown in Fig. 110. This is
+exactly what the earth does, and Newton calculated the bulging of it as
+fourteen miles all round the equator. Make an elastic globe revolve
+round a fixed centre outside itself, and it gets pulled into a prolate
+or lemon shape; the simplest illustrative experiment is to attach a
+string to an elastic bag or football full of water, and whirl it round
+and round. Its prolateness is readily visible.
+
+Now consider the earth and moon revolving round each other like a man
+whirling a child round. The child travels furthest, but the man cannot
+merely rotate, he leans back and thus also describes a small circle: so
+does the earth; it revolves round the common centre of gravity of earth
+and moon (_cf._ p. 212). This is a vital point in the comprehension of
+the tides: the earth's centre is not at rest, but is being whirled round
+by the moon, in a circle about 1/80 as big as the circle which the moon
+describes, because the earth weighs eighty times as much as the moon.
+The effect of the revolution is to make both bodies slightly protrude in
+the direction of the line joining them; they become slightly "prolate"
+as it is called--that is, lemon-shaped. Illustrating still by the man
+and child, the child's legs fly outwards so that he is elongated in the
+direction of a radius; the man's coat-tails fly out too, so that he too
+is similarly though less elongated. These elongations or protuberances
+constitute the tides.
+
+[Illustration: FIG. 111.--Earth and moon model, illustrating the
+production of statical or "equilibrium" tides when the whole is whirled
+about the point G.]
+
+Fig. 111 shows a model to illustrate the mechanism. A couple of
+cardboard disks (to represent globes of course), one four times the
+diameter of the other, and each loaded so as to have about the correct
+earth-moon ratio of weights, are fixed at either end of a long stick,
+and they balance about a certain point, which is their common centre of
+gravity. For convenience this point is taken a trifle too far out from
+the centre of the earth--that is, just beyond its surface. Through the
+balancing point G a bradawl is stuck, and on that as pivot the whole
+readily revolves. Now, behind the circular disks, you see, are four
+pieces of card of appropriate shape, which are able to slide out under
+proper forces. They are shown dotted in the figure, and are lettered A,
+B, C, D. The inner pair, B and C, are attached to each other by a bit of
+string, which has to typify the attraction of gravitation; the outer
+pair, A and D, are not attached to anything, but have a certain amount
+of play against friction in slots parallel to the length of the stick.
+The moon-disk is also slotted, so a small amount of motion is possible
+to it along the stick or bar. These things being so arranged, and the
+protuberant pieces of card being all pushed home, so that they are
+hidden behind their respective disks, the whole is spun rapidly round
+the centre of gravity, G. The result of a brief spin is to make A and D
+fly out by centrifugal force and show, as in the figure; while the moon,
+flying out too in its slot, tightens up the string, which causes B and C
+to be pulled out too. Thus all four high tides are produced, two on the
+earth and two on the moon, A and D being caused by centrifugal force, B
+and C by the attraction of gravitation. Each disk has become prolate in
+the same sort of fashion as yielding globes do. Of course the fluid
+ocean takes this shape more easily and more completely than the solid
+earth can, and so here are the very oceanic humps we have been talking
+about, and about three feet high (Fig. 112). If there were a sea on the
+_moon_, its humps would be a good deal bigger; but there probably is no
+sea there, and if there were, the earth's tides are more interesting to
+us, at any rate to begin with.
+
+[Illustration: FIG. 112.--Earth and moon (earth's rotation neglected).]
+
+The humps as so far treated are always protruding in the earth-moon
+line, and are stationary. But now we have to remember that the earth is
+spinning inside them. It is not easy to see what precise effect this
+spin will have upon the humps, even if the world were covered with a
+uniform ocean; but we can see at any rate that however much they may get
+displaced, and they do get displaced a good deal, they cannot possibly
+be carried round and round. The whole explanation we have given of their
+causes shows that they must maintain some steady aspect with respect to
+the moon--in other words, they must remain stationary as the earth spins
+round. Not that the same identical water remains stationary, for in that
+case it would have to be dragged over the earth's equator at the rate of
+1,000 miles an hour, but the hump or wave-crest remains stationary. It
+is a true wave, or form only, and consists of continuously changing
+individual particles. The same is true of all waves, except breaking
+ones.
+
+Given, then, these stationary humps and the earth spinning on its axis,
+we see that a given place on the earth will be carried round and round,
+now past a hump, and six hours later past a depression: another six
+hours and it will be at the antipodal hump, and so on. Thus every six
+hours we shall travel from the region in space where the water is high
+to the region where it is low; and ignoring our own motion we shall say
+that the sea first rises and then falls; and so, with respect to the
+place, it does. Thus the succession of high and low water, and the two
+high tides every twenty-four hours, are easily understood in their
+easiest and most elementary aspect. A more complete account of the
+matter it will be wisest not to attempt: suffice it to say that the
+difficulties soon become formidable when the inertia of the water, its
+natural time of oscillation, the varying obliquity of the moon to the
+ecliptic, its varying distance, and the disturbing action of the sun are
+taken into consideration. When all these things are included, the
+problem becomes to ordinary minds overwhelming. A great many of these
+difficulties were successfully attacked by Laplace. Others remained for
+modern philosophers, among whom are Sir George Airy, Sir William
+Thomson, and Professor George Darwin.
+
+     I may just mention that the main and simplest effect of including
+     the inertia or momentum of the water is to dislocate the obvious
+     and simple connexion between high water and high moon; inertia
+     always tends to make an effect differ in phase by a quarter period
+     from the cause producing it, as may be illustrated by a swinging
+     pendulum. Hence high water is not to be expected when the
+     tide-raising force is a maximum, but six hours later; so that,
+     considering inertia and neglecting friction, there would be low
+     water under the moon. Including friction, something nearer the
+     equilibrium state of things occurs. With _sufficient_ friction the
+     motion becomes dead-beat again, _i.e._ follows closely the force
+     that causes it.
+
+Returning to the elementary discussion, we see that the rotation of the
+earth with respect to the humps will not be performed in exactly
+twenty-four hours, because the humps are travelling slowly after the
+moon, and will complete a revolution in a month in the same direction as
+the earth is rotating. Hence a place on the earth has to catch them up,
+and so each high tide arrives later and later each day--roughly
+speaking, an hour later for each day tide; not by any means a constant
+interval, because of superposed disturbances not here mentioned, but on
+the average about fifty minutes.
+
+We see, then, that as a result of all this we get a pair of humps
+travelling all over the surface of the earth, about once a day. If the
+earth were all ocean (and in the southern hemisphere it is nearly all
+ocean), then they would go travelling across the earth, tidal waves
+three feet high, and constituting the mid-ocean tides. But in the
+northern hemisphere they can only thus journey a little way without
+striking land. As the moon rises at a place on the east shores of the
+Atlantic, for instance, the waters begin to flow in towards this place,
+or the tide begins to rise. This goes on till the moon is overhead and
+for some time afterwards, when the tide is at its highest. The hump then
+follows the moon in its apparent journey across to America, and there
+precipitates itself upon the coast, rushing up all the channels, and
+constituting the land tide. At the same time, the water is dragged away
+from the east shores, and so _our_ tide is at its lowest. The same thing
+repeats itself in a little more than twelve hours again, when the other
+hump passes over the Atlantic, as the moon journeys beneath the earth,
+and so on every day.
+
+     In the free Southern Ocean, where land obstruction is comparatively
+     absent, the water gets up a considerable swing by reason of its
+     accumulated momentum, and this modifies and increases the open
+     ocean tides there. Also for some reason, I suppose because of the
+     natural time of swing of the water, one of the humps is there
+     usually much larger than the other; and so places in the Indian and
+     other offshoots of the Southern Ocean get their really high tide
+     only once every twenty-four hours. These southern tides are in fact
+     much more complicated than those the British Isles receive. Ours
+     are singularly simple. No doubt some trace of the influence of the
+     Southern Ocean is felt in the North Atlantic, but any ocean
+     extending over 90 deg. of longitude is big enough to have its own
+     tides generated; and I imagine that the main tides we feel are thus
+     produced on the spot, and that they are simple because the
+     damping-out being vigorous, and accumulated effects small, we feel
+     the tide-producing forces more directly. But for authoritative
+     statements on tides, other books must be read. I have thought, and
+     still think, it best in an elementary exposition to begin by a
+     consideration of the tide-generating forces as if they acted on a
+     non-rotating earth. It is the tide generating forces, and not the
+     tides themselves, that are really represented in Figs. 112 and 114.
+     The rotation of the earth then comes in as a disturbing cause. A
+     more complete exposition would begin with the rotating earth, and
+     would superpose the attraction of the moon as a disturbing cause,
+     treating it as a problem in planetary perturbation, the ocean being
+     a sort of satellite of the earth. This treatment, introducing
+     inertia but ignoring friction and land obstruction, gives low water
+     in the line of pull, and high water at right angles, or where the
+     pull is zero; in the same sort of way as a pendulum bob is highest
+     where most force is pulling it down, and lowest where no force is
+     acting on it. For a clear treatment of the tides as due to the
+     perturbing forces of sun and moon, see a little book by Mr. T.K.
+     Abbott of Trinity College, Dublin. (Longman.)
+
+[Illustration: FIG. 113.--Maps showing how comparatively free from land
+obstruction the ocean in the Southern Hemisphere is.]
+
+If the moon were the only body that swung the earth round, this is all
+that need be said in an elementary treatment; but it is not the only
+one. The moon swings the earth round once a month, the sun swings it
+round once a year. The circle of swing is bigger, but the speed is so
+much slower that the protuberance produced is only one-third of that
+caused by the monthly whirl; _i.e._ the simple solar tide in the open
+sea, without taking momentum into account, is but a little more than a
+foot high, while the simple lunar tide is about three feet. When the two
+agree, we get a spring tide of four feet; when they oppose each other,
+we get a neap tide of only two feet. They assist each other at full moon
+and at new moon. At half-moon they oppose each other. So we have spring
+tides regularly once a fortnight, with neap tides in between.
+
+[Illustration: FIG. 114.--Spring and neap tides.]
+
+Fig. 114 gives the customary diagrams to illustrate these simple things.
+You see that when the moon and sun act at right angles (_i.e._ at every
+half-moon), the high tides of one coincide with the low tides of the
+other; and so, as a place is carried round by the earth's rotation, it
+always finds either solar or else lunar high water, and only experiences
+the difference of their two effects. Whereas, when the sun and moon act
+in the same line (as they do at new and full moon), their high and low
+tides coincide, and a place feels their effects added together. The tide
+then rises extra high and falls extra low.
+
+[Illustration: FIG. 115.--Tidal clock. The position of the disk B shows
+the height of the tide. The tide represented is a nearly high tide eight
+feet above mean level.]
+
+Utilizing these principles, a very elementary form of tidal-clock, or
+tide-predicter, can be made, and for an open coast station it really
+would not give the tides so very badly. It consists of a sort of clock
+face with two hands, one nearly three times as long as the other. The
+short hand, CA, should revolve round C once in twelve hours, and the
+vertical height of its end A represents the height of the solar tide on
+the scale of horizontal lines ruled across the face of the clock. The
+long hand, AB, should revolve round A once in twelve hours and
+twenty-five minutes, and the height of its end B (if A were fixed on the
+zero line) would represent the lunar tide. The two revolutions are made
+to occur together, either by means of a link-work parallelogram, or,
+what is better in practice, by a string and pulleys, as shown; and the
+height of the end point, B, of the third side or resultant, CB, read off
+on a scale of horizontal parallel lines behind, represents the
+combination or actual tide at the place. Every fortnight the two will
+agree, and you will get spring tides of maximum height CA + AB; every
+other fortnight the two will oppose, and you will get neap tides of
+maximum height CA-AB.
+
+Such a clock, if set properly and driven in the ordinary way, would then
+roughly indicate the state of the tide whenever you chose to look at it
+and read the height of its indicating point. It would not indeed be very
+accurate, especially for such an inclosed station as Liverpool is, and
+that is probably why they are not made. A great number of disturbances,
+some astronomical, some terrestrial, have to be taken into account in
+the complete theory. It is not an easy matter to do this, but it can be,
+and has been, done; and a tide-predicter has not only been constructed,
+but two of them are in regular work, predicting the tides for years
+hence--one, the property of the Indian Government, for coast stations of
+India; the other for various British and foreign stations, wherever the
+necessary preliminary observations have been made. These machines are
+the invention of Sir William Thomson. The tide-tables for Indian ports
+are now always made by means of them.
+
+[Illustration: FIG. 116.--Sir William Thomson (Lord Kelvin).]
+
+[Illustration: FIG. 117.--Tide-gauge for recording local tides, a
+pencil moved up and down by a float writes on a drum driven by
+clockwork.]
+
+The first thing to be done by any port which wishes its tides to be
+predicted is to set up a tide-gauge, or automatic recorder, and keep it
+working for a year or two. The tide-gauge is easy enough to understand:
+it marks the height of the tide at every instant by an irregular curved
+line like a barometer chart (Fig. 117). These observational curves so
+obtained have next to be fed into a fearfully complex machine, which it
+would take a whole lecture to make even partially intelligible, but Fig.
+118 shows its aspect. It consists of ten integrating machines in a row,
+coupled up and working together. This is the "harmonic analyzer," and
+the result of passing the curve through this machine is to give you all
+the constituents of which it is built up, viz. the lunar tide, the solar
+tide, and eight of the sub-tides or disturbances. These ten values are
+then set off into a third machine, the tide-predicter proper. The
+general mode of action of this machine is not difficult to understand.
+It consists of a string wound over and under a set of pulleys, which are
+each set on an excentric, so as to have an up-and-down motion. These
+up-and-down motions are all different, and there are ten of these
+movable pulleys, which by their respective excursions represent the
+lunar tide, the solar tide, and the eight disturbances already analyzed
+out of the tide-gauge curve by the harmonic analyzer. One end of the
+string is fixed, the other carries a pencil which writes a trace on a
+revolving drum of paper--a trace which represents the combined motion of
+all the pulleys, and so predicts the exact height of the tide at the
+place, at any future time you like. The machine can be turned quite
+quickly, so that a year's tides can be run off with every detail in
+about half-an-hour. This is the easiest part of the operation. Nothing
+has to be done but to keep it supplied with paper and pencil, and turn a
+handle as if it were a coffee-mill instead of a tide-mill. (Figs. 119
+and 120.)
+
+[Illustration: FIG. 118.--Harmonic analyzer; for analyzing out the
+constituents from a set of observational curves.]
+
+My subject is not half exhausted. I might go on to discuss the question
+of tidal energy--whether it can be ever utilized for industrial
+purposes; and also the very interesting question whence it comes. Tidal
+energy is almost the only terrestrial form of energy that does not
+directly or indirectly come from the sun. The energy of tides is now
+known to be obtained at the expense of the earth's rotation; and
+accordingly our day must be slowly, very slowly, lengthening. The tides
+of past ages have destroyed the moon's rotation, and so it always turns
+the same face to us. There is every reason to believe that in geologic
+ages the moon was nearer to us than it is now, and that accordingly our
+tides were then far more violent, rising some hundreds of feet instead
+of twenty or thirty, and sweeping every six hours right over the face of
+a country, ploughing down hills, denuding rocks, and producing a copious
+sedimentary deposit.
+
+[Illustration: FIG. 119.--Tide-predicter, for combining the ascertained
+constituents into a tidal curve for the future.]
+
+In thus discovering the probable violent tides of past ages, astronomy
+has, within the last few years, presented geology with the most powerful
+denuding agent known; and the study of the earth's past history cannot
+fail to be greatly affected by the modern study of the intricate and
+refined conditions attending prolonged tidal action on incompletely
+rigid bodies. [Read on this point the last chapter of Sir R. Ball's
+_Story of the Heavens_.]
+
+[Illustration: Fig. 120.--Weekly sheet of curves. Tides for successive
+days are predicted on the same sheet of paper, to economise space.]
+
+I might also point out that the magnitude of our terrestrial tides
+enables us to answer the question as to the internal fluidity of the
+earth. It used to be thought that the earth's crust was comparatively
+thin, and that it contained a molten interior. We now know that this is
+not the case. The interior of the earth is hot indeed, but it is not
+fluid. Or at least, if it be fluid, the amount of fluid is but very
+small compared with the thickness of the unyielding crust. All these,
+and a number of other most interesting questions, fringe the subject of
+the tides; the theoretical study of which, started by Newton, has
+developed, and is destined in the future to further develop, into one of
+the most gigantic and absorbing investigations--having to do with the
+stability or instability of solar systems, and with the construction and
+decay of universes.
+
+These theories are the work of pioneers now living, whose biographies it
+is therefore unsuitable for us to discuss, nor shall I constantly
+mention their names. But Helmholtz, and Thomson, are household words,
+and you well know that in them and their disciples the race of Pioneers
+maintains its ancient glory.
+
+
+
+
+NOTES FOR LECTURE XVIII
+
+
+Tides are due to incomplete rigidity of bodies revolving round each
+other under the action of gravitation, and at the same time spinning on
+their axes.
+
+Two spheres revolving round each other can only remain spherical if
+rigid; if at all plastic they become prolate. If either rotate on its
+axis, in the same or nearly the same plane as it revolves, that one is
+necessarily subject to tides.
+
+The axial rotation tends to carry the humps with it, but the pull of the
+other body keeps them from moving much. Hence the rotation takes place
+against a pull, and is therefore more or less checked and retarded. This
+is the theory of Von Helmholtz.
+
+The attracting force between two such bodies is no longer _exactly_
+towards the centre of revolution, and therefore Kepler's second law is
+no longer precisely obeyed: the rate of description of areas is subject
+to slight acceleration. The effect of this tangential force acting on
+the tide-compelling body is gradually to increase its distance from the
+other body.
+
+Applying these statements to the earth and moon, we see that tidal
+energy is produced at the expense of the earth's rotation, and that the
+length of the day is thereby slowly increasing. Also that the moon's
+rotation relative to the earth has been destroyed by past tidal action
+in it (the only residue of ancient lunar rotation now being a scarcely
+perceptible libration), so that it turns always the same face towards
+us. Moreover, that its distance from the earth is steadily increasing.
+This last is the theory of Professor G.H. Darwin.
+
+Long ago the moon must therefore have been much nearer the earth, and
+the day was much shorter. The tides were then far more violent.
+
+Halving the distance would make them eight times as high; quartering it
+would increase them sixty-four-fold. A most powerful geological denuding
+agent. Trade winds and storms were also more violent.
+
+If ever the moon were close to the earth, it would have to revolve round
+it in about three hours. If the earth rotated on its axis in three
+hours, when fluid or pasty, it would be unstable, and begin to separate
+a portion of itself as a kind of bud, which might then get detached and
+gradually pushed away by the violent tidal action. Hence it is possible
+that this is the history of the moon. If so, it is probably an
+exceptional history. The planets were not formed from the sun in this
+way.
+
+Mars' moons revolve round him more quickly than the planet rotates:
+hence with them the process is inverted, and they must be approaching
+him and may some day crash along his surface. The inner moon is now
+about 4,000 miles away, and revolves in 7-1/2 hours. It appears to be
+about 20 miles in diameter, and weighs therefore, if composed of rock,
+40 billion tons. Mars rotates in 24-1/2 hours.
+
+A similar fate may _possibly_ await our moon ages hence--by reason of
+the action of terrestrial tides produced by the sun.
+
+
+
+
+LECTURE XVIII
+
+THE TIDES, AND PLANETARY EVOLUTION
+
+
+In the last lecture we considered the local peculiarities of the tides,
+the way in which they were formed in open ocean under the action of the
+moon and the sun, and also the means by which their heights and times
+could be calculated and predicted years beforehand. Towards the end I
+stated that the subject was very far from being exhausted, and
+enumerated some of the large and interesting questions which had been
+left untouched. It is with some of these questions that I propose now to
+deal.
+
+I must begin by reminding you of certain well-known facts, a knowledge
+of which I may safely assume.
+
+And first we must remind ourselves of the fact that almost all the rocks
+which form the accessible crust of the earth were deposited by the
+agency of water. Nearly all are arranged in regular strata, and are
+composed of pulverized materials--materials ground down from
+pre-existing rocks by some denuding and grinding action. They nearly all
+contain vestiges of ancient life embedded in them, and these vestiges
+are mainly of marine origin. The strata which were once horizontal are
+now so no longer--they have been tilted and upheaved, bent and
+distorted, in many places. Some of them again have been metamorphosed by
+fire, so that their organic remains have been destroyed, and the traces
+of their aqueous origin almost obliterated. But still, to the eye of the
+geologist, all are of aqueous or sedimentary origin: roughly speaking,
+one may say they were all deposited at the bottom of some ancient sea.
+
+The date of their formation no man yet can tell, but that it was vastly
+distant is certain. For the geological era is not over. Aqueous action
+still goes on: still does frost chip the rocks into fragments; still do
+mountain torrents sweep stone and mud and _debris_ down the gulleys and
+watercourses; still do rivers erode their channels, and carry mud and
+silt far out to sea. And, more powerful than any of these agents of
+denudation, the waves and the tides are still at work along every
+coast-line, eating away into the cliffs, undermining gradually and
+submerging acre after acre, and making with the refuse a shingly, or a
+sandy, or a muddy beach--the nucleus of a new geological formation.
+
+Of all denuding agents, there can be no doubt that, to the land exposed
+to them, the waves of the sea are by far the most powerful. Think how
+they beat and tear, and drive and drag, until even the hardest rock,
+like basalt, becomes honeycombed into strange galleries and
+passages--Fingal's Cave, for instance--and the softer parts are crumbled
+away. But the area now exposed to the teeth of the waves is not great.
+The fury of a winter storm may dash them a little higher than usual, but
+they cannot reach cliffs 100 feet high. They can undermine such cliffs
+indeed, and then grind the fragments to powder, but their direct action
+is limited. Not so limited, however, as they would be without the tides.
+Consider for a moment the denudation import of the tides: how does the
+existence of tidal rise and fall affect the geological problem?
+
+The scouring action of the tidal currents themselves is not to be
+despised. It is the tidal ebb and flow which keeps open channel in the
+Mersey, for instance. But few places are so favourably situated as
+Liverpool in this respect, and the direct scouring action of the tides
+in general is not very great. Their geological import mainly consists in
+this--that they raise and lower the surface waves at regular intervals,
+so as to apply them to a considerable stretch of coast. The waves are a
+great planing machine attacking the land, and the tides raise and lower
+this planing machine, so that its denuding tooth is applied, now twenty
+feet vertically above mean level, now twenty feet below.
+
+Making all allowance for the power of winds and waves, currents, tides,
+and watercourses, assisted by glacial ice and frost, it must be apparent
+how slowly the work of forming the rocks is being carried on. It goes on
+steadily, but so slowly that it is estimated to take 6000 years to wear
+away one foot of the American continent by all the denuding causes
+combined. To erode a stratum 5000 feet thick will require at this rate
+thirty million years.
+
+The age of the earth is not at all accurately known, but there are many
+grounds for believing it not to be much older than some thirty million
+years. That is to say, not greatly more than this period of time has
+elapsed since it was in a molten condition. It may be as old as a
+hundred million years, but its age is believed by those most competent
+to judge to be more likely within this limit than beyond it. But if we
+ask what is the thickness of the rocks which in past times have been
+formed, and denuded, and re-formed, over and over again, we get an
+answer, not in feet, but in miles. The Laurentian and Huronian rocks of
+Canada constitute a stratum ten miles thick; and everywhere the rocks at
+the base of our stratified system are of the most stupendous volume and
+thickness.
+
+It has always been a puzzle how known agents could have formed these
+mighty masses, and the only solution offered by geologists was,
+unlimited time. Given unlimited time, they could, of course, be formed,
+no matter how slowly the process went on. But inasmuch as the time
+allowable since the earth was cool enough for water to exist on it
+except as steam is not by any means unlimited, it becomes necessary to
+look for a far more powerful engine than any now existing; there must
+have been some denuding agent in those remote ages--ages far more
+distant from us than the Carboniferous period, far older than any forms
+of life, fossil or otherwise, ages among the oldest known to geology--a
+denuding agent must have then existed, far more powerful than any we now
+know.
+
+Such an agent it has been the privilege of astronomy and physics, within
+the last ten years, to discover. To this discovery I now proceed to lead
+up.
+
+Our fundamental standard of time is the period of the earth's
+rotation--the length of the day. The earth is our one standard clock:
+all time is expressed in terms of it, and if it began to go wrong, or if
+it did not go with perfect uniformity, it would seem a most difficult
+thing to discover its error, and a most puzzling piece of knowledge to
+utilize when found.
+
+That it does not go much wrong is proved by the fact that we can
+calculate back to past astronomical events--ancient eclipses and the
+like--and we find that the record of their occurrence, as made by the
+old magi of Chaldaea, is in very close accordance with the result of
+calculation. One of these famous old eclipses was observed in Babylon
+about thirty-six centuries ago, and the Chaldaean astronomers have put on
+record the time of its occurrence. Modern astronomers have calculated
+back when it should have occurred, and the observed time agrees very
+closely with the actual, but not exactly. Why not exactly?
+
+Partly because of the acceleration of the moon's mean motion, as
+explained in the lecture on Laplace (p. 262). The orbit of the earth was
+at that time getting rounder, and so, as a secondary result, the speed
+of the moon was slightly increasing. It is of the nature of a
+perturbation, and is therefore a periodic not a progressive or
+continuous change, and in a sufficiently long time it will be reversed.
+Still, for the last few thousand years the moon's motion has been, on
+the whole, accelerated (though there seems to be a very slight retarding
+force in action too).
+
+Laplace thought that this fact accounted for the whole of the
+discrepancy; but recently, in 1853, Professor Adams re-examined the
+matter, and made a correction in the details of the theory which
+diminishes its effect by about one-half, leaving the other half to be
+accounted for in some other way. His calculations have been confirmed by
+Professor Cayley. This residual discrepancy, when every known cause has
+been allowed for, amounts to about one hour.
+
+     The eclipse occurred later than calculation warrants. Now this
+     would have happened from either of two causes, either an
+     acceleration of the moon in her orbit, or a retardation of the
+     earth in her diurnal rotation--a shortening of the month or a
+     lengthening of the day, or both. The total discrepancy being, say,
+     two hours, an acceleration of six seconds-per-century per century
+     will in thirty-six centuries amount to one hour; and this,
+     according to the corrected Laplacian theory, is what has occurred.
+     But to account for the other hour some other cause must be sought,
+     and at present it is considered most probably due to a steady
+     retardation of the earth's rotation--a slow, very slow, lengthening
+     of the day.
+
+     The statement that a solar eclipse thirty-six centuries ago was an
+     hour late, means that a place on the earth's surface came into the
+     shadow one hour behind time--that is, had lagged one twenty-fourth
+     part of a revolution. The earth, therefore, had lost this amount in
+     the course of 3600 x 365-1/4 revolutions. The loss per revolution
+     is exceedingly small, but it accumulates, and at any era the total
+     loss is the sum of all the losses preceding it. It may be worth
+     while just to explain this point further.
+
+     Suppose the earth loses a small piece of time, which I will call an
+     instant, per day; a locality on the earth will come up to a given
+     position one instant late on the first day after an event. On the
+     next day it would come up two instants late by reason of the
+     previous loss; but it also loses another instant during the course
+     of the second day, and so the total lateness by the end of that day
+     amounts to three instants. The day after, it will be going slower
+     from the beginning at the rate of two instants a day, it will lose
+     another instant on the fresh day's own account, and it started
+     three instants late; hence the aggregate loss by the end of the
+     third day is 1 + 2 + 3 = 6. By the end of the fourth day the whole
+     loss will be 1 + 2 + 3 + 4, and so on. Wherefore by merely losing
+     one instant every day the total loss in _n_ days is (1 + 2 + 3 +
+     ... + _n_) instants, which amounts to 1/2_n_ (_n_ + 1) instants;
+     or practically, when _n_ is big, to 1/2n^2. Now in thirty-six
+     centuries there have been 3600 x 365-1/4 days, and the total loss
+     has amounted to an hour; hence the length of "an instant," the loss
+     per diem, can be found from the equation 1/2(3600 x 365)^2 instants
+     = 1 hour; whence one "instant" equals the 240 millionth part of a
+     second. This minute quantity represents the retardation of the
+     earth per day. In a year the aggregate loss mounts up to 1/3600th
+     part of a second, in a century to about three seconds, and in
+     thirty-six centuries to an hour. But even at the end of the
+     thirty-six centuries the day is barely any longer; it is only 3600
+     x 365 instants, that is 1/180th of a second, longer than it was at
+     the beginning. And even a million years ago, unless the rate of
+     loss was different (as it probably was), the day would only be
+     thirty-five minutes shorter, though by that time the aggregate
+     loss, as measured by the apparent lateness of any perfectly
+     punctual event reckoned now, would have amounted to nine years.
+     (These numbers are to be taken as illustrative, not as precisely
+     representing terrestrial fact.)
+
+What can have caused the slowing down? Swelling of the earth by reason
+of accumulation of meteoric dust might do something, but probably very
+little. Contraction of the earth as it goes on cooling would act in the
+opposite direction, and probably more than counterbalance the dust
+effect. The problem is thus not a simple one, for there are several
+disturbing causes, and for none of them are the data enough to base a
+quantitative estimate upon; but one certain agent in lengthening the
+day, and almost certainly the main agent, is to be found in the tides.
+
+Remember that the tidal humps were produced as the prolateness of a
+sphere whirled round and round a fixed centre, like a football whirled
+by a string. These humps are pulled at by the moon, and the earth
+rotates on its axis against this pull. Hence it tends to be constantly,
+though very slightly, dragged back.
+
+In so far as the tidal wave is allowed to oscillate freely, it will
+swing with barely any maintaining force, giving back at one
+quarter-swing what it has received at the previous quarter; but in so
+far as it encounters friction, which it does in all channels where
+there is an actual ebb and flow of the water, it has to receive more
+than it gives back, and the balance of energy has to be made up to it,
+or the tides would cease. The energy of the tides is, in fact,
+continually being dissipated by friction, and all the energy so
+dissipated is taken from the rotation of the earth. If tidal energy were
+utilized by engineers, the machines driven would be really driven at the
+expense of the earth's rotation: it would be a mode of harnessing the
+earth and using the moon as fixed point or fulcrum; the moon pulling at
+the tidal protuberance, and holding it still as the earth rotates, is
+the mechanism whereby the energy is extracted, the handle whereby the
+friction brake is applied.
+
+     Winds and ocean currents have no such effect (as Mr. Fronde in
+     _Oceania_ supposes they have), because they are all accompanied by
+     a precisely equal counter-current somewhere else, and no internal
+     rearrangement of fluid can affect the motion of a mass as a whole;
+     but the tides are in different case, being produced, not by
+     internal inequalities of temperature, but by a straightforward pull
+     from an external body.
+
+The ultimate effect of tidal friction and dissipation of energy will,
+therefore, be to gradually retard the earth till it does not rotate with
+reference to the moon, _i.e._ till it rotates once while the moon
+revolves once; in other words, to make the day and the month equal. The
+same cause must have been in operation, but with eighty-fold greater
+intensity, on the moon. It has ceased now, because the rotation has
+stopped, but if ever the moon rotated on its axis with respect to the
+earth, and if it were either fluid itself or possessed any liquid ocean,
+then the tides caused by the pull of the earth must have been
+prodigious, and would tend to stop its rotation. Have they not
+succeeded? Is it not probable that this is _why_ the moon always now
+turns the same face towards us? It is believed to be almost certainly
+the cause. If so, there was a time when the moon behaved
+differently--when it rotated more quickly than it revolved, and
+exhibited to us its whole surface. And at this era, too, the earth
+itself must have rotated a little faster, for it has been losing speed
+ever since.
+
+We have thus arrived at this fact, that a thousand years ago the day was
+a trifle shorter than it is now. A million years ago it was, perhaps, an
+hour shorter. Twenty million years ago it must have been much shorter.
+Fifty million years ago it may have been only a few hours long. The
+earth may have spun round then quite quickly. But there is a limit. If
+it spun too fast it would fly to pieces. Attach shot by means of wax to
+the whirling earth model, Fig. 110, and at a certain speed the cohesion
+of the wax cannot hold them, so they fly off. The earth is held together
+not by cohesion but by gravitation; it is not difficult to reckon how
+fast the earth must spin for gravity at its surface to be annulled, and
+for portions to fly off. We find it about one revolution in three hours.
+This is a critical speed. If ever the day was three hours long,
+something must have happened. The day can never have been shorter than
+that; for if it were, the earth would have a tendency to fly in pieces,
+or, at least, to separate into two pieces. Remember this, as a natural
+result of a three-hour day, which corresponds to an unstable state of
+things; remember also that in some past epoch a three-hour day is a
+probability.
+
+     If we think of the state of things going on in the earth's
+     atmosphere, if it had an atmosphere at that remote date, we shall
+     recognize the existence of the most fearful tornadoes. The trade
+     winds, which are now peaceful agents of commerce, would then be
+     perpetual hurricanes, and all the denudation agents of the
+     geologist would be in a state of feverish activity. So, too, would
+     the tides: instead of waiting six hours between low and high tide,
+     we should have to wait only three-quarters of an hour. Every
+     hour-and-a-half the water would execute a complete swing from high
+     tide to high again.
+
+Very well, now leave the earth, and think what has been happening to the
+moon all this while.
+
+We have seen that the moon pulls the tidal hump nearest to it back; but
+action and reaction are always equal and opposite--it cannot do that
+without itself getting pulled forward. The pull of the earth on the moon
+will therefore not be quite central, but will be a little in advance of
+its centre; hence, by Kepler's second law, the rate of description of
+areas by its radius vector cannot be constant, but must increase (p.
+208). And the way it increases will be for the radius vector to
+lengthen, so as to sweep out a bigger area. Or, to put it another way,
+the extra speed tending to be gained by the moon will fling it further
+away by extra centrifugal force. This last is not so good a way of
+regarding the matter; though it serves well enough for the case of a
+ball whirled at the end of an elastic string. After having got up the
+whirl, the hand holding the string may remain almost fixed at the centre
+of the circle, and the motion will continue steadily; but if the hand be
+moved so as always to pull the string a little in advance of the centre,
+the speed of whirl will increase, the elastic will be more and more
+stretched, until the whirling ball is describing a much larger circle.
+But in this case it will likewise be going faster--distance and speed
+increase together. This is because it obeys a different law from
+gravitation--the force is not inversely as the square, or any other
+single power, of the distance. It does not obey any of Kepler's laws,
+and so it does not obey the one which now concerns us, viz. the third;
+which practically states that the further a planet is from the centre
+the slower it goes; its velocity varies inversely with the square root
+of its distance (p. 74).
+
+If, instead of a ball held by elastic, it were a satellite held by
+gravity, an increase in distance must be accompanied by a diminution in
+speed. The time of revolution varies as the square of the cube root of
+the distance (Kepler's third law). Hence, the tidal reaction on the
+moon, having as its primary effect, as we have seen, the pulling the
+moon a little forward, has also the secondary or indirect effect of
+making it move slower and go further off. It may seem strange that an
+accelerating pull, directed in front of the centre, and therefore always
+pulling the moon the way it is going, should retard it; and that a
+retarding force like friction, if such a force acted, should hasten it,
+and make it complete its orbit sooner; but so it precisely is.
+
+Gradually, but very slowly, the moon is receding from us, and the month
+is becoming longer. The tides of the earth are pushing it away. This is
+not a periodic disturbance, like the temporary acceleration of its
+motion discovered by Laplace, which in a few centuries, more or less,
+will be reversed; it is a disturbance which always acts one way, and
+which is therefore cumulative. It is superposed upon all periodic
+changes, and, though it seems smaller than they, it is more inexorable.
+In a thousand years it makes scarcely an appreciable change, but in a
+million years its persistence tells very distinctly; and so, in the long
+run, the month is getting longer and the moon further off. Working
+backwards also, we see that in past ages the moon must have been nearer
+to us than it is now, and the month shorter.
+
+Now just note what the effect of the increased nearness of the moon was
+upon our tides. Remember that the tide-generating force varies inversely
+as the cube of distance, wherefore a small change of distance will
+produce a great difference in the tide-force.
+
+The moon's present distance is 240 thousand miles. At a time when it was
+only 190 thousand miles, the earth's tides would have been twice as high
+as they are now. The pushing away action was then a good deal more
+violent, and so the process went on quicker. The moon must at some time
+have been just half its present distance, and the tides would then have
+risen, not 20 or 30 feet, but 160 or 200 feet. A little further back
+still, we have the moon at one-third of its present distance from the
+earth, and the tides 600 feet high. Now just contemplate the effect of a
+600-foot tide. We are here only about 150 feet above the level of the
+sea; hence, the tide would sweep right over us and rush far away inland.
+At high tide we should have some 200 feet of blue water over our heads.
+There would be nothing to stop such a tide as that in this neighbourhood
+till it reached the high lands of Derbyshire. Manchester would be a
+seaport then with a vengeance!
+
+The day was shorter then, and so the interval between tide and tide was
+more like ten than twelve hours. Accordingly, in about five hours, all
+that mass of water would have swept back again, and great tracts of sand
+between here and Ireland would be left dry. Another five hours, and the
+water would come tearing and driving over the country, applying its
+furious waves and currents to the work of denudation, which would
+proceed apace. These high tides of enormously distant past ages
+constitute the denuding agent which the geologist required. They are
+very ancient--more ancient than the Carboniferous period, for instance,
+for no trees could stand the furious storms that must have been
+prevalent at this time. It is doubtful whether any but the very lowest
+forms of life then existed. It is the strata at the bottom of the
+geological scale that are of the most portentous thickness, and the only
+organism suspected in them is the doubtful _Eozoon Canadense_. Sir
+Robert Ball believes, and several geologists agree with him, that the
+mighty tides we are contemplating may have been coaeval with this ancient
+Laurentian formation, and others of like nature with it.
+
+But let us leave geology now, and trace the inverted progress of events
+as we recede in imagination back through the geological era, beyond,
+into the dim vista of the past, when the moon was still closer and
+closer to the earth, and was revolving round it quicker and quicker,
+before life or water existed on it, and when the rocks were still
+molten.
+
+Suppose the moon once touched the earth's surface, it is easy to
+calculate, according to the principles of gravitation, and with a
+reasonable estimate of its size as then expanded by heat, how fast it
+must then have revolved round the earth, so as just to save itself from
+falling in. It must have gone round once every three hours. The month
+was only three hours long at this initial epoch.
+
+Remember, however, the initial length of the day. We found that it was
+just possible for the earth to rotate on its axis in three hours, and
+that when it did so, something was liable to separate from it. Here we
+find the moon in contact with it, and going round it in this same
+three-hour period. Surely the two are connected. Surely the moon was a
+part of the earth, and was separating from it.
+
+That is the great discovery--the origin of the moon.
+
+Once, long ages back, at date unknown, but believed to be certainly as
+much as fifty million years ago, and quite possibly one hundred million,
+there was no moon, only the earth as a molten globe, rapidly spinning on
+its axis--spinning in about three hours. Gradually, by reason of some
+disturbing causes, a protuberance, a sort of bud, forms at one side, and
+the great inchoate mass separates into two--one about eighty times as
+big as the other. The bigger one we now call earth, the smaller we now
+call moon. Round and round the two bodies went, pulling each other into
+tremendously elongated or prolate shapes, and so they might have gone on
+for a long time. But they are unstable, and cannot go on thus: they must
+either separate or collapse. Some disturbing cause acts again, and the
+smaller mass begins to revolve less rapidly. Tides at once
+begin--gigantic tides of molten lava hundreds of miles high; tides not
+in free ocean, for there was none then, but in the pasty mass of the
+entire earth. Immediately the series of changes I have described begins,
+the speed of rotation gets slackened, the moon's mass gets pushed
+further and further away, and its time of revolution grows rapidly
+longer. The changes went on rapidly at first, because the tides were so
+gigantic; but gradually, and by slow degrees, the bodies get more
+distant, and the rate of change more moderate. Until, after the lapse of
+ages, we find the day twenty-four hours long, the moon 240,000 miles
+distant, revolving in 27-1/3 days, and the tides only existing in the
+water of the ocean, and only a few feet high. This is the era we call
+"to-day."
+
+The process does not stop here: still the stately march of events goes
+on; and the eye of Science strives to penetrate into the events of the
+future with the same clearness as it has been able to descry the events
+of the past. And what does it see? It will take too long to go into full
+detail: but I will shortly summarize the results. It sees this
+first--the day and the month both again equal, but both now about 1,400
+hours long. Neither of these bodies rotating with respect to each
+other--the two as if joined by a bar--and total cessation of
+tide-generating action between them.
+
+The date of this period is one hundred and fifty millions of years
+hence, but unless some unforeseen catastrophe intervenes, it must
+assuredly come. Yet neither will even this be the final stage; for the
+system is disturbed by the tide-generating force of the sun. It is a
+small effect, but it is cumulative; and gradually, by much slower
+degrees than anything we have yet contemplated, we are presented with a
+picture of the month getting gradually shorter than the day, the moon
+gradually approaching instead of receding, and so, incalculable myriads
+of ages hence, precipitating itself upon the surface of the earth whence
+it arose.
+
+Such a catastrophe is already imminent in a neighbouring planet--Mars.
+Mars' principal moon circulates round him at an absurd pace, completing
+a revolution in 7-1/2 hours, and it is now only 4,000 miles from his
+surface. The planet rotates in twenty-four hours as we do; but its tides
+are following its moon more quickly than it rotates after them; they are
+therefore tending to increase its rate of spin, and to retard the
+revolution of the moon. Mars is therefore slowly but surely pulling its
+moon down on to itself, by a reverse action to that which separated our
+moon. The day shorter than the month forces a moon further away; the
+month shorter than the day tends to draw a satellite nearer.
+
+This moon of Mars is not a large body: it is only twenty or thirty miles
+in diameter, but it weighs some forty billion tons, and will ultimately
+crash along the surface with a velocity of 8,000 miles an hour. Such a
+blow must produce the most astounding effects when it occurs, but I am
+unable to tell you its probable date.
+
+So far we have dealt mainly with the earth and its moon; but is the
+existence of tides limited to these bodies? By no means. No body in the
+solar system is rigid, no body in the stellar universe is rigid. All
+must be susceptible of some tidal deformation, and hence, in all of
+them, agents like those we have traced in the history of the earth and
+moon must be at work: the motion of all must be complicated by the
+phenomena of tides. It is Prof. George Darwin who has worked out the
+astronomical influence of the tides, on the principles of Sir William
+Thomson: it is Sir Robert Ball who has extended Mr. Darwin's results to
+the past history of our own and other worlds.[32]
+
+     Tides are of course produced in the sun by the action of the
+     planets, for the sun rotates in twenty-five days or thereabouts,
+     while the planets revolve in much longer periods than that. The
+     principal tide-generating bodies will be Venus and Jupiter; the
+     greater nearness of one rather more than compensating for the
+     greater mass of the other.
+
+     It may be interesting to tabulate the relative tide-producing
+     powers of the planets on the sun. They are as follows, calling that
+     of the earth 1,000:--
+
+  RELATIVE TIDE-PRODUCING POWERS OF THE PLANETS
+  ON THE SUN.
+
+  Mercury         1,121
+  Venus           2,339
+  Earth           1,000
+  Mars              304
+  Jupiter         2,136
+  Saturn          1,033
+  Uranus             21
+  Neptune             9
+
+     The power of all of them is very feeble, and by acting on different
+     sides they usually partly neutralize each other's action; but
+     occasionally they get all on one side, and in that case some
+     perceptible effect may be produced; the probable effect seems
+     likely to be a gentle heaving tide in the solar surface, with
+     breaking up of any incipient crust; and such an effect may be
+     considered as evidenced periodically by the great increase in the
+     number of solar spots which then break out.
+
+     The solar tides are, however, much too small to appreciably push
+     any planet away, hence we are not to suppose that the planets
+     originated by budding from the sun, in contradiction of the nebular
+     hypothesis. Nor is it necessary to assume that the satellites, as a
+     class, originated in the way ours did; though they may have done
+     so. They were more probably secondary rings. Our moon differs from
+     other satellites in being exceptionally large compared with the
+     size of its primary; it is as big as some of the moons of Jupiter
+     and Saturn. The earth is the only one of the small planets that has
+     an appreciable moon, and hence there is nothing forced or unnatural
+     in supposing that it may have had an exceptional history.
+
+     Evidently, however, tidal phenomena must be taken into
+     consideration in any treatment of the solar system through enormous
+     length of time, and it will probably play a large part in
+     determining its future.
+
+When Laplace and Lagrange investigated the question of the stability or
+instability of the solar system, they did so on the hypothesis that the
+bodies composing it were rigid. They reached a grand conclusion--that
+all the mutual perturbations of the solar system were periodic--that
+whatever changes were going on would reach a maximum and then begin to
+diminish; then increase again, then diminish, and so on. The system was
+stable, and its changes were merely like those of a swinging pendulum.
+
+But this conclusion is not final. The hypothesis that the bodies are
+rigid is not strictly true: and directly tidal deformation is taken into
+consideration it is perceived to be a potent factor, able in the long
+run to upset all their calculations. But it is so utterly and
+inconceivably minute--it only produces an appreciable effect after
+millions of years--whereas the ordinary perturbations go through their
+swings in some hundred thousand years or so at the most. Granted it is
+small, but it is terribly persistent; and it always acts in one
+direction. Never does it cease: never does it begin to act oppositely
+and undo what it has done. It is like the perpetual dropping of water.
+There may be only one drop in a twelvemonth, but leave it long enough,
+and the hardest stone must be worn away at last.
+
+*      *      *      *      *
+
+We have been speaking of millions of years somewhat familiarly; but
+what, after all, is a million years that we should not speak familiarly
+of it? It is longer than our lifetime, it is true. To the ephemeral
+insects whose lifetime is an hour, a year might seem an awful period,
+the mid-day sun might seem an almost stationary body, the changes of the
+seasons would be unknown, everything but the most fleeting and rapid
+changes would appear permanent and at rest. Conversely, if our
+life-period embraced myriads of aeons, things which now seem permanent
+would then appear as in a perpetual state of flux. A continent would be
+sometimes dry, sometimes covered with ocean; the stars we now call fixed
+would be moving visibly before our eyes; the earth would be humming on
+its axis like a top, and the whole of human history might seem as
+fleeting as a cloud of breath on a mirror.
+
+Evolution is always a slow process. To evolve such an animal as a
+greyhound from its remote ancestors, according to Mr. Darwin, needs
+immense tracts of time; and if the evolution of some feeble animal
+crawling on the surface of this planet is slow, shall the stately
+evolution of the planetary orbs themselves be hurried? It may be that we
+are able to trace the history of the solar system for some thousand
+million years or so; but for how much longer time must it not have a
+history--a history, and also a future--entirely beyond our ken?
+
+Those who study the stars have impressed upon them the existence of the
+most immeasurable distances, which yet are swallowed up as nothing in
+the infinitude of space. No less are we compelled to recognize the
+existence of incalculable aeons of time, and yet to perceive that these
+are but as drops in the ocean of eternity.
+
+
+FOOTNOTES:
+
+[1] The following account of Mars's motion is from the excellent small
+manual of astronomy by Dr. Haughton of Trinity College, Dublin:--(P.
+151) "Mars's motion is very unequal; when he first appears in the
+morning emerging from the rays of the sun, his motion is direct and
+rapid; it afterwards becomes slower, and he becomes stationary when at
+an elongation of 137 deg. from the sun; then his motion becomes retrograde,
+and its velocity increases until he is in opposition to the sun at 180 deg.;
+at this time the retrograde motion is most rapid, and afterwards
+diminishes until he is 137 deg. distant from the sun on the other side, when
+Mars again becomes stationary; his motion then becomes direct, and
+increases in velocity until it reaches a maximum, when the planet is
+again in conjunction with the sun. The retrograde motion of this planet
+lasts for 73 days: and its arc of retrogradation is 16 deg.."
+
+[2] It is not so easy to plot the path of the sun among the stars by
+direct observation, as it is to plot the path of a planet; because sun
+and stars are not visible together. Hipparchus used the moon as an
+intermediary; since sun and moon are visible together, and also moon and
+stars.
+
+[3] This is, however, by no means the whole of the matter. The motion is
+not a simple circle nor has it a readily specifiable period. There are
+several disturbing causes. All that is given here is a first rough
+approximation.
+
+[4] The proof is easy, and ought to occur in books on solid geometry. By
+a "regular" solid is meant one with all its faces, edges, angles, &c.,
+absolutely alike: it is of these perfectly symmetrical bodies that there
+are only five. Crystalline forms are practically infinite in number.
+
+[5] Best known to us by his Christian name, as so many others of that
+time are known, _e.g._ Raphael Sanzio, Dante Alighieri, Michael Angelo
+Buonarotti. The rule is not universal. Tasso and Ariosto are surnames.
+
+[6] It would seem that the fact that all bodies of every material tend
+to fall at the same rate is still not clearly known. Confusion is
+introduced by the resistance of the air. But a little thought should
+make it clear that the effect of the air is a mere disturbance, to be
+eliminated as far as possible, since the atmosphere has nothing to do
+with gravitation. The old fashioned "guinea and feather experiment"
+illustrates that in a vacuum things entirely different in specific
+gravity or surface drop at the same pace.
+
+[7] Karl von Gebler (Galileo), p. 13.
+
+[8] It is of course the "silver lining" of clouds that outside observers
+see.
+
+[9] L.U.K., _Life of Galileo_, p. 26.
+
+[10] _Note added September, 1892._ News from the Lick Observatory makes
+a very small fifth satellite not improbable.
+
+[11] They remained there till this century. In 1835 they were quietly
+dropped.
+
+[12] It was invented by van Helmont, a Belgian chemist, who died in
+1644. He suggested two names _gas_ and _blas_, and the first has
+survived. Blas was, I suppose, from _blasen_, to blow, and gas seems to
+be an attempt to get at the Sanskrit root underlying all such words as
+_geist_.
+
+[13] Such as this, among many others:--The duration of a flame under
+different conditions is well worth determining. A spoonful of warm
+spirits of wine burnt 116 pulsations. The same spoonful of spirits of
+wine with addition of one-sixth saltpetre burnt 94 pulsations. With
+one-sixth common salt, 83; with one-sixth gunpowder, 110; a piece of wax
+in the middle of the spirit, 87; a piece of _Kieselstein_, 94; one-sixth
+water, 86; and with equal parts water, only 4 pulse-beats. This, says
+Liebig, is given as an example of a "_licht-bringende Versuch_."
+
+[14] Draper, _History of Civilization in Europe_, vol. ii. p. 259.
+
+[15] Professor Knight's series of Philosophical Classics.
+
+[16] To explain why the entire system, horse and cart together, move
+forward, the forces acting on the ground must be attended to.
+
+[17] The distance being proportional to the _square_ of the time, see p.
+82.
+
+[18] The following letter, recently unearthed and published in _Nature_,
+May 12, 1881, seems to me well worth preserving. The feeling of a
+respiratory interval which it describes is familiar to students during
+the too few periods of really satisfactory occupation. The early guess
+concerning atmospheric electricity is typical of his extraordinary
+instinct for guessing right.
+
+  "LONDON, _Dec. 15, 1716_.
+
+"DEAR DOCTOR,--He that in ye mine of knowledge deepest diggeth, hath,
+like every other miner, ye least breathing time, and must sometimes at
+least come to terr. alt. for air.
+
+"In one of these respiratory intervals I now sit down to write to you,
+my friend.
+
+"You ask me how, with so much study, I manage to retene my health. Ah,
+my dear doctor, you have a better opinion of your lazy friend than he
+hath of himself. Morpheous is my last companion; without 8 or 9 hours of
+him yr correspondent is not worth one scavenger's peruke. My practices
+did at ye first hurt my stomach, but now I eat heartily enou' as y' will
+see when I come down beside you.
+
+"I have been much amused at ye singular [Greek: _phenomena_] resulting
+from bringing of a needle into contact with a piece of amber or resin
+fricated on silke clothe. Ye flame putteth me in mind of sheet lightning
+on a small--how very small--scale. But I shall in my epistles abjure
+Philosophy whereof when I come down to Sakly I'll give you enou'. I
+began to scrawl at 5 mins. from 9 of ye clk. and have in writing consmd.
+10 mins. My Ld. Somerset is announced.
+
+"Farewell, Gd. bless you and help yr sincere friend.
+
+  "ISAAC NEWTON.
+
+  "_To_ DR. LAW, Suffolk."
+
+
+
+[19] Kepler's laws may be called respectively, the law of path, the law
+of speed, and the relationship law. By the "mass" of a body is meant the
+number of pounds or tons in it: the amount of matter it contains. The
+idea is involved in the popular word "massive."
+
+[20] The equation we have to verify is
+
+         4[pi]^2r^3
+  gR^2 = -----------,
+            T^2
+
+with the data that _r_, the moon's distance, is 60 times R, the earth's
+radius, which is 3,963 miles; while T, the time taken to complete the
+moon's orbit, is 27 days, 13 hours, 18 minutes, 37 seconds. Hence,
+suppose we calculate out _g_, the intensity of terrestrial gravity, from
+the above equation, we get
+
+          4[pi]^2               39.92 x 216000 x 3963 miles
+  _g_ = ---------- x (60)^3R = -----------------------------
+             T                  (27 days, 13 hours, &c.)^2
+
+           = 32.57 feet-per-second per second,
+
+which is not far wrong.
+
+[21] The two motions may be roughly compounded into a single motion,
+which for a few centuries may without much error be regarded as a
+conical revolution about a different axis with a different period; and
+Lieutenant-Colonel Drayson writes books emphasizing this simple fact,
+under the impression that it is a discovery.
+
+[22] Members of the Accademia dei Lyncei, the famous old scientific
+Society established in the time of Cosmo de Medici--older than our own
+Royal Society.
+
+[23] Newton suspected that the moon really did so oscillate, and so it
+may have done once; but any real or physical libration, if existing at
+all, is now extremely minute.
+
+[24] An interesting picture in the New Gallery this year (1891),
+attempting to depict "Earth-rise in Moon-land," unfortunately errs in
+several particulars. First of all, the earth does not "rise," but is
+fixed relatively to each place on the moon; and two-fifths of the moon
+never sees it. Next, the earth would not look like a map of the world
+with a haze on its edge. Lastly, whatever animal remains the moon may
+contain would probably be rather in the form of fossils than of
+skeletons. The skeleton is of course intended as an image of death and
+desolation. It is a matter of taste: but a skeleton, it seems to me,
+speaks too recently of life to be as appallingly weird and desolate as a
+blank stone or ice landscape, unshaded by atmosphere or by any trace of
+animal or plant life, could be made.
+
+[25] Five of Jupiter's revolutions occupy 21,663 days; two of Saturn's
+revolutions occupy 21,526 days.
+
+[26] _Excircularity_ is what is meant by this term. It is called
+"excentricity" because the foci (not the centre) of an ellipse are
+regarded as the representatives of the centre of a circle. Their
+distance from the centre, compared with the radius of the unflattened
+circle, is called the excentricity.
+
+[27] A curve of the _n_th degree has 1/2_n_(_n_+3) arbitrary constants
+in its equation, hence this number of points specifically determine it.
+But special points, like focus or vertex, count as two ordinary ones.
+Hence three points plus the focus act as five points, and determine a
+conic or curve of the second degree. Three observations therefore fix an
+orbit round the sun.
+
+[28] Its name suggests a measure of the diameter of the sun's disk, and
+this is one of its functions; but it can likewise measure planetary and
+other disks; and in general behaves as the most elaborate and expensive
+form of micrometer. The Koenigsberg instrument is shewn in fig. 92.
+
+[29] It may be supposed that the terms "minute" and "second" have some
+necessary connection with time, but they are mere abbreviations for
+_partes minutae_ and _partes minutae secundae_, and consequently may be
+applied to the subdivision of degrees just as properly as to the
+subdivision of hours. A "second" of arc means the 3600th part of a
+degree, just as a second of time means the 3600th part of an hour.
+
+[30] A group of flying particles, each one invisible, obstructs light
+singularly little, even when they are close together, as one can tell by
+the transparency of showers and snowstorms. The opacity of haze may be
+due not merely to dust particles, but to little eddies set up by
+radiation above each particle, so that the air becomes turbulent and of
+varying density. (See a similar suggestion by Mr. Poynting in _Nature_,
+vol. 39, p. 323.)
+
+[31] The moon ought to be watched during the next great shower, if the
+line of fire happens to take effect on a visible part of the dark
+portion.
+
+[32] Address to Birmingham Midland Institute, "A Glimpse through the
+Corridors of Time."
+
+
+
+
+INDEX
+
+INDEX
+
+
+A
+
+Abbott, T.K., on tides, 369
+
+Adams, John Couch, 193, 217, 302, 323, 324, 325, 327, 329, 330, 352, 385
+
+Airy, Sir George, 193, 244, 302, 323, 324, 327, 367
+
+Anaxagoras, 15
+
+Appian, 218
+
+Arabs, the, form a link between the old and new science, 9
+
+Archimedes, 7, 8, 84, 87, 144, 177
+
+Aristarchus, 34
+
+Aristotle, 66, 69, 88, 94, 99, 167.
+  He taught that the earth was a sphere, 16;
+  his theories did not allow of the earth's motion, 34;
+  he was regarded as inspired, 89
+
+
+B
+
+Bacon, Francis, 142, 143, 144, 145.
+  His _Novum Organum_, 141
+
+Bacon, Roger, 96, 139, 140.
+  The herald of the dawn of science, 9
+
+Brahe, George, uncle of Tycho Brahe, 39
+
+Brahe, Steno, brother of Tycho Brahe, 39
+
+Brahe, Tycho, 37, 39, 40, 44, 45, 49, 51, 53, 54, 55, 58, 63, 64, 65, 66,
+ 68, 71, 72, 74, 75, 77, 78, 86, 94, 117, 137, 155, 165, 166, 200, 244,
+ 281, 288.
+  He tried to adopt the main features of the Copernican theory without
+   admitting the motion of the earth, 37;
+  he was a poor theorist but a great observer, 38;
+  his medicine, 44;
+  his personal history, 39, _seq._;
+  his observatory, Uraniburg, 47;
+  his greatest invention, 50, note;
+  his maniac Lep, 52;
+  his kindness to Kepler, 63
+
+Ball, Sir R., 391, 394;
+  his _Story of the Heavens_, 377
+
+Barrow, Dr., 165, 187
+
+Bessel, 288, 310, 311, 313, 315, 316, 318, 323
+
+Biela, 345, 346, 347
+
+Bode's Law, 60, 296, 298, 299, 326
+
+Boyle, 139, 188
+
+Bradley, Prof. James, 233, 246, 247, 249, 252, 253, 308, 319
+
+Bremiker, 328, 329
+
+Brewster, on Kepler, 78
+
+Brinkley, 308
+
+Bruno, Giordano, 108, 127
+
+
+C
+
+Castelli, 112, 133
+
+Cayley, Prof., 385
+
+Challis, Prof., 328, 329
+
+Clairut, 193, 216, 217, 219, 234, 341
+
+Clark, Alvan and Sons, 316
+
+Columbus, 9, 144
+
+Copernicus, 7, 10, _seq._, 14, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38,
+ 62, 66, 68, 70, 78, 93, 95, 100, 108, 111, 121, 122, 137, 155, 166, 223,
+ 234, 247, 307;
+  his _De Revolutionibus Orbium Coelestium_, 11, 75, 138;
+  he _proved_ that the earth went round the sun, 13;
+  the influence of his theory on the Church, 13, _seq._;
+  his life-work summarised, 30;
+  his Life by Mr. E.J.C. Morton, 31
+
+Copernican tables, 40;
+  Copernican theory, 59, 60, 125, 144, 167
+
+Copernik, Nicolas; see Copernicus
+
+Cornu, 238
+
+Croll, Dr., his _Climate and Time_, 264
+
+
+D
+
+D'Alembert, 193, 234
+
+Darwin, Charles, 134, 138, 397
+
+Darwin, Prof. George, 367, 394
+
+Delambre, 253
+
+Descartes, 145, 146, 148, 151, 153, 156, 158, 164, 165, 167, 178, 181,
+ 224, 227;
+  his _Discourse on Method_, 142;
+  his dream, 147;
+  his system of algebraic geometry, 149, _seq._;
+  his doctrine of vortices, 151, _seq._;
+  his _Principia Mathematica_, 154;
+  his Life by Mr. Mahaffy, 154
+
+
+E
+
+Earth, the difficulties in the way of believing that it moved, 34, _seq._
+
+"Earth-rise in Moon-land," 258, note
+
+Encke, 345, 346
+
+Epicyclic orbits explained, 23, _seq._
+
+Equinoxes, their precession discovered by Hipparchus, 27
+
+Eudoxus, 19
+
+Euler, 193, 234
+
+
+F
+
+Faraday, 84
+
+Fizeau, 238, 239
+
+Flamsteed, 215, 246, 284, 308, 319
+
+Fraunhofer, 311
+
+Froude, Prof.; his _Oceania_, 387
+
+
+G
+
+Galen, 87
+
+Galileo, Galilei, 63, 75, 84, 88, 90, 92, 93, 97, 98, 101, 104, 106, 107,
+ 108, 109, 110, 112, 114, 116, 117, 118, 120, 121, 122, 123, 125, 127,
+ 133, 134, 137, 144, 145, 153, 154, 157, 165, 166, 167, 168, 177, 188,
+ 200, 224, 227, 256, 281, 288, 309, 361;
+  his youth, 85;
+  his discovery of the pendulum, 86;
+  his first observations about falling bodies, 88, _seq._;
+  he invents a telescope, 95;
+  he adopts the Copernican theory, 94;
+  he conceives "earth-shine," 100;
+  he discovers Jupiter's moons, 103;
+  he studies Saturn, 114, _seq._;
+  his _Dialogues on the Ptolemaic and Copernican Systems_, 124;
+  his abjuration, 130;
+  he becomes blind, 132;
+  he discovered the Laws of Motion, 167, _seq._;
+  he guessed that sight was not instantaneous, 236, 237
+
+Galle, Dr., 245, 329
+
+Gauss, 299, 300
+
+Gilbert, Dr., 139, 140, 157, 188;
+  his _De Magnete_, 140, 144
+
+Greeks, their scientific methods, 7
+
+Groombridge's Catalogue, 315
+
+
+H
+
+Hadley, 185
+
+Halley, 192, 193, 194, 195, 197, 215, 218, 219, 246, 258, 260, 261, 340,
+ 341;
+  he discovered the _Principia_, 194
+
+Harvey, 144, 149
+
+Haughton, Dr., 321;
+  his manual on Astronomy, 21, note
+
+Heliometer, described, 311
+
+Helmholtz, 378
+
+Helmont, Van, invented the word "gas," 141
+
+Henderson, 310, 314
+
+Herschel, Alexander, 275, 277, 278, 279
+
+Herschel, Caroline, 275, 276, 279, 286, 345;
+  her journal quoted, 277, _seq._;
+  her work with William H. described, 284
+
+Herschel, Sir John, 283, 285, 327, 329
+
+Herschel, William, 185, 234, 235, 244, 249, 274, 275, 280, 281, 282, 284,
+ 288, 289, 290, 293, 295, 305, 309, 310, 318, 319, 327;
+  he "sweeps" the heavens, 280;
+  his discovery of Uranus, 281, 287;
+  his artificial Saturn, 281, 282;
+  his methods of work with his sister, described, 284;
+  he founded the science of Astronomy, 287
+
+Hind, 300
+
+Hipparchus, 7, 18, 20, 27, 28, and note, 30, 40, 66, 223, 253;
+  an explanation of his discovery of the precession of the equinoxes,
+  27, seq.
+
+Hippocrates, 87
+
+Homeric Cosmogony, 15, _seq._
+
+Hooke, 139, 188, 192, 193, 196, 197, 308
+
+Hopital, Marquis de l', 228
+
+Horkey, Martin, 106
+
+Horrebow, 244
+
+Huxley, Prof., 149
+
+Huyghens, 86, 166, 185
+
+
+K
+
+Kant, 267, 270
+
+Kelvin, Lord, see Thomson, Sir W.
+
+Kepler, John, 59, 60, 63, 64, 65, 66, 70, 72, 73, 75, 77, 79, 84, 93,
+ 94, 95, 104, 106, 107, 110, 122, 137, 145, 153, 158, 164, 165, 166,
+ 167, 192, 200, 208, 209, 210, 211, 212, 214, 218, 224, 227, 253, 256,
+ 259, 260, 262, 288, 295, 296, 332, 338, 361, 389;
+  he replaced epicycles by an ellipse, 27;
+  he was a pupil of Tycho Brahe, 54;
+  he was a speculator more than an observer, 58;
+  his personal life, 58, _seq._;
+  his theories about the numbers and distances of the planets, 60, 62;
+  he was helped by Tycho, 63;
+  his main work, 65, _seq._;
+  he gave up circular motion, 69;
+  his _Mysterium Cosmographicon_, 105;
+  his Laws, 71, 74, 173, 174, 176, 179, 180, 206, _seq._
+
+
+L
+
+Lagrange, 193, 234, 255, 256, 257, 258, 263
+
+Lagrange and Laplace, 258, 266, 395;
+  they laid the foundations of the planetary theory, 259
+
+Laplace, 68, 193, 218, 234, 255, 261, 262, 267, 268, 269, 270, 272,
+ 288, 301, 317, 384, 385, 390;
+  his nebular hypothesis, 267, 292;
+  his _Mecanique Celeste_, 323
+
+Lassell, Mr., 283, 284
+
+Leibnitz, 192, 197, 233
+
+Le Monnier, 319
+
+Leonardo, see Vinci, Leonardo da
+
+Leverrier, 193, 327, 328, 329, 330, 352
+
+Lippershey, Hans, 95
+
+
+M
+
+Maskelyne, 281
+
+Maxwell, Clerk, 302, 303
+
+Molyneux, 248, 249
+
+Morton, Mr. E.J. C, his Life of Copernicus, 31
+
+
+N
+
+Newton, Prof. H.A., 347
+
+Newton, Sir Isaac, 7, 30, 79, 138, 139, 144, 145, 149, 153, 157, 158,
+ 165, 166, 167, 174, 176, 184, 187, 188, 189, 191, 192, 194, 196, 198,
+ 199, 201, 213, 216, 219, 220, 221, 224, 226, 227, 228, 233, 242, 253,
+ 255, 256, 274, 288, 317, 340, 378;
+  his _Principia_, 191, 192, 193, 194, 195, 196, 197, 207, 214, 216, 218,
+   228, 233, 242, 253;
+  his early life, 161, _seq._;
+  his first experiments, 163;
+  his work at Cambridge, 164;
+  his Laws, 168;
+  his application of the Laws of Gravity to Astronomy, 177, 178, 179, 185,
+   190;
+  his reticence, 178;
+  his discoveries in Optics, 181, _seq._;
+  his work summarised, 186;
+  his _Optics_, 189;
+  anecdotes of him, 191;
+  his appearance in a Court of Justice, 195;
+  some of his manuscripts very recently discovered, 217;
+  his theories of the Equinoxes and tides, 223, _seq._, 225, 363, _seq._
+
+
+O
+
+Olbers, 299, 300
+
+
+P
+
+Peters, Prof., 300, 316
+
+Piazzi, 298, 299, 308, 313
+
+Picard, 190, 242, 244, 247
+
+Pioneers, genuine, 7
+
+Planets and days of the week, 18
+
+Poynting, 332
+
+Printing, 9
+
+Ptolemy, 18, 20, 27, 38, 153, 155, 166, 214;
+  his system of the Heavens simplified by Copernicus, 11, 30;
+  his system described, 19, _seq._;
+  his system taught, 34;
+  his harmonies, 74
+
+Pythagoras, 19, 20, 34
+
+
+Q
+
+Quadrant, an early, 42, 43
+
+
+R
+
+Rheiter, 107
+
+Ricci, Ostillio, 86, 87
+
+Roberts, Isaac, 268
+
+Roemer, 239, 240, 242, 244, 249, 251, 308
+
+Rosse, Lord, his telescope, 186, 268
+
+Rudolphine tables, 65
+
+
+S
+
+Scheiner, 107
+
+Sizzi, Francesca, an orthodox astronomer, 106
+
+Snell, Willebrod, and the law of refraction, 65
+
+Solar system, its fate, 265
+
+Stars, a list of, 307
+
+Struve, 308, 310, 311, 313
+
+Stuart, Prof., quoted, 52
+
+
+T
+
+Tatius, 296
+
+Telescopes, early, 96
+
+Thales, 7, 140, 317
+
+Thomson, Sir William, 367, 372, 373, 378, 394
+
+Tide-gauge, described, 373, _seq._
+
+Tides, 354, _seq._
+
+Time, is not exactly uniform, 384
+
+Torricelli, 133, 168
+
+Tycho, see Brahe, Tycho
+
+
+V
+
+Vinci, Leonardo da, 9, 100, 144, 184
+
+Viviani, 133, 168
+
+Voltaire, 181
+
+
+W
+
+Watson, Prof., 300
+
+Whewell, 227
+
+Wren, Sir Christopher, 188, 192, 193, 197
+
+
+Z
+
+Zach, Von, 296, 299
+
+Zone of Asteroids, 300, _seq._
+
+
+  THE END.
+
+  RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.
+
+
+
+
+
+End of the Project Gutenberg EBook of Pioneers of Science, by Oliver Lodge
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author	Roger Frank <rfrank@pglaf.org>	2025-10-15 02:38:57 -0700
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