





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 BRAHÉ 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 NEBULÆ                                         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 KÖNIGSBERG 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 _vivâ 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 Brahé     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 Brahé'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 BRAHÉ 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 Brahé.

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
Brahé, 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 Brahé, 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 _protégé_ 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
Würtemberg. 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
Tübingen. 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 Brahé, 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 Tübingen, 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 Brahé 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
Würtemberg. 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 Brahé, 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 chimæra 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 Brahé.

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
nebulæ. 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 Brahé 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 nebulæ.]

[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 _protégé_ 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 cathedrâ_ 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 nebulæ 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 _ansæ_.

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 Brahé 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 mediævalism 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 primæval 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 Brahé, 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.

_René 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 Ælia
     Lælia 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 formulæ 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
_Encyclopædia 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, René 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 æsthetic 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 René
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, _née_ 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 Brahé.

     (_c_) The annual equation, discovered by Tycho Brahé.

     (_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'Hôpital, 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 Brahé, 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 Brahé. 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 _Mécanique Céleste_
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 Chaldæan 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 nebulæ. Some of the nebulæ 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 nebulæ 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, nebulæ, 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
Brahé 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 nebulæ. I went to bed about one o'clock,
     and up to that time he had found that night four or five new
     nebulæ. 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 nebulæ 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 nebulæ. 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 nebulæ 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 MÜLLER, 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] Lyræ 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 nebulæ. 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 nebulæ 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 nebulæ 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 nebulæ 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 _éclât_ 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 Chaldæan
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
     _vivâ 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             Boötes
  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 Königsberg, which, by his system,
zeal, and genius, he rapidly made a place of the first importance.

Struve at Dorpat, Bessel at Königsberg, 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 "Königsberg 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] Lyræ (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] Lyræ, 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
_Mécanique Céleste_.

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 École
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 Astræa; 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 nebulæ, 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 aërolites, 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 aërolites 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 Cæsar 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 _débris_ 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 Chaldæa, 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 Chaldæan 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 coæval 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 æons, 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 æons 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 Königsberg 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 minutæ_ and _partes minutæ secundæ_, 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

Brahé, George, uncle of Tycho Brahé, 39

Brahé, Steno, brother of Tycho Brahé, 39

Brahé, 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

Hôpital, 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 Brahé, 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 _Mécanique Céleste_, 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 Brahé, 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.





