Kepler by Walter W. Bryant
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[Illustration: KEPLER]
Pioneers of Progress
Men of Science
Edited by S. Chapman, M.A., D.Sc., F.R.S.
KEPLER
by
WALTER W. BRYANT
of the Royal Observatory, Greenwich
1920
CONTENTS.
I. Astronomy Before Kepler
II. Early Life of Kepler
III. Tycho Brahe
IV. Kepler Joins Tycho
V. Kepler's Laws
VI. Closing Years
Appendix I.--List of Dates
Appendix II.--Bibliography
Glossary
CHAPTER I.
ASTRONOMY BEFORE KEPLER.
In order to emphasise the importance of the reforms introduced into
astronomy by Kepler, it will be well to sketch briefly the history of
the theories which he had to overthrow. In very early times it must have
been realised that the sun and moon were continually changing their
places among the stars. The day, the month, and the year were obvious
divisions of time, and longer periods were suggested by the tabulation
of eclipses. We can imagine the respect accorded to the Chaldaean sages
who first discovered that eclipses could be predicted, and how the
philosophers of Mesopotamia must have sought eagerly for evidence of
fresh periodic laws. Certain of the stars, which appeared to wander, and
were hence called planets, provided an extended field for these
speculations. Among the Chaldaeans and Babylonians the knowledge
gradually acquired was probably confined to the priests and utilised
mainly for astrological prediction or the fixing of religious
observances. Such speculations as were current among them, and also
among the Egyptians and others who came to share their knowledge, were
almost entirely devoted to mythology, assigning fanciful terrestrial
origins to constellations, with occasional controversies as to how the
earth is supported in space. The Greeks, too, had an elaborate mythology
largely adapted from their neighbours, but they were not satisfied with
this, and made persistent attempts to reduce the apparent motions of
celestial objects to geometrical laws. Some of the Pythagoreans, if not
Pythagoras himself, held that the earth is a sphere, and that the
apparent daily revolution of the sun and stars is really due to a motion
of the earth, though at first this motion of the earth was not supposed
to be one of rotation about an axis. These notions, and also that the
planets on the whole move round from west to east with reference to the
stars, were made known to a larger circle through the writings of Plato.
To Plato moreover is attributed the challenge to astronomers to
represent all the motions of the heavenly bodies by uniformly described
circles, a challenge generally held responsible for a vast amount of
wasted effort, and the postponement, for many centuries, of real
progress. Eudoxus of Cnidus, endeavouring to account for the fact that
the planets, during every apparent revolution round the earth, come to
rest twice, and in the shorter interval between these "stationary
points," move in the opposite direction, found that he could represent
the phenomena fairly well by a system of concentric spheres, each
rotating with its own velocity, and carrying its own particular planet
round its own equator, the outermost sphere carrying the fixed stars. It
was necessary to assume that the axes about which the various spheres
revolved should have circular motions also, and gradually an increased
number of spheres was evolved, the total number required by Aristotle
reaching fifty-five. It may be regarded as counting in Aristotle's
favour that he did consider the earth to be a sphere and not a flat
disc, but he seems to have thought that the mathematical spheres of
Eudoxus had a real solid existence, and that not only meteors, shooting
stars and aurora, but also comets and the milky way belong to the
atmosphere. His really great service to science in collating and
criticising all that was known of natural science would have been
greater if so much of the discussion had not been on the exact meaning
of words used to describe phenomena, instead of on the facts and causes
of the phenomena themselves.
Aristarchus of Samos seems to have been the first to suggest that the
planets revolved not about the earth but about the sun, but the idea
seemed so improbable that it was hardly noticed, especially as
Aristarchus himself did not expand it into a treatise.
About this time the necessity for more accurate places of the sun and
moon, and the liberality of the Ptolemys who ruled Egypt, combined to
provide regular observations at Alexandria, so that, when Hipparchus
came upon the scene, there was a considerable amount of material for him
to use. His discoveries marked a great advance in the science of
astronomy. He noted the irregular motion of the sun, and, to explain it,
assumed that it revolved uniformly not exactly about the earth but
about a point some distance away, called the "excentric".[1] The line
joining the centre of the earth to the excentric passes through the
apses of the sun's orbit, where its distance from the earth is greatest
and least. The same result he could obtain by assuming that the sun
moved round a small circle, whose centre described a larger circle about
the earth; this larger circle carrying the other was called the
"deferent": so that the actual motion of the sun was in an epicycle. Of
the two methods of expression Hipparchus ultimately preferred the
second. He applied the same process to the moon but found that he could
depend upon its being right only at new and full moon. The irregularity
at first and third quarters he left to be investigated by his
successors. He also considered the planetary observations at his
disposal insufficient and so gave up the attempt at a complete planetary
theory. He made improved determinations of some of the elements of the
motions of the sun and moon, and discovered the Precession of the
Equinoxes, from the Alexandrian observations which showed that each year
as the sun came to cross the equator at the vernal equinox it did so at
a point about fifty seconds of arc earlier on the ecliptic, thus
producing in 150 years an unmistakable change of a couple of degrees, or
four times the sun's diameter. He also invented trigonometry. His star
catalogue was due to the appearance of a new star which caused him to
search for possible previous similar phenomena, and also to prepare for
checking future ones. No advance was made in theoretical astronomy for
260 years, the interval between Hipparchus and Ptolemy of Alexandria.
Ptolemy accepted the spherical form of the earth but denied its rotation
or any other movement. He made no advance on Hipparchus in regard to the
sun, though the lapse of time had largely increased the errors of the
elements adopted by the latter. In the case of the moon, however,
Ptolemy traced the variable inequality noticed sometimes by Hipparchus
at first and last quarter, which vanished when the moon was in apogee or
perigee. This he called the evection, and introduced another epicycle
to represent it. In his planetary theory he found that the places given
by his adopted excentric did not fit, being one way at apogee and the
other at perigee; so that the centre of distance must be nearer the
earth. He found it best to assume the centre of distance half-way
between the centre of the earth and the excentric, thus "bisecting the
excentricity". Even this did not fit in the case of Mercury, and in
general the agreement between theory and observation was spoilt by the
necessity of making all the orbital planes pass through the centre of
the earth, instead of the sun, thus making a good accordance practically
impossible.
[Footnote 1: See Glossary for this and other technical terms.]
After Ptolemy's time very little was heard for many centuries of any
fresh planetary theory, though advances in some points of detail were
made, notably by some of the Arab philosophers, who obtained improved
values for some of the elements by using better instruments. From time
to time various modifications of Ptolemy's theory were suggested, but
none of any real value. The Moors in Spain did their share of the work
carried on by their Eastern co-religionists, and the first independent
star catalogue since the time of Hipparchus was made by another
Oriental, Tamerlane's grandson, Ulugh Begh, who built a fine observatory
at Samarcand in the fifteenth century. In Spain the work was not
monopolised by the Moors, for in the thirteenth century Alphonso of
Castile, with the assistance of Jewish and Christian computers, compiled
the Alphonsine tables, completed in 1252, in which year he ascended the
throne as Alphonso X. They were long circulated in MS. and were first
printed in 1483, not long before the end of the period of stagnation.
Copernicus was born in 1473 at Thorn in Polish Prussia. In the course of
his studies at Cracow and at several Italian universities, he learnt all
that was known of the Ptolemaic astronomy and determined to reform it.
His maternal uncle, the Bishop of Ermland, having provided him with a
lay canonry in the Cathedral of Frauenburg, he had leisure to devote
himself to Science. Reviewing the suggestions of the ancient Greeks, he
was struck by the simplification that would be introduced by reviving
the idea that the annual motion should be attributed to the earth itself
instead of having a separate annual epicycle for each planet and for the
sun. Of the seventy odd circles or epicycles required by the latest form
of the Ptolemaic system, Copernicus succeeded in dispensing with rather
more than half, but he still required thirty-four, which was the exact
number assumed before the time of Aristotle. His considerations were
almost entirely mathematical, his only invasion into physics being in
defence of the "moving earth" against the stock objection that if the
earth moved, loose objects would fly off, and towers fall. He did not
break sufficiently away from the old tradition of uniform circular
motion. Ptolemy's efforts at exactness were baulked, as we have seen, by
the supposed necessity of all the orbit planes passing through the
earth, and if Copernicus had simply transferred this responsibility to
the sun he would have done better. But he would not sacrifice the old
fetish, and so, the orbit of the earth being clearly not circular with
respect to the sun, he made all his planetary planes pass through the
centre of the earth's orbit, instead of through the sun, thus
handicapping himself in the same way though not in the same degree as
Ptolemy. His thirty-four circles or epicycles comprised four for the
earth, three for the moon, seven for Mercury (on account of his highly
eccentric orbit) and five each for the other planets.
It is rather an exaggeration to call the present accepted system the
Copernican system, as it is really due to Kepler, half a century after
the death of Copernicus, but much credit is due to the latter for his
successful attempt to provide a real alternative for the Ptolemaic
system, instead of tinkering with it. The old geocentric system once
shaken, the way was gradually smoothed for the heliocentric system,
which Copernicus, still hampered by tradition, did not quite reach. He
was hardly a practical astronomer in the observational sense. His first
recorded observation, of an occultation of Aldebaran, was made in 1497,
and he is not known to have made as many as fifty astronomical
observations, while, of the few he did make and use, at least one was
more than half a degree in error, which would have been intolerable to
such an observer as Hipparchus. Copernicus in fact seems to have
considered accurate observations unattainable with the instruments at
hand. He refused to give any opinion on the projected reform of the
calendar, on the ground that the motions of the sun and moon were not
known with sufficient accuracy. It is possible that with better data he
might have made much more progress. He was in no hurry to publish
anything, perhaps on account of possible opposition. Certainly Luther,
with his obstinate conviction of the verbal accuracy of the Scriptures,
rejected as mere folly the idea of a moving earth, and Melanchthon
thought such opinions should be prohibited, but Rheticus, a professor at
the Protestant University of Wittenberg and an enthusiastic pupil of
Copernicus, urged publication, and undertook to see the work through the
press. This, however, he was unable to complete and another Lutheran,
Osiander, to whom he entrusted it, wrote a preface, with the apparent
intention of disarming opposition, in which he stated that the
principles laid down were only abstract hypotheses convenient for
purposes of calculation. This unauthorised interpolation may have had
its share in postponing the prohibition of the book by the Church of
Rome.
According to Copernicus the earth is only a planet like the others, and
not even the biggest one, while the sun is the most important body in
the system, and the stars probably too far away for any motion of the
earth to affect their apparent places. The earth in fact is very small
in comparison with the distance of the stars, as evidenced by the fact
that an observer anywhere on the earth appears to be in the middle of
the universe. He shows that the revolution of the earth will account for
the seasons, and for the stationary points and retrograde motions of the
planets. He corrects definitely the order of the planets outwards from
the sun, a matter which had been in dispute. A notable defect is due to
the idea that a body can only revolve about another body or a point, as
if rigidly connected with it, so that, in order to keep the earth's axis
in a constant direction in space, he has to invent a third motion. His
discussion of precession, which he rightly attributes to a slow motion
of the earth's axis, is marred by the idea that the precession is
variable. With all its defects, partly due to reliance on bad
observations, the work showed a great advance in the interpretation of
the motions of the planets; and his determinations of the periods both
in relation to the earth and to the stars were adopted by Reinhold,
Professor of Astronomy at Wittenberg, for the new Prutenic or Prussian
Tables, which were to supersede the obsolete Alphonsine Tables of the
thirteenth century.
In comparison with the question of the motion of the earth, no other
astronomical detail of the time seems to be of much consequence. Comets,
such as from time to time appeared, bright enough for naked eye
observation, were still regarded as atmospheric phenomena, and their
principal interest, as well as that of eclipses and planetary
conjunctions, was in relation to astrology. Reform, however, was
obviously in the air. The doctrine of Copernicus was destined very soon
to divide others besides the Lutheran leaders. The leaven of inquiry was
working, and not long after the death of Copernicus real advances were
to come, first in the accuracy of observations, and, as a necessary
result of these, in the planetary theory itself.
CHAPTER II.
EARLY LIFE OF KEPLER.
On 21st December, 1571, at Weil in the Duchy of Wurtemberg, was born a
weak and sickly seven-months' child, to whom his parents Henry and
Catherine Kepler gave the name of John. Henry Kepler was a petty officer
in the service of the reigning Duke, and in 1576 joined the army serving
in the Netherlands. His wife followed him, leaving her young son in his
grandfather's care at Leonberg, where he barely recovered from a severe
attack of smallpox. It was from this place that John derived the
Latinised name of Leonmontanus, in accordance with the common practice
of the time, but he was not known by it to any great extent. He was sent
to school in 1577, but in the following year his father returned to
Germany, almost ruined by the absconding of an acquaintance for whom he
had become surety. Henry Kepler was obliged to sell his house and most
of his belongings, and to keep a tavern at Elmendingen, withdrawing his
son from school to help him with the rough work. In 1583 young Kepler
was sent to the school at Elmendingen, and in 1584 had another narrow
escape from death by a violent illness. In 1586 he was sent, at the
charges of the Duke, to the monastic school of Maulbronn; from whence,
in accordance with the school regulations, he passed at the end of his
first year the examination for the bachelor's degree at Tuebingen,
returning for two more years as a "veteran" to Maulbronn before being
admitted as a resident student at Tuebingen. The three years thus spent
at Maulbronn were marked by recurrences of several of the diseases from
which he had suffered in childhood, and also by family troubles at his
home. His father went away after a quarrel with his wife Catherine, and
died abroad. Catherine herself, who seems to have been of a very
unamiable disposition, next quarrelled with her own relatives. It is not
surprising therefore that Kepler after taking his M.A. degree in August,
1591, coming out second in the examination lists, was ready to accept
the first appointment offered him, even if it should involve leaving
home. This happened to be the lectureship in astronomy at Gratz, the
chief town in Styria. Kepler's knowledge of astronomy was limited to the
compulsory school course, nor had he as yet any particular leaning
towards the science; the post, moreover, was a meagre and unimportant
one. On the other hand he had frequently expressed disgust at the way in
which one after another of his companions had refused "foreign"
appointments which had been arranged for them under the Duke's scheme of
education. His tutors also strongly urged him to accept the lectureship,
and he had not the usual reluctance to leave home. He therefore
proceeded to Gratz, protesting that he did not thereby forfeit his claim
to a more promising opening, when such should appear. His astronomical
tutor, Maestlin, encouraged him to devote himself to his newly adopted
science, and the first result of this advice appeared before very long
in Kepler's "Mysterium Cosmographicum". The bent of his mind was towards
philosophical speculation, to which he had been attracted in his
youthful studies of Scaliger's "Exoteric Exercises". He says he devoted
much time "to the examination of the nature of heaven, of souls, of
genii, of the elements, of the essence of fire, of the cause of
fountains, the ebb and flow of the tides, the shape of the continents
and inland seas, and things of this sort". Following his tutor in his
admiration for the Copernican theory, he wrote an essay on the primary
motion, attributing it to the rotation of the earth, and this not for
the mathematical reasons brought forward by Copernicus, but, as he
himself says, on physical or metaphysical grounds. In 1595, having more
leisure from lectures, he turned his speculative mind to the number,
size, and motion of the planetary orbits. He first tried simple
numerical relations, but none of them appeared to be twice, thrice, or
four times as great as another, although he felt convinced that there
was some relation between the motions and the distances, seeing that
when a gap appeared in one series, there was a corresponding gap in the
other. These gaps he attempted to fill by hypothetical planets between
Mars and Jupiter, and between Mercury and Venus, but this method also
failed to provide the regular proportion which he sought, besides being
open to the objection that on the same principle there might be many
more equally invisible planets at either end of the series. He was
nevertheless unwilling to adopt the opinion of Rheticus that the number
six was sacred, maintaining that the "sacredness" of the number was of
much more recent date than the creation of the worlds, and could not
therefore account for it. He next tried an ingenious idea, comparing the
perpendiculars from different points of a quadrant of a circle on a
tangent at its extremity. The greatest of these, the tangent, not being
cut by the quadrant, he called the line of the sun, and associated with
infinite force. The shortest, being the point at the other end of the
quadrant, thus corresponded to the fixed stars or zero force;
intermediate ones were to be found proportional to the "forces" of the
six planets. After a great amount of unfinished trial calculations,
which took nearly a whole summer, he convinced himself that success did
not lie that way. In July, 1595, while lecturing on the great planetary
conjunctions, he drew quasi-triangles in a circular zodiac showing the
slow progression of these points of conjunction at intervals of just
over 240 deg. or eight signs. The successive chords marked out a smaller
circle to which they were tangents, about half the diameter of the
zodiacal circle as drawn, and Kepler at once saw a similarity to the
orbits of Saturn and Jupiter, the radius of the inscribed circle of an
equilateral triangle being half that of the circumscribed circle. His
natural sequence of ideas impelled him to try a square, in the hope that
the circumscribed and inscribed circles might give him a similar
"analogy" for the orbits of Jupiter and Mars. He next tried a pentagon
and so on, but he soon noted that he would never reach the sun that way,
nor would he find any such limitation as six, the number of "possibles"
being obviously infinite. The actual planets moreover were not even six
but only five, so far as he knew, so he next pondered the question of
what sort of things these could be of which only five different figures
were possible and suddenly thought of the five regular solids.[2] He
immediately pounced upon this idea and ultimately evolved the following
scheme. "The earth is the sphere, the measure of all; round it describe
a dodecahedron; the sphere including this will be Mars. Round Mars
describe a tetrahedron; the sphere including this will be Jupiter.
Describe a cube round Jupiter; the sphere including this will be Saturn.
Now, inscribe in the earth an icosahedron, the sphere inscribed in it
will be Venus: inscribe an octahedron in Venus: the circle inscribed in
it will be Mercury." With this result Kepler was inordinately pleased,
and regretted not a moment of the time spent in obtaining it, though to
us this "Mysterium Cosmographicum" can only appear useless, even without
the more recent additions to the known planets. He admitted that a
certain thickness must be assigned to the intervening spheres to cover
the greatest and least distances of the several planets from the sun,
but even then some of the numbers obtained are not a very close fit for
the corresponding planetary orbits. Kepler's own suggested explanation
of the discordances was that they must be due to erroneous measures of
the planetary distances, and this, in those days of crude and infrequent
observations, could not easily be disproved. He next thought of a
variety of reasons why the five regular solids should occur in precisely
the order given and in no other, diverging from this into a subtle and
not very intelligible process of reasoning to account for the division
of the zodiac into 360 deg.. The next subject was more important, and dealt
with the relation between the distances of the planets and their times
of revolution round the sun. It was obvious that the period was not
simply proportional to the distance, as the outer planets were all too
slow for this, and he concluded "either that the moving intelligences of
the planets are weakest in those that are farthest from the sun, or that
there is one moving intelligence in the sun, the common centre, forcing
them all round, but those most violently which are nearest, and that it
languishes in some sort and grows weaker at the most distant, because of
the remoteness and the attenuation of the virtue". This is not so near a
guess at the theory of gravitation as might be supposed, for Kepler
imagined that a repulsive force was necessary to account for the planets
being sometimes further from the sun, and so laid aside the idea of a
constant attractive force. He made several other attempts to find a law
connecting the distances and periods of the planets, but without success
at that time, and only desisted when by unconsciously arguing in a
circle he appeared to get the same result from two totally different
hypotheses. He sent copies of his book to several leading astronomers,
of whom Galileo praised his ingenuity and good faith, while Tycho Brahe
was evidently much struck with the work and advised him to adapt
something similar to the Tychonic system instead of the Copernican. He
also intimated that his Uraniborg observations would provide more
accurate determinations of the planetary orbits, and thus made Kepler
eager to visit him, a project which as we shall see was more than
fulfilled. Another copy of the book Kepler sent to Reymers the Imperial
astronomer with a most fulsome letter, which Tycho, who asserted that
Reymers had simply plagiarised his work, very strongly resented, thus
drawing from Kepler a long letter of apology. About the same time Kepler
had married a lady already twice widowed, and become involved in
difficulties with her relatives on financial grounds, and with the
Styrian authorities in connection with the religious disputes then
coming to a head. On account of these latter he thought it expedient,
the year after his marriage, to withdraw to Hungary, from whence he sent
short treatises to Tuebingen, "On the magnet" (following the ideas of
Gilbert of Colchester), "On the cause of the obliquity of the ecliptic"
and "On the Divine wisdom as shown in the Creation". His next important
step makes it desirable to devote a chapter to a short notice of Tycho
Brahe.