<p>It is often said that a coincidence like this only happens to somebody
who “deserves his luck,” but this simply means that recognition is
essential to the coincidence. In the same way the appearance of one of a
large number of people mentioned is hailed as a case of the old adage
“Talk of the devil, etc.,” ignoring all the people who failed to
appear. No one, however, will consider Kepler unduly favoured. His
genius, in his case certainly “an infinite capacity for taking pains,”
enabled him out of his medley of hypotheses, mainly unsound, by dint of
enormous labour and patience, to arrive thus at the first two of the
laws which established his title of “Legislator of the Heavens”.</p>
<h3><SPAN name="figs" id="figs">Figures Explanatory of Kepler’s Theory of the Motion of Mars.</SPAN></h3>
<div class="illustration">
<ANTIMG src="images/fig1.png" alt="[Illustration: Fig. 1.]" title="Fig. 1" class="figure" id="fig1" />
<div class="caption"><span style="font-variant: small-caps">Fig. 1.</span></div>
</div>
<div class="illustration">
<ANTIMG src="images/fig2.png" alt="[Illustration: Fig. 2.]" title="Fig. 2" class="figure" id="fig2" />
<div class="caption"><span style="font-variant: small-caps">Fig. 2.</span></div>
</div>
<p><span style="font-variant: small-caps">Fig. 1.</span>—In Ptolemy’s excentric theory, A may be taken to represent the
earth, C the centre of a planet’s orbit, and E the equant, P (perigee)
and Q (apogee) being the apses of the orbit. Ptolemy’s idea was that
uniform motion in a circle must be provided, and since the motion was
not uniform about the earth, A could not coincide with C; and since the
motion still failed to be uniform about A or C, some point E must be
found about which the motion should be uniform.</p>
<p><span style="font-variant: small-caps">Fig. 2.</span>—This is not drawn to scale, but is intended to illustrate
Kepler’s modification of Ptolemy’s excentric. Kepler found velocities at
P and Q proportional not to AP and AQ but to AQ and AP, or to EP and EQ
if EC = CA (bisection of the excentricity). The velocity at M was wrong,
and AM appeared too great. Kepler’s first ellipse had M moved too near
C. The distance AC is much exaggerated in the figure, as also is MN.
AN = CP, the radius of the circle. MN should be .00429 of the radius, and
MC / NC should be 1.00429. The velocity at N appeared to be proportional
to EN ( = AN). Kepler concluded that Mars moved round PNQ, so that the
area described about A (the sun) was equal in equal times, A being the
focus of the ellipse PNQ. The angular velocity is not quite constant
about E, the equant or empty focus, but the difference could hardly have
been detected in Kepler’s time.</p>
<p>Kepler’s improved determination of the earth’s orbit was obtained by
plotting the different positions of the earth corresponding to
successive rotations of Mars, i.e. intervals of 687 days. At each of
these the date of the year would give the angle MSE (Mars-Sun-Earth),
and Tycho’s observation the angle MES. So the triangle could be solved
except for scale, and the ratio of SE to SM would give the distance of
Mars from the sun in terms of that of the earth. Measuring from a fixed
position of Mars (e.g. perihelion), this gave the variation of SE,
showing the earth’s inequality. Measuring from a fixed position of the
earth, it would give similarly a series of positions of Mars, which,
though lying not far from the circle whose diameter was the axis of
Mars’ orbit, joining perihelion and aphelion, always fell inside the
circle except at those two points. It was a long time before it dawned
upon Kepler that the simplest figure falling within the circle except at
the two extremities of the diameter, was an ellipse, and it is not clear
why his first attempt with an ellipse should have been just as much too
narrow as the circle was too wide. The fact remains that he recognised
suddenly that halving this error was tantamount to reducing the circle
to the ellipse whose eccentricity was that of the old theory, i.e. that
in which the sun would be in one focus and the equant in the other.</p>
<p>Having now fitted the ends of both major and minor axes of the ellipse,
he leaped to the conclusion that the orbit would fit everywhere.</p>
<p>The practical effect of his clearing of the “second inequality” was to
refer the orbit of Mars directly to the sun, and he found that the area
between successive distances of Mars from the sun (instead of the sum of
the distances) was strictly proportional to the time taken, in short,
equal areas were described in equal times (2nd Law) when referred to the
sun in the focus of the ellipse (1st Law).</p>
<p>He announced that (1) The planet describes an ellipse, the sun being in
one focus; and (2) The straight line joining the planet to the sun
sweeps out equal areas in any two equal intervals of time. These are
Kepler’s first and second Laws though not discovered in that order, and
it was at once clear that Ptolemy’s “bisection of the excentricity”
simply amounted to the fact that the centre of an ellipse bisects the
distance between the foci, the sun being in one focus and the angular
velocity being uniform about the empty focus. For so many centuries had
the fetish of circular motion postponed discovery. It was natural that
Kepler should assume that his laws would apply equally to all the
planets, but the proof of this, as well as the reason underlying the
laws, was only given by Newton, who approached the subject from a
totally different standpoint.</p>
<p>This commentary on Mars was published in 1609, the year of the invention
of the telescope, and Kepler petitioned the Emperor for further funds to
enable him to complete the study of the other planets, but once more
there was delay; in 1612 Rudolph died, and his brother Matthias who
succeeded him, cared very little for astronomy or even astrology, though
Kepler was reappointed to his post of Imperial Mathematician. He left
Prague to take up a permanent professorship at the University of Linz.
His own account of the circumstances is gloomy enough. He says, “In the
first place I could get no money from the Court, and my wife, who had
for a long time been suffering from low spirits and despondency, was
taken violently ill towards the end of 1610, with the Hungarian fever,
epilepsy and phrenitis. She was scarcely convalescent when all my three
children were at once attacked with smallpox. Leopold with his army
occupied the town beyond the river just as I lost the dearest of my
sons, him whose nativity you will find in my book on the new star. The
town on this side of the river where I lived was harassed by the
Bohemian troops, whose new levies were insubordinate and insolent; to
complete the whole, the Austrian army brought the plague with them into
the city. I went into Austria and endeavoured to procure the situation
which I now hold. Returning in June, I found my wife in a decline from
her grief at the death of her son, and on the eve of an infectious
fever, and I lost her also within eleven days of my return. Then came
fresh annoyance, of course, and her fortune was to be divided with my
step-sisters. The Emperor Rudolph would not agree to my departure; vain
hopes were given me of being paid from Saxony; my time and money were
wasted together, till on the death of the Emperor in 1612, I was named
again by his successor, and suffered to depart to Linz.”</p>
<p>Being thus left a widower with a ten-year-old daughter Susanna, and a
boy Louis of half her age, he looked for a second wife to take charge of
them. He has given an account of eleven ladies whose suitability he
considered. The first, an intimate friend of his first wife, ultimately
declined; one was too old, another an invalid, another too proud of her
birth and quarterings, another could do nothing useful, and so on.
Number eight kept him guessing for three months, until he tired of her
constant indecision, and confided his disappointment to number nine, who
was not impressed. Number ten, introduced by a friend, Kepler found
exceedingly ugly and enormously fat, and number eleven apparently too
young. Kepler then reconsidered one of the earlier ones, disregarding
the advice of his friends who objected to her lowly station. She was the
orphan daughter of a cabinetmaker, educated for twelve years by favour
of the Lady of Stahrenburg, and Kepler writes of her: “Her person and
manners are suitable to mine; no pride, no extravagance; she can bear to
work; she has a tolerable knowledge of how to manage a family;
middle-aged and of a disposition and capability to acquire what she
still wants”.</p>
<p>Wine from the Austrian vineyards was plentiful and cheap at the time of
the marriage, and Kepler bought a few casks for his household. When the
seller came to ascertain the quantity, Kepler noticed that no proper
allowance was made for the bulging parts, and the upshot of his
objections was that he wrote a book on a new method of gauging—one of
the earliest specimens of modern analysis, extending the properties of
plane figures to segments of cones and cylinders as being “incorporated
circles”. He was summoned before the Diet at Ratisbon to give his
opinion on the Gregorian Reform of the Calendar, and soon afterwards was
excommunicated, having fallen foul of the Roman Catholic party at Linz
just as he had previously at Gratz, the reason apparently being that he
desired to think for himself. Meanwhile his salary was not paid any more
regularly than before, and he was forced to supplement it by publishing
what he called a “vile prophesying almanac which is scarcely more
respectable than begging unless it be because it saves the Emperor’s
credit, who abandons me entirely, and with all his frequent and recent
orders in council, would suffer me to perish with hunger”.</p>
<p>In 1617 he was invited to Italy to succeed Magini as Professor of
Mathematics at Bologna. Galileo urged him to accept the post, but he
excused himself on the ground that he was a German and brought up among
Germans with such liberty of speech as he thought might get him into
trouble in Italy. In 1619 Matthias died and was succeeded by Ferdinand
III, who again retained Kepler in his post. In the same year Kepler
reprinted his “Mysterium Cosmographicum,” and also published his
“Harmonics” in five books dedicated to James I of England. “The first
geometrical, on the origin and demonstration of the laws of the figures
which produce harmonious proportions; the second, architectonical, on
figurate geometry and the congruence of plane and solid regular figures;
the third, properly Harmonic, on the derivation of musical proportions
from figures, and on the nature and distinction of things relating to
song, in opposition to the old theories; the fourth, metaphysical,
psychological, and astrological, on the mental essence of Harmonics, and
of their kinds in the world, especially on the harmony of rays emanating
on the earth from the heavenly bodies, and on their effect in nature and
on the sublunary and human soul; the fifth, astronomical and
metaphysical, on the very exquisite Harmonics of the celestial motions
and the origin of the excentricities in harmonious proportions.” The
extravagance of his fancies does not appear until the fourth book, in
which he reiterates the statement that he was forced to adopt his
astrological opinions from direct and positive observation. He despises
“The common herd of prophesiers who describe the operations of the stars
as if they were a sort of deities, the lords of heaven and earth, and
producing everything at their pleasure. They never trouble themselves to
consider what means the stars have of working any effects among us on
the earth whilst they remain in the sky and send down nothing to us
which is obvious to the senses, except rays of light.” His own notion is
“Like one who listens to a sweet melodious song, and by the gladness of
his countenance, by his voice, and by the beating of his hand or foot
attuned to the music, gives token that he perceives and approves the
harmony: just so does sublunary nature, with the notable and evident
emotion of the bowels of the earth, bear like witness to the same
feelings, especially at those times when the rays of the planets form
harmonious configurations on the earth,” and again “The earth is not an
animal like a dog, ready at every nod; but more like a bull or an
elephant, slow to become angry, and so much the more furious when
incensed.” He seems to have believed the earth to be actually a living
animal, as witness the following: “If anyone who has climbed the peaks
of the highest mountains, throw a stone down their very deep clefts, a
sound is heard from them; or if he throw it into one of the mountain
lakes, which beyond doubt are bottomless, a storm will immediately
arise, just as when you thrust a straw into the ear or nose of a
ticklish animal, it shakes its head, or runs shudderingly away. What so
like breathing, especially of those fish who draw water into their
mouths and spout it out again through their gills, as that wonderful
tide! For although it is so regulated according to the course of the
moon, that, in the preface to my ‘Commentaries on Mars,’ I have
mentioned it as probable that the waters are attracted by the moon, as
iron by the loadstone, yet if anyone uphold that the earth regulates its
breathing according to the motion of the sun and moon, as animals have
daily and nightly alternations of sleep and waking, I shall not think
his philosophy unworthy of being listened to; especially if any flexible
parts should be discovered in the depths of the earth, to supply the
functions of lungs or gills.”</p>
<p>In the same book Kepler enlarges again on his views in reference to the
basis of astrology as concerned with nativities and the importance of
planetary conjunctions. He gives particulars of his own nativity.
“Jupiter nearest the nonagesimal had passed by four degrees the trine of
Saturn; the Sun and Venus in conjunction were moving from the latter
towards the former, nearly in sextiles with both: they were also
removing from quadratures with Mars, to which Mercury was closely
approaching: the moon drew near to the trine of the same planet, close
to the Bull’s Eye even in latitude. The 25th degree of Gemini was
rising, and the 22nd of Aquarius culminating. That there was this triple
configuration on that day—namely the sextile of Saturn and the Sun, the
sextile of Mars and Jupiter, and the quadrature of Mercury and Mars, is
proved by the change of weather; for after a frost of some days, that
very day became warmer, there was a thaw and a fall of rain.” This
alleged “proof” is interesting as it relies on the same principle which
was held to justify the correction of an uncertain birth-time, by
reference to illnesses, etc., met with later. Kepler however goes on to
say, “If I am to speak of the results of my studies, what, I pray, can I
find in the sky, even remotely alluding to it? The learned confess that
several not despicable branches of philosophy have been newly extricated
or amended or brought to perfection by me: but here my constellations
were, not Mercury from the East in the angle of the seventh, and in
quadratures with Mars, but Copernicus, but Tycho Brahe, without whose
books of observations everything now set by me in the clearest light
must have remained buried in darkness; not Saturn predominating Mercury,
but my lords the Emperors Rudolph and Matthias, not Capricorn the house
of Saturn but Upper Austria, the house of the Emperor, and the ready and
unexampled bounty of his nobles to my petition. Here is that corner, not
the western one of the horoscope, but on the earth whither, by
permission of my Imperial master, I have betaken myself from a too
uneasy Court; and whence, during these years of my life, which now
tends towards its setting, emanate these Harmonics and the other matters
on which I am engaged.”</p>
<p>The fifth book contains a great deal of nonsense about the harmony of
the spheres; the notes contributed by the several planets are gravely
set down, that of Mercury having the greatest resemblance to a melody,
though perhaps more reminiscent of a bugle-call. Yet the book is not all
worthless for it includes Kepler’s Third Law, which he had diligently
sought for years. In his own words, “The proportion existing between the
periodic times of any two planets is exactly the sesquiplicate
proportion of the mean distances of the orbits,” or as generally given,
“the squares of the periodic times are proportional to the cubes of the
mean distances.” Kepler was evidently transported with delight and
wrote, “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 ‘Harmonics’—what I had
promised my friends in the title of this book, which I named before I
was sure of my discovery,—what sixteen years ago I urged as a thing to
be sought,—that for which I joined Tycho Brahe, for which I settled in
Prague, for which I have devoted the best part of my life to
astronomical computations, at length I have brought to light, and have
recognised its truth beyond my most sanguine expectations. Great as is
the absolute nature of Harmonics, with all its details as set forth in
my third book, it is all found among the celestial motions, not indeed
in the manner which I imagined (that is not the least part of my
delight), but in another very different, and yet most perfect and
excellent. It is now 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 on, burst out upon me. Nothing holds me; I
will indulge in 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.” He gives the date 15th May,
1618, for the completion of his discovery. In his “Epitome of the
Copernican Astronomy,” he gives his own idea as to the reason for this
Third Law. “Four causes concur for lengthening the periodic time. First,
the length of the path; secondly, the weight or quantity of matter to be
carried; thirdly, the degree of strength of the moving virtue; fourthly,
the bulk or space into which is spread out the matter to be moved. The
orbital paths of the planets are in the simple ratio of the distances;
the weights or quantities of matter in different planets are in the
subduplicate ratio of the same distances, as has been already proved; so
that with every increase of distance a planet has more matter and
therefore is moved more slowly, and accumulates more time in its
revolution, requiring already, as it did, more time by reason of the
length of the way. The third and fourth causes compensate each other in
a comparison of different planets; the simple and subduplicate
proportion compound the sesquiplicate proportion, which therefore is the
ratio of the periodic times.” The only part of this “explanation” that
is true is that the paths are in the simple ratio of the distances, the
“proof” so confidently claimed being of the circular kind commonly known
as “begging the question”. It was reserved for Newton to establish the
Laws of Motion, to find the law of force that would constrain a planet
to obey Kepler’s first and second Laws, and to prove that it must
therefore also obey the third.</p>
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