KEPLER'S LAWS.
When Gilbert of Colchester, in his "New Philosophy," founded on his
researches in magnetism, was dealing with tides, he did not suggest that
the moon attracted the water, but that "subterranean spirits and
humours, rising in sympathy with the moon, cause the sea also to rise
and flow to the shores and up rivers". It appears that an idea,
presented in some such way as this, was more readily received than a
plain statement. This so-called philosophical method was, in fact, very
generally applied, and Kepler, who shared Galileo's admiration for
Gilbert's work, adopted it in his own attempt to extend the idea of
magnetic attraction to the planets. The general idea of "gravity"
opposed the hypothesis of the rotation of the earth on the ground that
loose objects would fly off: moreover, the latest refinements of the old
system of planetary motions necessitated their orbits being described
about a mere empty point. Kepler very strongly combated these notions,
pointing out the absurdity of the conclusions to which they tended, and
proceeded in set terms to describe his own theory.
"Every corporeal substance, so far forth as it is corporeal, has a
natural fitness for resting in every place where it may be situated by
itself beyond the sphere of influence of a body cognate with it. Gravity
is a mutual affection between cognate bodies towards union or
conjunction (similar in kind to the magnetic virtue), so that the earth
attracts a stone much rather than the stone seeks the earth. Heavy
bodies (if we begin by assuming the earth to be in the centre of the
world) are not carried to the centre of the world in its quality of
centre of the world, but as to the centre of a cognate round body,
namely, the earth; so that wheresoever the earth may be placed, or
whithersoever it may be carried by its animal faculty, heavy bodies will
always be carried towards it. If the earth were not round, heavy bodies
would not tend from every side in a straight line towards the centre of
the earth, but to different points from different sides. If two stones
were placed in any part of the world near each other, and beyond the
sphere of influence of a third cognate body, these stones, like two
magnetic needles, would come together in the intermediate point, each
approaching the other by a space proportional to the comparative mass of
the other. If the moon and earth were not retained in their orbits by
their animal force or some other equivalent, the earth would mount to
the moon by a fifty-fourth part of their distance, and the moon fall
towards the earth through the other fifty-three parts, and they would
there meet, assuming, however, that the substance of both is of the same
density. If the earth should cease to attract its waters to itself all
the waters of the sea would he raised and would flow to the body of the
moon. The sphere of the attractive virtue which is in the moon extends
as far as the earth, and entices up the waters; but as the moon flies
rapidly across the zenith, and the waters cannot follow so quickly, a
flow of the ocean is occasioned in the torrid zone towards the westward.
If the attractive virtue of the moon extends as far as the earth, it
follows with greater reason that the attractive virtue of the earth
extends as far as the moon and much farther; and, in short, nothing
which consists of earthly substance anyhow constituted although thrown
up to any height, can ever escape the powerful operation of this
attractive virtue. Nothing which consists of corporeal matter is
absolutely light, but that is comparatively lighter which is rarer,
either by its own nature, or by accidental heat. And it is not to be
thought that light bodies are escaping to the surface of the universe
while they are carried upwards, or that they are not attracted by the
earth. They are attracted, but in a less degree, and so are driven
outwards by the heavy bodies; which being done, they stop, and are kept
by the earth in their own place. But although the attractive virtue of
the earth extends upwards, as has been said, so very far, yet if any
stone should be at a distance great enough to become sensible compared
with the earth's diameter, it is true that on the motion of the earth
such a stone would not follow altogether; its own force of resistance
would be combined with the attractive force of the earth, and thus it
would extricate itself in some degree from the motion of the earth." The
above passage from the Introduction to Kepler's "Commentaries on the
Motion of Mars," always regarded as his most valuable work, must have
been known to Newton, so that no such incident as the fall of an apple
was required to provide a necessary and sufficient explanation of the
genesis of his Theory of Universal Gravitation. Kepler's glimpse at such
a theory could have been no more than a glimpse, for he went no further
with it. This seems a pity, as it is far less fanciful than many of his
ideas, though not free from the "virtues" and "animal faculties," that
correspond to Gilbert's "spirits and humours". We must, however, proceed
to the subject of Mars, which was, as before noted, the first important
investigation entrusted to Kepler on his arrival at Prague.
The time taken from one opposition of Mars to the next is decidedly
unequal at different parts of his orbit, so that many oppositions must
be used to determine the mean motion. The ancients had noticed that what
was called the "second inequality," due as we now know to the orbital
motion of the earth, only vanished when earth, sun, and planet were in
line, i.e. at the planet's opposition; therefore they used oppositions
to determine the mean motion, but deemed it necessary to apply a
correction to the true opposition to reduce to mean opposition, thus
sacrificing part of the advantage of using oppositions. Tycho and
Longomontanus had followed this method in their calculations from
Tycho's twenty years' observations. Their aim was to find a position of
the "equant," such that these observations would show a constant angular
motion about it; and that the computed positions would agree in latitude
and longitude with the actual observed positions. When Kepler arrived he
was told that their longitudes agreed within a couple of minutes of arc,
but that something was wrong with the latitudes. He found, however, that
even in longitude their positions showed discordances ten times as great
as they admitted, and so, to clear the ground of assumptions as far as
possible, he determined to use true oppositions. To this Tycho objected,
and Kepler had great difficulty in convincing him that the new move
would be any improvement, but undertook to prove to him by actual
examples that a false position of the orbit could by adjusting the
equant be made to fit the longitudes within five minutes of arc, while
giving quite erroneous values of the latitudes and second inequalities.
To avoid the possibility of further objection he carried out this
demonstration separately for each of the systems of Ptolemy, Copernicus,
and Tycho. For the new method he noticed that great accuracy was
required in the reduction of the observed places of Mars to the
ecliptic, and for this purpose the value obtained for the parallax by
Tycho's assistants fell far short of the requisite accuracy. Kepler
therefore was obliged to recompute the parallax from the original
observations, as also the position of the line of nodes and the
inclination of the orbit. The last he found to be constant, thus
corroborating his theory that the plane of the orbit passed through the
sun. He repeated his calculations no fewer than seventy times (and that
before the invention of logarithms), and at length adopted values for
the mean longitude and longitude of aphelion. He found no discordance
greater than two minutes of arc in Tycho's observed longitudes in
opposition, but the latitudes, and also longitudes in other parts of the
orbit were much more discordant, and he found to his chagrin that four
years' work was practically wasted. Before making a fresh start he
looked for some simplification of the labour; and determined to adopt
Ptolemy's assumption known as the principle of the bisection of the
excentricity. Hitherto, since Ptolemy had given no reason for this
assumption, Kepler had preferred not to make it, only taking for granted
that the centre was at some point on the line called the excentricity
(see Figs. 1, 2).
A marked improvement in residuals was the result of this step, proving,
so far, the correctness of Ptolemy's principle, but there still remained
discordances amounting to eight minutes of arc. Copernicus, who had no
idea of the accuracy obtainable in observations, would probably have
regarded such an agreement as remarkably good; but Kepler refused to
admit the possibility of an error of eight minutes in any of Tycho's
observations. He thereupon vowed to construct from these eight minutes a
new planetary theory that should account for them all. His repeated
failures had by this time convinced him that no uniformly described
circle could possibly represent the motion of Mars. Either the orbit
could not be circular, or else the angular velocity could not be
constant about any point whatever. He determined to attack the "second
inequality," i.e. the optical illusion caused by the earth's annual
motion, but first revived an old idea of his own that for the sake of
uniformity the sun, or as he preferred to regard it, the earth, should
have an equant as well as the planets. From the irregularities of the
solar motion he soon found that this was the case, and that the motion
was uniform about a point on the line from the sun to the centre of the
earth's orbit, such that the centre bisected the distance from the sun
to the "Equant"; this fully supported Ptolemy's principle. Clearly then
the earth's linear velocity could not be constant, and Kepler was
encouraged to revive another of his speculations as to a force which was
weaker at greater distances. He found the velocity greater at the nearer
apse, so that the time over an equal arc at either apse was proportional
to the distance. He conjectured that this might prove to be true for
arcs at all parts of the orbit, and to test this he divided the orbit
into 360 equal parts, and calculated the distances to the points of
division. Archimedes had obtained an approximation to the area of a
circle by dividing it radially into a very large number of triangles,
and Kepler had this device in mind. He found that the sums of successive
distances from his 360 points were approximately proportional to the
times from point to point, and was thus enabled to represent much more
accurately the annual motion of the earth which produced the second
inequality of Mars, to whose motion he now returned. Three points are
sufficient to define a circle, so he took three observed positions of
Mars and found a circle; he then took three other positions, but
obtained a different circle, and a third set gave yet another. It thus
began to appear that the orbit could not be a circle. He next tried to
divide into 360 equal parts, as he had in the case of the earth, but the
sums of distances failed to fit the times, and he realised that the sums
of distances were not a good measure of the area of successive
triangles. He noted, however, that the errors at the apses were now
smaller than with a central circular orbit, and of the opposite sign, so
he determined to try whether an oval orbit would fit better, following a
suggestion made by Purbach in the case of Mercury, whose orbit is even
more eccentric than that of Mars, though observations were too scanty to
form the foundation of any theory. Kepler gave his fancy play in the
choice of an oval, greater at one end than the other, endeavouring to
satisfy some ideas about epicyclic motion, but could not find a
satisfactory curve. He then had the fortunate idea of trying an ellipse
with the same axis as his tentative oval. Mars now appeared too slow at
the apses instead of too quick, so obviously some intermediate ellipse
must be sought between the trial ellipse and the circle on the same
axis. At this point the "long arm of coincidence" came into play.
Half-way between the apses lay the mean distance, and at this position
the error was half the distance between the ellipse and the circle,
amounting to .00429 of a radius. With these figures in his mind, Kepler
looked up the greatest optical inequality of Mars, the angle between the
straight lines from Mars to the Sun and to the centre of the circle.[3]
The secant of this angle was 1.00429, so that he noted that an ellipse
reduced from the circle in the ratio of 1.00429 to 1 would fit the
motion of Mars at the mean distance as well as the apses.
[Footnote 3: This is clearly a maximum at AMC in Fig. 2, when its
tangent AC/CM = the eccentricity.]
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".
FIGURES EXPLANATORY OF KEPLER'S THEORY OF THE MOTION OF MARS.
[Illustration: FIG. 1.]
_______
/ \
/ \
| |
|___________|
Q| E C A |P
| |
\ /
\_______/
[Illustration: FIG. 2.]
___M___
/___|\__\
// N|\\ \\
|/ | \\ \|
|_____|__\\_|
Q| E C A |P
|\ | /|
\\___|___//
\___|___/
[Transcriber's Note: Approximate renditions of these figures are
provided. Fig. 1 is a circle. Fig. 2 is a circle which contains an
ellipse, tangent to the circle at Q and P. Line segments from M (on the
circle) and N (on the ellipse) meet at point A.]
FIG. 1.--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.
FIG. 2.--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.
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.
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.
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).
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.
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."
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".
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".
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."
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."
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.