<h1><SPAN name="ch-5" id="ch-5">Chapter V.</SPAN></h1>
<h2>Kepler’s Laws.</h2>
<p>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.</p>
<p>“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.</p>
<p>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 <SPAN href="#figs" class="link">Figs. 1, 2</SPAN>).</p>
<p>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.<span class="fn-marker"><SPAN href="#fn-3" class="link">[3]</SPAN></span>
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.</p>
<div class="footnote">
<SPAN name="fn-3" id="fn-3">
<span class="fn-label">Footnote 3:</span>
This is clearly a maximum at AMC in Fig. 2, when its tangent AC / CM = the eccentricity.</SPAN></div>
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