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The following discussion begins a long journey, along a
pathway of astronomical paradoxes leading from our discussion
of "the simplest discovery," via the revolutionary work of
Johannes Kepler, to the birth of a physics characterized by
non-algebraic, elliptic and hypergeometric functions.
In his "Commentaries on Mars" (also known as "Astronomia
Nova"), Kepler locates the origin of astronomy itself, in a
paradox going back to the most ancient times:
"The testimony of the ages confirms that the motions of the
heavenly bodies are in circular orbs. It is an immediate
presumption of reason, reflected in experience, that their
gyrations are perfect circles. For among figures it is circles,
and among bodies the heavens, that are considered the most
perfect. However, when experience is seen to teach something
different to those who pay careful attention, namely, that the
planets deviate from a simple circular path, it gives rise to a
powerful sense of wonder, which at length drives men to look into
causes. It is just this from which astronomy arose among men."
Indeed, in our previous discussion of "the simplest
discovery," the hypothetical prehistoric astronomer, observing
the cycle of day and night, came upon the paradox of a growing
discrepancy between the Sun's motion and that of the
constellation of stars. While the stars pursue what appear to be
perfectly circular orbits, the pathway of the Sun, as recorded
(for example) from week-to-week and month-to-month on the surface
of a large spherical sundial, has the form of a tightly-wound
coil. Each day the Sun completes one loop, making a slightly
different loop the next day. In the course of a year, the spiral
runs forward and then backward, doubling back on itself. More
complicated still than the path of the Sun, are the motions of
the Moon and planets. The latter displaying irregular, even
bizarre behavior when mapped against the background of the stars.
Kepler continues:
"The first adumbration of astronomy explains no causes, but
consists solely of the experience of the eyes, extremely slowly
acquired. It cannot be explained in figures or numbers, nor can
it be extrapolated into the future, since it is always different
from itself, to the extent that no spiral is equal to any other
in elapsed time ... Nevertheless, there are some people today who,
riding roughshod over 2,000 years' work, care, erudition
and knowledge, are trying to revive this, gaining admiration of
themselves from the mob ... Those with more experience consider
them with good reason to be incompetent....
"For it was very helpful to astronomers to understand that
two simple motions, the first and the second ones, the common and
the proper, are mixed together, and that from this confusion
there necessarily follows the continuous series of conglomerated
motions."
Indeed, to make some sense out of the motions of the Sun and
the planets, it is necessary to disentangle them from the daily
apparent rotation of the heavens ("the First Motion"). This is
most easily done, by recording the positions of the planets
relative to the stars and their constellations, rather than
relative to the horizon of the observer on the Earth. In other
words, we plot the positions of the planets against a "map" of
the stars (the so-called siderial positions). The resulting
motions of the planets relative to the background of stars,
became known as the "Second Motions." The first and second
motions combine together to give the observed motions.
In the case of the Sun, we have to overcome the difficulty,
that its illumination masks the weaker light of the stars, so the
Sun's position among the stars cannot be observed directly. But
there are many ways to adduce it indirectly; for example, we can
observe the positions of the constellations visible in the
still-dark side of the sky opposite to the Sun at the moment of
sunrise or sunset, and use the relevant angular measurements to
reconstruct the exact position the Sun must have on the stellar
map. The result of plotting the Sun's motion against the "dome"
of the stars, is very beautiful: The Sun is found to move along a
great circle in the heavens, called the ecliptic, whose
circumference is traditionally divided into twelve parts named by
stellar constellations ("signs of the zodiac").
For the planets, however, the siderial motions turn out to
be surprisingly complicated, and even bizarre. Kepler explains:
"Now that the first and diurnal motion had thus been set
aside, and those motions that are apprehended by comparison over
a period of days, and that belong to the planets individually,
had been considered in themselves, there appeared in these
motions a much greater confusion than before, when the diurnal
and common motion was still mixed in. For although this residual
confusion was there before, it was less observed, less striking
to the eyes, because the diurnal motion was very swift ... (In
particular) it was apparent that the three superior planets,
Saturn, Jupiter, and Mars, attune their motions to their
proximity to the Sun. For when the Sun approaches them, they move
forward and are swifter than usual, and when the Sun somes to the
sign opposite the planets, they retrace with crab-like steps the
road they had just covered."
What could be the reason for this bizarre "crab-like"
behavior of planetary motions, even forming doubled-back loops in
the case of the planet Mars? Where is the simple circular motion,
which would supposedly constitute the elementary, self-evident
form of action in the Universe?
Don't rush to supply answers from what you were taught in
the past, thus cheating yourself out of the joy of reliving some
earth-shaking discoveries. Let's stop and think about this.
Remember first Plato's parable of shadows in the cave. Are
we seeing, in the bizarre motions of Mars and other planets, mere
shadows of the real process? Assuming, for example, that we are
seeing only a projection of the real planetary motions in space,
how could we discover the "true motions" of the planets?
Reflecting on this challenge, we soon find ourselves confronted
with a seemingly formidable array of interconnected paradoxes.
First, given that astronomers were restricted (until recent
decades) to observations made only from the Earth, how could we
determine the exact location of a planet in space? In particular,
how could we even determine its distance from us?
To see the elementary difficulty involved, pose the task in
more general terms. Imagine an observer, located at any arbitrary
point in space. In respect to distant objects, the Universe
appears to that observer as if projected onto the surface of a
large sphere centered at the observer -- the so-called "celestial
sphere." The principle of the projection is very simple: Imagine
a distant object, such as a star, emitting rays of light in all
directions. The rays which reach the observer, form a very thin
cone, which intersects the sphere in a tiny circle (assuming the
star itself has a spherical cross-section). Now, from the
standpoint of what the observer sees, the star has the same
appearance as if it were a light source of appropriate size,
brightness, and color and so forth, fixed to the surface of the
sphere. Or, again, if we were to compare the given star to
another star, at twice the distance, but also twice as large,
how could the observer tell the difference? Furthermore, in the
case of distant stars (and to some extent even planets, when
observed by the naked eye), the ratio of the object's diameter to
distance is so small, and the cone of rays so thin, that these
objects are seen as hardly more than mere points; evidently their
distances could be varied over a considerable range, without the
observer being able to detect the difference.
The situation becomes even more complicated, when we
consider the effect of motion. First, consider the case of a
distant planet moving at constant velocity in a circular orbit
around the observer. As seen from the observer, the planet's
motion over any given interval of time will appear to describe a
circular arc on the celestial sphere. It is easy to see, that the
same apparent motion would be caused by a planet moving twice as
fast, on a circular orbit of twice the radius around our
observer.
Actually, the ambiguity is much greater! Construct a plane
passing through the original circular orbit. That plane passes
through the location of the observer, and cuts the sphere in a
great circle. Now draw any arbitrary curve on that plane, only
subject to the condition, that it encloses the observer without
folding back on itself. Then it is easy to construct a
hypothetical motion of a planet on that curve, which would
present exactly the same appearance to the observer as original
planet moving in a constant circular orbit! All we have to do is
construct a ray from the observer to the location of the original
planet on its circular orbit. That ray intersects the arbitrary
curve in some point P. As the ray follows the motion of the
original planet, rotating at constant speed, the point P moves
along our arbitrary curve. If we now attach a hypothetical planet
to the moving point P, its motion, as seen from the standpoint of
the observer, will seem to coincide with that of the original
planet. Note, that although the observed imagine will appear to
move always at a constant rate around the observer, the actual
speed of the hypothetical planet on the arbitrary curve will be
highly variable; in fact, the planet will be accelerating or
decelerating at each point where the arbitrary curve deviates from
a perfect circle around the observer. Consider, for example, the
case where the curve is an elongated ellipse with the observer at
one focus.
The problem becomes more complicated still, if we admit the
possibility, that the observer himself might be moving. The
paradox already hits us with full force, when we observe the
nightly motion of the stars. Are the stars orbitting around us, or
are the stars fixed and the earth is rotating, in the opposite
direction? Or some combination of both? Supposing the stars are
fixed, and the Earth is rotating, what about the Sun? When we
"clean away" the effect of the Earth's rotation, by plotting the
Sun's apparent motion against the "sphere of the fixed stars,"
the Sun is seen to move on a circle, the ecliptic. Is the center
of the Earth fixed relative to the stars, and the Sun orbiting
around that center? Or, is the Sun fixed, and the Earth orbitting
with the same speed, but the opposite direction around the Sun,
on a circle of the same radius? In each case, and in countless
other imaginable combinations and variations, the observed
phenomena would seem to be the same!
These arguments would appear to demonstrate the complete
futility of determining the actual orbit and speed of a planet
from its observed motion as seen from the Earth! We seem to be
confronting Kant's famous "Ding an sich" -- the pessimistic
notion, that Man can never know reality "as it really is." Can we
accept such a standpoint? Were God so cruel, as to create such a
hermetic barrier to Reason's participation in His universe?
During centuries of debate about the motion of the Earth and
the celestial bodies, there were those who rejected even the
concept of "true motions" as opposed to "apparent" ones, and
maintained that only observations -- i.e., sense perceptions --
are real. From that sort of radical-positivist standpoint, it
makes no difference whether we assume the Earth is fixed and the
Sun is moving, or vice-versa; these are merely two among an
infinity of mathematically equivalent opinions, none of which
have any particular claim to truth.
One of the notable advocates of this kind of indifferentism,
sharply and repeatedly denounced by Kepler, was one Petrus Ramus
(1515-1572). Ramus was a leading "anti-Aristotelian" of the
species of the later Paolo Sarpi. (In other words, he was more
Aristotelian than Aristotle!) Ramus held a prestigious
Professorship at the College de France and was known for works on
philosophy, law and mathematics. In his famous book on elementary
mathematics, Ramus banned incommensurables, eliminated the
axiomatic approach of Euclid, and rejected the regular solids as
insignificant and useless. He went over from the Catholic Church
to Calvinism and found his end during the famous "St.
Bartholemeus night." Kepler put his polemic against Ramus on the
very first page of the "Astronomia Nova," quoting Ramus' demand
for an "astronomy without hypotheses," and then giving his own,
devastating reply:
Petrus Ramus, Scholae Mathematica, Book II:
"Thus, the contrivance of hypotheses is absurd; nevertheless,
in Eudoxus, Aristotle, and Callippus, the contrivance is simpler,
as they supposed the hypotheses to be true -- indeed, they have
been venerated as if they were the gods of the starless orbs. In
later times, on the other hand, the tale is by far the most
absurd, the demonstration of the truth of natural phenomena
through false causes. For this reason, Logic above all, as well
as the Mathematical elements of Arithmetic and Geometry, will
provide the greatest assistance in establishing the purity and
dignity of the most noble art [Astronomy - JT]. Would that
Copernicus had been more inclined towards this idea of
establishing an astronomy without hypotheses! For it would have
been far easier for him to describe an astronomy corresponding to
the truth about the stars, than to move the Earth, a task like
the labor of some giant, so that in consequence of the earth's
being moved, we might observe the stars at rest ... I will solemnly
promise you the Regius Professorship at Paris as a prize for an
astronomy constructed without hypotheses, and will fulfill this
promise with the greatest pleasure, even by resigning our
professorship."
The author [Kepler - JT] to Ramus:
"Conveniently for you, Ramus, you have abandoned this surety
by departing both from life and professorship. Had you still held
the latter, I would, in my judgement, have won it indeed,
inasmuch as, in this work, I have at length succeeded, even by
the judgement of your own logic. As you ask the assistance of
Logic and Mathematics for the noblest art, I would only ask you
not to exclude the support of Physics, which it can by no means
forego ... It is a most absurd business, I admit, to demonstrate
natural phenomena through false causes, but this is not what is
happening in Copernicus. For he too considered his hypotheses
true, no less than those whom you mentioned considered their old
ones true, but he did not just consider them true, but
demonstrates it; as evidence of which I offer this work.... Thus,
Copernicus does not mythologize, but seriously presents
paradoxes; that is, he philosophizes. Which is what you wish of
the astronomer."
What is wrong with our arguments? Provoked by Kepler's
remarks, reflect for a moment on the paradox of "unknowability"
of the true planetary motions, presented above. Is the Universe
really unknowable in that way? Or might it not rather be the
case, that our reasoning contains some pervasive, false
assumption, which is the root of the trouble?
(Note: This discussion begins a longer series, which will
not run consecutively, but will nonethless constitute a coherent
whole.)
In Part I of this series (which readers should review before
proceeding further here), I presented a series of arguments,
purporting to demonstrate that there is no way to determine the
actual movement of a planet in space, from observations made on
the Earth. To this effect, I showed how, for any given pattern of
observed motions, to construct an infinity of hypothetical
motions in space, each of which would present exactly the same
apparent motions as seen from the Earth. Short of leaving the
Earth's surface -- an option not available to Kepler and his
contemporaries -- the effort to determine the actual orbits of
the planets would appear to be nothing but useless speculation.
Actually, similar sorts of arguments could be used to "prove" the
futility of Man's gaining any solid knowledge at all about the
outside world, beyond the mere data of sense perception per se!
But wait! Man's history of sustained, orders-of-magnitude
increases in per-capita power over Nature since the Pleistocene,
demonstrates exactly the opposite: The human mind is able, by
the method of hypothesis, to overleap the bounds of empiricism,
and attain increasingly efficient knowledge of the ordering of
the Universe. Kepler's own, brilliantly successful pathway of
discovery, in unravelling the form and ordering of the planetary
orbits, provides a most instructive case in point. The conclusion
is unavoidable: the arguments I presented earlier in favor of a
supposed "unknowability" of the planetary motions, must contain
some fundamental error!
Kepler's emphasis on physics and the method of
hypothesis, as opposed to the impotence of mere "mathematics and
logic," should help us to sniff out the sophistry embedded in
those arguments.
Did we not, in constructing a multiplicity of hypothetical
orbits consistent with given observations, implicitly assume
that those motions took place in a non-existent, empty
mathematical space of the Sarpi-Galileo-Descartes-Newton type,
rather than the real Universe? Did we not implicity collapse the
"observer" to an inert mathematical point, ignoring the crucial
factor of curvature in the infinitesimally small? Didn't we
overlook the inseparable connection between human knowledge,
hypothesis, and change?
Human knowledge is not a contemplative matter of fitting
plausible interpretations to an array of sense perceptions. On
the contrary, knowledge develops through human intervention to
change the Universe -- a process which involves not only
generating scientific hypotheses, but above all acting on them.
The "infinitesimal" is no mathematical point; it possesses an
internal curvature which is in demonstrable correspondence with
the curvature of the Universe as a whole. That relationship
centers on the role of the sovereign, creative human individual
as God's helper in the ongoing process of Creation.
For example, even the most banal application of
"triangulation" in elementary geometry, reflects the principle of
change. Rather than impotently staring at a distant object X (for
example, a distant mountain peak, or an enemy position in war)
from a fixed location A, "triangulation" relies on change of
position from A to a second vantage-point B and so on, measuring
the corresponding angular shifts in X's apparent position
relative to other landmarks and the "baseline" A-B.
Notice, that when we shift from A to B, we not only change the
apparent angle to X, but we change the entire array of
relationships to every other visible object in the field of view
of A and B. Taken at face value, the two spherical-projected
images of the world, as seen from A and as seen from B, are
formally contradictory. They define a paradox which can only
be solved by hypothesis. So, we conceptualize an additional
dimension, a "depth" which is not represented in any single
projection per se. The same metaphorical principle is already
built into the binocular organization of our own visual
apparatus. Compare this with the more advanced principle of
Eratosthenes' measurement of the Earth's curvature, and the
methods developed by Aristarchus and others to estimate the
Earth-Moon and Earth-Sun distances.
The circumstance, that even our sensory apparatus (including
the relevant cortical functions) is organized in this way, once
again underlines the fallacies embedded in Kant's claimed
unknowability of the "Ding an sich."
It was Kepler himself, who first used the combination of
Mars, the Earth and the Sun -- without leaving the Earth's
surface! -- to unfold a "nested" series of triangulations which
definitively established the elliptical functions of the
planetary orbits and their overall organization within the solar
system. But, as we shall see, the key to Kepler's method was not
simple triangulation in the sense of elementary geometry, but
rather his shift away from naive Euclidean geometry, toward a
revolutionary conception of physical geometry.
Turn back now to the paradoxes of planetary motion as seen
from the Earth, particularly Mars, Jupiter, and Saturn. Mapping
the motion of these planets against the stars, we find that they
travel around the ecliptic circle (or more precisely, in a
band-like region around the ecliptic), but not at a uniform
rate. Although the predominant motion is forward in the same
direction as the Sun, at periodic intervals these planets are
seen to slow down and reverse their motion, making a rather flat
"loop" in the sky, and then reverting to forward motion once
again. This process of retrograde motion and "looping",
invariably occurs around the time of the so-called opposition
with the Sun, i.e., when the positions of the Sun and the given
planet, as mapped on the "sphere of the stars," are approaching
opposite poles relative to each other. Curiously, around that
same time, the planet appears the brightest and largest, while in
the opposite relative position -- near the so-called conjunction
with the Sun -- the planet appears smaller and weaker, while at
the same time displaying its most rapid apparent motion!
Although the "looping" of Mars (for example) recurs at
roughly equal intervals of time, and is evidently closely
correlated with the motion of the Sun, the period of recurrence
is not equal to a year, nor is it the same for Jupiter and
Saturn, as for Mars! The so-called synodic period of Mars -- the
period between the successive oppositions of Mars to the Sun,
which coincides with the period between successive "loops" in
Mars' orbit -- is observed to be approximately 780 days. In the
case of Jupiter, on the other hand, the opposition to the Sun and
formation of a loop, occur at intervals of about 399 days, or
roughly once every 13 months.
But there is an additional complication. The planet does not
come back to its original position in the stars (its siderial
position) after a synodic period! The locus of the "loop",
relative to the stars, changes with each cycle of recurrence.
After ending its retrograde motion and completing a loop, Mars
proceeds to travel something more than a full circuit forward
along the ecliptic, before the looping process begins again. Long
observation, shows that the locus of each loop is shifted an
average of about 49 degrees forward along the ecliptic, relative
to the preceeding one.
Our experiments on the behavior of epicycloids, strongly
suggest, that what we are looking at is some sort of
epicyloid-like combination of two (or more) astronomical cycles!
If so, then one of them would be the one producing the "looping,"
and having a cycle length equal to 780 days, the synodic period
of Mars. The other cycle -- which cannot be observed directly,
because it is strongly disturbed and distorted by the looping --
would be the one determining the overall, net "forward" motion of
Mars along the ecliptic. The fact, that Mars travels 360 plus 49
degrees along the ecliptic, before "looping" recurs, suggests
that the cycle determing the looping has a somewhat longer
period, than the cycle responsible for the net forward motion. In
fact, the synodic cycle would have to be about 13.5% longer than
the other cycle, to give the shift of 49 degrees forward from
loop to loop. Or, to put it differently: if the hypothetical
cycle of forward motion along the ecliptic, generates an angle of
360 plus 49 degrees in the time between successive loops -- i.e.,
780 days -- then the time needed by that same forward motion to
complete a full cycle of exactly 360 degrees, would be 687 days,
or about 1.88 years. Of course, this whole reasoning assumes that
each cycle progresses at a uniform, constant rate.
Let's stop to reflect for a moment. On the basis of
assumptions which, admittedly, require further examination, we
have just adduced the existence of a 1.88-year cycle of Mars, a
cycle which is not directly observable. Firstly, as Kepler
remarked, Mars' apparent trajectory never closes! Evidently we
have a phenomenon of "incommensurability" of cycles. Moreover,
the Mars trajectory itself does not lie exactly on the ecliptic
circle, but winds around it in a band-like region like a coil.
When Mars returns after going around the eliptic, it does not
return to the same precise positions. So, where is the cycle? If
we leave aside the deviations from the ecliptic, and just count
the number of days Mars needs to make a single circuit within the
ecliptic "band," we get many different answers, depending on when
and where in the "looping" cycle, we begin to count. Again, the
observed motion of Mars is not strictly periodic. The 1.88-year
cycle is born of hypothesis, not of direct empirical observation.
Historically (and as per the discussion in Kepler's
"Astronomia Nova," the adduced 1.88-year cycle was referred to as
the "first inequality" of Mars, while the cycle governing the
"looping" phenomenon, was called the "second inequality."
Now, our analysis up to now has been based on the
assumption, that the underlying motion of a "cycle," is uniform
circular motion. That assumption dominated astronomical thinking
up to the time of Kepler, and not without good reasons. After
all, didn't the approach of combining circular motions prove
rather successful, earlier, in unravelling the motion of the Sun?
We found that the Sun's apparent motion can be understood as a
combination of two circular motions: a daily rotation of the
entire sphere of the stars, and a yearly motion of the Sun along
a great-circle path (the ecliptic) on that stellar sphere. In the
case of Mars (and the other outer planets), we evidently are
dealing with a combination of three degrees of rotation: the
daily stellar rotation; the "first inequality" with a period of
690 days; and the "second inequality" with a period of 780 days.
We are not finished, however. As Kepler would have
emphasized, "the devil is in the detail." To undercover a new set
of anomalies, we must drive the fundamental hypothesis which has
been the basis of our reasoning up to now -- the hypothesis of
uniform circular motion as elementary -- to its limits. This is
exactly what Kepler does in his {Astronomia Nova}. As his point
of departure, he reviews the three main methods, developed up to
that time, to construct the observed motions from a combination
of simple circular motions. These were: 1) the method of
epicycles associated with Ptolemeus, but actually developed by
Greek astronomers centuries earlier; 2) the method of concentric
circles, associated with Copernicus, but which had been put
forward 14 centuries earlier by Aristarchos, and probably even by
the original Pythagoreans; and finally 3) the method favored by
Kepler's elder collaborator, Tycho Brahe, which combines elements
of both.
The differences between the constructions of Ptolemy,
Copernicus and Tycho Brahe do not concern their common assumption
of simple circular motion as elementary; at first glance, they
merely differ in the way they combine circular motions to produce
the observed trajectories.
(Readers should construct models to illustrate the following
constructions!)
In the simplest form of Ptolemy's construction, the Earth is
the center of motion of the Sun and the primary center of motion
of all the planets. The "first inequality" (of Mars, Jupiter or
Saturn) is represented by motion on a large circle, C1 (called
the "eccentric"), centered at the Earth, while the planet itself
is carried along on the circumference of a second, smaller circle
C2 (called the "epicycle"), whose center moves along C1. That
motion of the planet on the second circle, corresponds to the
"second inequality." In the case of Mars, for example, the planet
makes one circuit of the second circle in 780 days, while at the
same time the center of the second circle moves along the first
circle at a rate corresponding to one revolution in 690 days. It
is easy to see how the phenomenon of retrograde motion is
produced: At the time when the planet is located on the portion
of its epicycle closest to the Earth, its motion on the epicycle
is opposite to the motion on the first circle, and somewhat
faster, yielding a net retrograde motion. From the angle
described by the retrograde motions we can conclude the ratio of
the radii of the two circles. To account for the transverse
component of motion in a loop according ot this hypothesis, we
must assume that the plane of the second circle is slightly
skewed to that of the first circle. Ptolemy used an somewhat
different, but analogous construction to account for the apparent
motions of the "inner" planets Mercury and Venus.
In the simplest form of the so-called Copernican
construction, the circular motions are assumed to be essentially
concentric, centered at the Sun, although in slightly different
planes. The apparent yearly motion of the Sun is assumed to
result from a yearly motion of the Earth around the Sun. As for
Mars, we represent its "first inequality" by a circle around the
Sun, upon which Mars is assumed to move directly. The "second
inequality," on the other hand, now appears as a mere artifact,
arising from the combined effect of the supposed,
concentric-circular motions of the Earth and Mars. Since the
Earth's period is shorter than that of Mars, the Earth
periodically catches up with and passes Mars on its "inside
track." At that moment of passing, Mars will appear from the
Earth as if it were moving backwards relative to the stars. On
the other hand, as the Earth approaches the position opposite to
Mars on the other side of the Sun, Mars will attain its fastest
apparent forward motion relative to the stars, the latter being
exaggerated by the effect of the Earth's motion in the opposite
direction.
In Tycho Brahe's construction, the planets (except the
Earth) are supposed to move on circular orbits around the Sun,
while the Sun itself (together with its swarm of planets, some
closer, some farther away than the Earth) is carried around the
Earth in an annual orbit.
Now, in his discussion in {Astronomia Nova}, Kepler
emphasized that the three constructions, when carried out in
detail, produce exactly the same apparent motions. From a
purely formal standpoint, it would seem there could be no basis
for deciding in favor of the one or the other. Yet, from a
conceptual standpoint, the three are entirely different. And
since Man does not merely contemplate his hypotheses, but acts
on them, every conceptual difference -- insofar as it bears on
axiomatics -- is eminently physical at the same time, even if
the effect appears first only as an "infinitesimal shift" in the
mind of a single human being.
Next week, by pushing the theories of Ptolemy, Copernicus,
and Brahe to their limit, Kepler will evoke from the Universe a
most remarkable response: All three approaches are false!
"It is true that a divine voice, which enjoins humans to
study astronomy, is expressed in the world itself, not in words
or syllables, but in things themselves and in the conformity of
the human intellect and senses with the ordering of the celestial
bodies and their motions. Nevertheless, there is also a kind of
fate, by whose invisible agency various individuals are driven to
take up various arts, which makes them certain that, just as they
are a part of the work of creation, they likewise also partake to
some extent in divine providence....
"I therefore once again think it to have happened by divine
arrangement, that I arrived at the same time in which he (Tycho
Brahe) was concentrated on Mars, whose motions provide the only
possible access to the hidden secrets of astronomy, without which
we should forever remain ignorant of those secrets." (Kepler,
Astronomia nova, Chapter 7).
Last week we briefly reviewed the three main competing
approaches to understanding the apparent planetary motions,
examined by Kepler: those of Ptolemy, Copernicus, and Tycho Brahe.
Kepler emphasized the purely formal equivalence of the three
approaches, at least in their simplest versions, but he pointed
out crucial differences in their physical (i.e.,
ontological-axiomatic) character, while also noting some deeper,
common assumptions of all three. Kepler first of all attacked
Ptolemy's method, on the grounds of its arbitrary assumptions,
which reject the principle of reason:
"Ptolemy made his opinions correspond to the data and to
geometry, and has failed to sustain our admiration. For the
question still remains, what cause it is that connects all the
epicycles of the planets to the Sun..." (My emphasis - JT).
"Copernicus, with the most ancient Pythagoreans and
Aristarchus, and I along with them, say that this second
inequality does not belong to the planet's own motion, but only
appears to do so, and is really a byproduct of the Earth's annual
motion around the motionless Sun."
In his Mysterium Cosmograpium, Kepler had pointed out:
"For, to turn from astronomy to physics or cosmography,
these hypotheses of Copernicus not only do not offend against
Nature, but assist her all the more. She loves simplicity, she
loves unity. Nothing ever exists in her which is superfluous, but
more often she uses one cause for many effects. Now under the
customary hypothesis there is no end to the invention of circles;
but under Copernicus a great many motions follow from a few
circles."
In the Ptolemaic construction, each planet has at least two
cycles, and not only the "first inequality," but also the "second
inequality" is different for each one. Not only does the
hypothesis of Aristarchus eliminate the need for many "second
equalities" -- deriving them all, as effects, from the single
cycle of the Earth -- but countless other specifics of the
apparent planetary motions begin to become intelligible.
Truth, however, does not lie in the simplicity of an
explanation per se. Indeed, very often the "simplest"
explanation, one in which everything appears to fit together
effortlessly, and all irritating singularities disappear, is the
farthest from the truth! When things become too easy, too banal,
watch out! To get at the truth, we must always generate a new
level of paradox, by pushing our hypotheses to their
breaking-points. This Kepler does, by focussing on the
implications of certain irregularities in the planetary motions
-- overlooked in our discussion up to now -- which would be
virtually incomprehensible, if the cycles of the "first and
second inequalities" were based only on simple circular action.
Indeed, on closer examination, we find that the "loops" of
the planet Mars (for example) are not identical in shape, but
vary somewhat from one synodic cycle to the next! Nor is the
displacement of each loop, relative to the preceeding one,
exactly equal from cycle to cycle. Furthermore, even the motion
of the Sun itself along the ecliptic circle, upon close study,
reveals itself to be alternately speed up and slow down
significantly in the course of a year, contrary to our tacit
assumption up to now.
Indeed, already in ancient times astronomers wondered at the
paradoxical "inequality" of the Sun's yearly motion. In fact,
when we carefully map the Sun's motion relative to the "sphere of
the fixed stars," we find, that although the Sun progresses along
the ecliptic at an average rate of 360 degrees per year, the
angular motion is actually about 7% faster in early January
(about 0.95 degrees per day) than in July (about 1.02 degrees per
day). This variation causes quite noticeable differences in the
lengths of the seasons, as these are defined in terms of a solar
calender. Indeed, the four seasons correspond to a division of
the ecliptic circle into four congruent arcs, the division-points
being the two equinoxes (the intersection-points of the ecliptic
with the celestial equator) and the two solstices (the points on
the ecliptic midway between the equinox points, marking the
extremes of displacement from the celestial equator and thereby
also the positions of the Sun on the longest and shortest days of
the year). Due to the changes in the Sun's angular velocity along
the ecliptic, those four arcs are traversed in different times.
In fact, the lengths of the seasons, so determined, are as
follows (we refer to the seasons in northern hemisphere, which
are reversed in the southern hemisphere):
Spring: 92 days and 22 hours;
Summer: 93 days and 14 hours;
Fall: 89 days and 17 hours;
Winter: 89 days and 1 hour.
This unevenness in the solar motion confronts us with a
striking paradox: How could we have a "perfect" circular
trajectory, as the Sun's path (the ecliptic) appears to be, and
yet the motion on that trajectory not be uniform? That would seem
to violate the very nature of the circle. Or shall we assume,
that some "outside" force could alternately accelerate or
decelerate the Sun (or Earth, if we take Copernicus' standpoint),
without leaving any trace in the shape of the trajectory itself?
Furthermore, how are we to comprehend this variation, if we hold
to the hypothesis, that the elementary form of action in
astronomy is uniform circular motion? On the other hand, if we
give up uniform circular motion as the basis for constructing all
forms of motion, then we seem to open up a Pandora's box of a
unlimited array of conceivable motions, with no criterion or
principle to guide us.
One "way out" -- which only shifts the paradox to another
place, however --, would be to keep the assumption, that the
Earth's motion (and that of the other planets) is uniform
circular motion, but to suppose that the center of the orbit is
not located exactly at the Sun's position. This notion of a
displaced circular orbit was known as an "eccentric"; both
Ptolemy and Copernicus employed it in the detailed elaboration of
their theories, to account for the mentioned irregularities in
planetary motions. Assuming such orbits really exist, it is not
hard to interpret the speeding-up and slowing-down of the Sun's
apparent motion as a kind of illusion due to projection, in the
following way: Taking Copernicus' approach for example, the
"true" motion of the Earth would be a uniform circular one; but
the Sun, being located off of the center of the Earth's orbit,
would appear from the Earth to be moving faster when the Earth is
located on the portion of its eccentric closest to the Sun, and
slower at the opposite end. On this asssumption, it is not hard
to calculate, by geometry, how far the center of the eccentric
would have to be displaced from the Sun, in order to account for
the 7% difference in observed angular speeds between the
perihelion (closest distance) and aphelion (farthest distance) of
the eccentric.
From the standpoint of this construction, the "true" motion
of the Sun (or the Earth, in Copernicus' theory) would be that
corresponding exactly to the mean or average motion of 360
degrees per year, while the apparent motion would vary according
to the varying distance between Earth and Sun. Accordingly, Tycho
Brahe and Copernicus elaborated their analyses of the apparent
planetary motions on the basis of the assumed "true" circular
motion of the Sun (or Earth).
This exact point becamce a focus of debate between Kepler
and Tycho Brahe. Kepler writes:
"The occasion of ... the whole first part (of Astronomia
nova) is this. When I first came to Brahe, I became aware that in
company with Ptolemy and Copernicus, he reckoned the second
inequality of a planet in relation to the mean motion of the
Sun ... So, when this point came up in discussion between us,
Brahe said in opposition to me, that when he used the mean Sun he
accounted for all the appearances of the first inequality. I
replied that this would not prevent my accounting for the same
observations of the first inequality using the Sun's apparent
motion, and thus it would be in the second inequality that we
would see which was more nearly correct."
This challenge eventually led to the breakthroughs which
Kepler announced in the title of Part II of his Astronomia Nova:
"Investigation of the second inequality, that is, of the motions
of the sun or earth, or the key to a deeper astronomy, wherein
there is much on the physical causes of the motions."
Kepler had reason to be suspicious about the assumption of
perfect circular orbits as "elementary." On the one hand, Kepler
was a follower of Nicolaus of Cusa, who had written, in the
famous Section 11 of Docta Ignorantia,
"What do I say? In the course of their motion, neither the
Sun, nor the Moon nor the Earth nor any sphere -- although the
opposite appears true to us -- can describe a true circle ... It
is impossible to give a circle for which one could not give one
even more perfect; and a heavenly body never moves at a given
moment exactly the same way as at some other moment, and never
describes a truly perfect circle, regardless of appearances."
On the other hand, already Ptolemy knew that the tactic of
uniform motion on displaced, "eccentric" circles, fails to fully
account for irregularities turning up in the "first inequality"
of the planets Venus, Mars, Jupiter, and Saturn (particularly
Mars). To explain the accelerations and decelerations of the
planets, which still remain after the effect of the "second
inequality" is removed, and to reconcile those with other
features of the apparent motions, it was not sufficient to merely
displace the circle of the "first inequality" from the observer
on the Earth. Ptolemy (or whoever actually did the work)
accordingly introduced a new artifice, called the "equant": On
this modifed hypothesis, the motion along the circumference of
the eccentric circle, instead of being itself uniform and
constant, would be driven forward by a uniform angular rotation
around a fixed point called the "equant," located at some
distance from the center of the circle. In the case of Mars, for
example, the Earth and the equant would be located on opposite
sides of the circle's center. This would result in a real
acceleration of the planet going toward its nearest point to the
Earth (and deceleration moving toward the opposite end), adding
to the effect of viewing this from the Earth. Actually, on the
basis of the "equant" construction, Ptolemy and his followers,
were able to make relatively precise calculations for all the
planets (except Mercury). It was first using the more precise
observations of Tycho Brahe, that Kepler could finally give
Ptolemy the "coup de grace."
Copernicus rejected the "equant," essentially on the grounds
that it de facto instituted "irregular" motions (i.e.,
non-circular motion) into astronomy. To avoid this, Copernicus
and Brahe invented still another circular cycle (in addition to
the "second inequality") to modify the supposed uniform motion on
the eccentric circle. We seem to be headed into a monstrous "bad
infinity."
But, isn't there something absurd and wholly artificial
about the idea of a planet orbiting in a circle around a mere
abstract mathematical point as center? And being propelled by an
abstract ray pivotting on another mathematical point? Kepler
writes:
"A mathematical point, whether or not it is the center of
the world, can neither effect the motion of heavy bodies nor act
as an object towards which they tend ... Let the physicists prove
that natural things have a sympathy for that which is nothing."
The same objection applies also, of course, to the device of
the epicycle, whose center is supposed to be a mere mathematical
point. Later Kepler adds:
"It is incredible in itself that an immaterial power reside
in a non-body, move in space and time, but have no subject ...
And I am making these absurd assumptions in order to establish in
the end the impossibility that every cause of the planet's
motions inhere in its body or somewhere else in its orb ... I
have presented these models hypothetically, the hypothesis being
astronomy's testimony, that the planet's path is a perfect
eccentric circle such as was described. If astronomy should
discover something different, the physical theories will also
change."
Aha! While seeking means to accurately determine the real
spatial trajectory, Kepler explores the notion, that something
like the effect of the "equant" might actually exist, as a new
mode of physical action:
"About center B let an eccentric DE be described, with
eccentricity BA, A being the place of the observer. The line
drawn through AB will indicate the apogee at D and the perigee at
F. Upon this line, above B, let another segment be extended,
equal to BA. C will be the point of the equant, that is, the
point about which the planet completes equal angles in equal
times, even though the circle is set up around B rather than C
..." Copernicus notes this hypothesis among other things in this
respect, that it offends against physical principles by
instituting "irregular celestial motions ... the entire solid orb
is now fast, now slow." This Copernicus rejects as absurd.
"Now I, too, for good reasons, would reject as absurd the
notion that the moving power should preside over a solid orb,
everywhere uniform, rather than over the unadorned planet. But
because there are no solid orbs, consider now the physical
evidence of this hypothesis when very slight changes are made, as
described below. This hypothesis, it should be added, requires
two motive powers to move the planet (Ptolemy was unaware of
this). It places one of these in the body A (which, in the
reformed astronomy will be the very Sun itself), and says that
this power endeavors to drive the planet around itself, but
possesses an infinite number of degrees corresponding to the
infinite number of points of the ray from A. Thus, as AD is the
longest, and AF the shortest, the planet is slowest at D and
fastest at F... The hypothesis attributes another motive power to
the planet itself, by which it works to adjust its approach to
and recession from the Sun, either by strength of the angles or
by intuition of the increase or decrease of the solar diameter,
and to make the difference between the mean distance and the
longest and shortest equal to AB. Therefore, the point of the
equant is nothing but a geometrical short cut for computing the
equations from an hypothesis that is clearly physical. But if, in
addition, the planet's path is a perfect circle, as Ptolemy
certainly thought, the planet also has to have some perception of
the swiftness and slowness by which it is carried along by the
other external power, in order to adjust its own approach and
recession in such accord with the power's prescriptions, that the
path DE itself is made to be a circle. It therefore requires both
an intellectual comprehension of the circle and a desire to
realize it...
"However, if the demonstrations of astronomy, founded upon
observations, should testify that the path of the planet is not
quite circular, contrary to what this hypothesis asserts, then
this physical account too will be constructed differently, and
the planet's power will be freed from these rather troublesome
requirements."
Kepler's hypothesis (which undergoes rapid evolution across
the pages of "Astronomia Nova") means throwing away the notion,
that the action underlying the solar system has the form of
"gear-box"-like mechanical-kinematic generation of motions.
Instead, Kepler references a notion of "power" and a constant
activity which generates dense singularities in every interval.
While for the moment, the circle remains a circle in outward
form, we have radically transformed the concept of the underlying
process of generation. In a sense, that shift in conception
amounts to an infinitesmal deformation of the hypothetical
circular orbit, which implicitly changes the entire universe. The
successful measurement of deviation of a planet's path from a
circular orbit, would constitute a unique experiment for the
hypothesis of a new, non-kinematic principle of action. That is
the "deeper astronomy" of Kepler!
So we come back to the problem: How to determine the precise
trajectory of a planet in space, given observations made only
from the Earth, and taking into account the fact, that the Earth
itself is moving? Having identified the "second inequality" as
the crux of the problem of apparent planetary motions, Kepler
turns the tables on the whole preceeding discussion, and uses
Mars and the Sun as "observation posts" to determine orbit of the
planet whose motion is the most difficult of all to "see" -- the
Earth itself!
But, how can we use Mars as an observation-post? Mars is
moving. No matter! Let us assume that part the hypothesis of
Aristarchus remains true, namely that the planets have closed
orbits, and that motion along those orbits is what produces the
so-called "first inequality" determined by the ancients. In that
case, Mars -- regardless of whether or not its orbit is
circular! -- periodically returns to any given locus in its
orbit. Furthermore, we already know the period-length of that
recurrence: it is the 1.88-year cycle which we adduced last week,
by indirect means, from the study of Mars' bizarre apparent
motions.
So, make a series of observations of the apparent positions
of Mars and the Sun, relative to the stars, at successive
intervals 1.88 years apart! If our reasoning is sound, Mars will
occupy (at least roughly) the same actual position in space,
relative to the assumed "fixed" Sun and stars, at each of those
times. On the other hand, at intervals corresponding to integral
multiples of 1.88 -- 0, 1.88, 3.76, 5.64, 7.54 years etc, -- the
Earth will occupy unequal positions, distributed more and more
densely around its orbit, the longer the series is continued (the
phenomenon of relative incommensurability).
Now make two "nested" types of triangulations. Assuming
first that the orbit of the Earth is very roughly circular, use
the observations of Mars's apparent position, as seen from two or
more of those positions of the Earth, to "triangulate" Mars'
location in space. Next, use that adduced location of Mars, plus
the angles defined by the apparent positions of Mars and the
Sun relative to the stars, to triangulate the position of the
Earth in space at each of the times 0, 1.88, 3.76 years etc.
Then use these adduced positions of the earth to develop an
improved {hypothesis} of the earth orbit. Apply the improved
knowledge of earth's orbit to correct the triangulation of Mars'
position. Use the improved localization of Mars to revise and
correct the values for the Earth's positions. Finally, use the
adduced knowledge of the Earth's orbital motion to "triangulate"
a series of positions of Mars, and other planets!
The experiment was successful. Ramus, Aristotle, and Kant were
demolished. The door was kicked open for a revolution in physics,
and a new mathematics of non-algebraic, non-kinematic functions.
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