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- How Kepler Changed The Laws of the Universe, Part I, by
Jonathan Tennenbaum

- How Kepler Changed The Laws of the Universe, Part II, by
Jonathan Tennenbaum

- How Kepler Changed The Laws of the Universe, Part III, by
Jonathan Tennenbaum

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.)

Part II of an Extended Pedagogical Discussion

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!

Part III of an extended pedagogical discussion

"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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