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.)