Riemann for Anti-Dummies: Part 4

when Kepler demonstrated the non-linear characteristic of the solar system, and consequentially, the entire physical universe, he set in motion a revolution in thinking, that to this day, is either hated or misunderstood, by scientists and laymen alike. Witness the discussion with a Baby Boomer mathematician who works for NASA, that took place at a recent chapter meeting. After a presentation on the congruence between LaRouche’s successful forecasting of the current systemic financial breakdown, and Gauss’ determination of the orbit of Ceres, the well-educated specialist asked, “Did Gauss know of the elliptical orbits? If so, then he must have had the inverse-square law.” When the historical illiteracy of his assertion was pointed out, the specialist replied, “Oh. I always thought the elliptical orbits were a consequence of the inverse square law. I never knew otherwise.”

Or, take the case of the slave of today’s popular opinion who confidently gauges his or her economic well-being from the standpoint of their own personal financial situation. Like the brother of the protagonist of Poe’s “Descent into the Maelstrom”, such fools are doomed to sink ever deeper into the abyss, hanging desperately onto the ship, when safety is easily won by leaving the temporary security of the ship’s greater bulk, for the seeming insecurity of a light barrel, which the whirlpool eagerly rejects.

Those who wish not to be counted among these legendary fools, find themselves compelled to actually comprehend Kepler’s great discovery.

It is commonly misunderstood, that Kepler’s discovery was the result of a numerical discrepancy between the observed positions of the planets, and the positions predicted by Ptolemy, Brahe, and Copernicus. While such discrepancies were certainly a marker that something was wrong in the state of astronomy, the paradox that provoked Kepler’s passion was not a numerical one. Rather, it was an epistemological one: If, God composed the solar system, so that action occurred in perfect circles, or inversely, in Gallileo’s straight lines, why then was the planets’ motion non-uniform? In other words, it was not a paradox in the domain of perception, but a paradox in the domain of Mind. The paradox arose from the irreconcilability of two ideas concerning the relationship between man and nature. On the one hand, the idea that physical action occurs according to perfect circles and straight lines, while seemingly more sensible to the naive mind, requires the universe to perform an irrational dance, in order to conform to its dictates. On the other hand, the concept that action in the physical universe was actually non-uniform, seemed to require man to embrace a less perfect geometry, but conformed more to planets’ actual motion. The former implicitly assumes that either man, nature, or both were irrational. The latter acknowledges, initially, a less precise geometrical construction , but in accepting a less simple mathematics, it hopes to gain a more perfect one.

Recall to mind again, the opening words of Kepler’s “New Astronomy”:

“The testimony of the ages confirms that the motions of the planets are orbicular. 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. Astronomy’s aim is considered to be to show why the star’ motions appear to be irregular on earth, despite their being exceedingly well ordered in heaven, and to investigate the circles wherein the stars may be moved, that their positions and appearances at any given time may thereby be predicted.”

The specific difficulty is this: Once, Kepler insisted that the Sun moved the planets by a force whose effect diminished with distance, the irregular speed of the planet could be known as the result of an eccentric orbit. This is because in an eccentric orbit, the distance between the planet and the Sun, is always changing, getting either longer or shorter. Thus, as the planet moves around the Sun, the effect of the Sun’s force is always diminishing or increasing, which slows the planet down, or speeds it up. (See the figures on pages 26, 27, and 33 of the Summer 1998 Fidelio.)

[See also http://csep10.phys.utk.edu/astr162/lect/binaries/visual/kepleroldframe.html]

This is distinct from the characteristic of a circular orbit, in which the distance between the planet and the Sun is always constant, and so a planet moving in such an orbit, will move at a constant rate. (This required the imposition of irrational demigods to speed the planet up or slow it down as it moved in this circular path.)

The question which struck fear in the hearts and minds of Aristoteleans to this day was: “How could the planet know to stay on this eccentric orbit?”

Since in an eccentric orbit, the planets’ distance from the Sun is always changing, the planet is have to constantly re-define its path, in every interval of action no matter how small. There is no way for the planet to define this path from the standpoint of simple straight line action between the planet and the Sun. Rather, the planet’s orbit must be defined by something outside the orbit itself. That “something” is what Kepler referred to in his “Epilogue” as the relationship of the Sun to the each planet individually and the relationship of the Sun to all the planets, which is the same as the relationship of noos to dianoia.

But, how can this “something” be measured? For this, Kepler had to settle for what he acknowledged was an imperfect method.

As Kepler recounts in the “New Astronomy”:

“My first error was to suppose that the path of the planet is a perfect circle, a supposition that was all the more noxious a thief of time the more it was endowed with the authority of all philosophers, and the more convenient it was for metaphysics in particular. Accordingly, let the path of the planet be a perfect eccentric…”

“Since, therefore, the times of a planet over equal parts of the eccentric are to one another as the distances of those parts, and since the individual points of the entire semicircle of the eccentric are all at different distances, it was no easy task I set myself when I sought to find how the sums of the individual distances may be obtained. For unless we can find the sum of all of them (and the are infinite in number) we cannot say how much time has elapsed for any one of them. Thus the equation will not be known. For the whole sum of the distances is to the whole periodic time as any partial sum of the distances is to its corresponding time.” (Diagram 5.3 on page 27 of the Fidelio, illustrates Kepler’s problem.)

In other words, while the speeds and distances are constantly changing, Kepler looks to the “sum” of the distances, or the area swept out, which remains constant for equal intervals of time. [See figure.] Thus, with respect to the relationship of the Sun to the individual planets, the “something” expresses itself in this proportionality. However, Kepler ran into a serious problem here. For while if he knew two positions of the planet, he could determine the time it took the planet to move from one position to the other, by calculating the area swept out, he was unable to do the inverse. That is, calculate the positions of the planet, if given the time elapsed. (In terms of diagram 5.4, for example, Kepler could calculate the area swept out (time elapsed) between position P1 and P2, but he was unable to determine a position P3, such that the time elapsed (area swept out) from P1 to P2 is equal to that of P2 to P3. See Fidelio for a complete review of what has since become known as the “Kepler Problem”.)

This begins to answer the question, “How could the planet know to stay on this eccentric orbit?”, but, it begs the next question, “How does the planet know to stay on this eccentric, and not some other?”

For this Kepler turned to the relationship of the Sun to all the planets. All the eccentric orbits were constituted, Kepler found, so that the square of the periodic time was equal to the cube of the average distance to the Sun. (See chapter 7 of the Fidelio.)

Now we have two characteristics of this “something” else, that determines the planetary orbits. Each orbit, though always changing, has its own constant of area to time elapsed, and all the planets have the same constant of the square of the periodic time and the cube of the distance from the Sun. These combined with the ordering according to the five Platonic solids, and the harmonic intervals of the planet’s extreme velocities, characterizes that “something” by which the planet knows to stay in its non-constant orbit.

And that “something else”, is what Gauss and Riemann would later call a function.