by Bruce Director

Before the excitement from his stunning determination of the orbit of Ceres had begun to diminish, Gauss found, in the tiny variations of the orbit of the newly discovered asteroids, a further means for extending Kepler's discovery of the non-linear harmonic ordering of the solar system. As those who have worked through Gauss' method know, the re-discovery of Ceres, in December 1801, accomplished much more than the opportunity to renew observations of Piazzi's newly found object. Gauss' accomplishment represented a triumph of Kepler's successful application of Plato's and Cusa's method. Gauss had demonstrated anew what Kepler himself had shown; that the harmonic ordering principles (Kepler's so-called three laws), existed and could be discovered in every small interval of action in the solar system.

There was another aspect of Kepler's principles that did not come directly into play in Gauss' initial determination of the Ceres orbit. In his {Harmonies of World}, Kepler investigated the interaction among the planets themselves, expressed in terms of musical intervals. These intervals were formed by the angular velocity of the planets at their extreme distances from the Sun. It was in these relationships, that Kepler, in fact, had forecast the existence of an exploded planet in the same region of space in which Gauss had determined Ceres to be. Significantly, Kepler noted a small discrepancy, between the calculated values of these angular velocity, and the musical intervals to which they corresponded.

It had since been observed, with the finer measurements of Gauss and his colleagues, that there were small deviations in the observed positions of the planets from purely elliptical orbits, known as perturbations, due to the interaction among the planets themselves.

In the {Theoria Motus}, Gauss says, "The perturbations which the motions of planets suffer from the influence of other planets, are so small and so slow, that they only become sensible after a long interval of time; within a shorter time, or even within one or several entire revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it would not be worthwhile to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been accurately observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion."

As Gauss indicates, this is another self-reflexive problem: "Since the determination of the elliptic elements with which, in order that the observations may be exactly represented, the perturbations are to be combined, supposes a knowledge of the latter; so, inversely, the theory of the perturbations cannot be accurately settled unless the elements are already very nearly known; ..."

The observable effect of these perturbations take two forms. One is the short term perturbations, small deviations from a purely elliptical path, the other is the so-called secular perturbations, that have the effect of changing the size, shape, and position of the orbit itself. In other words, while the planets are moving about the Sun in Keplerian orbits, the orbits themselves are changing positions with respect to each other.

Because these perturbations are so small, they are very hard to measure in the larger planets known before the discovery of Ceres. But, in the much smaller Ceres, and the other newly-discovered asteroid, Pallas, these perturbations were more easily measured. For this reason, Gauss jumped on the opportunity to investigate these tiny changes. By June 1802, Gauss already wrote to his friend Olbers concerning his ideas about calculating the perturbations of Ceres and Pallas. His aim was not simply to better trace the orbital motion of these asteroids, but to investigate a whole new set of cycles that were present in the orbits of every planet. The significance of these perturbation cycles are similar to Pythagoras' commas or Kepler's small discrepancy in the musical intervals. Gauss found in these investigations, the means to elaborate a new mathematical metaphor of the arithmetic-geometric mean and hypergeometric functions.

Just as Gauss' determination of the Ceres orbit itself was a devastating refutation of the fraud of Newton, Euler, and Laplace, the investigation of the perturbation cycles nails their coffins. Kepler had demonstrated that it was the anti-entropic ordering principle of the solar system as a whole, that governed the motion of the individual planets. The oligarchy's attack focussed on imposing the Newton/Sarpi formalism of the pair-wise inverse square law, that in effect, reduced the harmonic motion of the planets, into a fixed Aristotelian order. A fraud from the beginning that was only maintained through pure political thuggery. Any attempt to calculate the interaction of two planets and the Sun using Newton's inverse square law, becomes a mathematical impossibility known as the three-body problem. Now, if the interactions of the planets among themselves could no longer be ignored, the whole edifice of Newton's pair-wise formalism falls apart.

As Kepler showed, the planets are not interacting in a pair-wise manner, but rather comprise a nonlinear harmonic process. However, these harmonic relations are not limited to the motion of the planets around the Sun. To grasp the interaction of the moving planets on each other, while they move about the Sun in Keplerian orbits, requires the mind to be able to think in terms of cycles moving cycles, or modular functions.

One may feel a tiresome tug in response to this thought. When do these cycles ever end? Is there not one cycle of cycles, by which all these motions can be brought into a One? Each new discovery, seems to leave something incomplete. None of the cycles ever close.

To get in the right frame of mind to approach these questions, it is appropriate to look to a passage from Friedrich Schiller's Aesthetical Lectures:

"I know of no more suitable image for the ideal of beautiful behavior, than a well-performed English dance, composed from many complicated figures. A spectator from the gallery sees innumerable movements, which cross one another most vividly and alter their direction briskly and playfully, and yet never knock into one another. Everything is so ordered, that the one has already made room, when the other arrives; everyone fits so skillfully and yet again so artlessly into one another, that each seems to follow only his own head and yet never steps in the way of the other. It is the most suitable emblem of the asserted self-freedom and the spared freedom of the other."

To dance with the planets, one must think of a dance in which the steps of the dancers alter the curvature of the dance floor, which in turn changes the motions of the dancers.

Gauss developed a beautiful approach to this dance, that we'll examine next week.

by Bruce Director

The motion of the newly-discovered asteroids, Ceres and Pallas, presented to Gauss a renewed opportunity to advance the Platonic method of astrophysical investigation that Kepler had presented to the world nearly 200 years earlier. In the intervening 200 years, the oligarchy, through their Leporellos, Euler and Newton, had perpetrated a hoax, to undo Kepler's accomplishment, not by locking up what Kepler wrote, but by locking up the minds of those who would read it, and not wish to be on the wrong side of popular opinion. Gauss, with his determination of the orbit of Ceres, wielded the power of the universe against those oligarchical mind games, and, like a good commanding general, pursued the enemy, with the investigation into the tiny perturbations of the Ceres and Pallas orbits.

This oligarchical mind control exists to the present day. Last month, Roger Ham forwarded a copy of the Summer Fidelio to one Brian Marsden who specializes in the determination of orbits at the Smithsonian Astrophysical Observatory of Harvard University. Within 3 days time, the professor replied, "I fully agree that Gauss' method of orbit determination has great elegance and power, and we certainly make use of it in our own work, including what we did on 1997 XF11. Nevertheless, that is not the whole story, and it was quite improper of the Fidelio article to criticize Newton and Euler in the way it did...."

Another professor, MIT's Laurence Taff, displays a similar ignorance in his 1985, text "Celestial Mechanics--A Computational Guide for the Practitioner." In the section on the determination of planetary orbits, Taff quotes extensively from Gauss' Theoria Motus. Then, after a brief description, albeit in formal terms, of Gauss measurement of non-linear curvature in the small, Taff states, "Although this picture of the Gaussian process is interesting, it is also irrelevant." Under the heading, "The Historical Myth--Ceres," Taff says, "We do not know how Gauss actually computed the orbit of Ceres or any of the other big four minor planets. We do know that however he did it."

Both men trumpet the superiority of the statistical methods, rejected by Gauss, because those methods are more practical, especially with the use of high speed computers. Their obsession, however, is not driven by the quality of their results, but by the emptiness of their minds. To them, there are those methods of elegance and power, and then there are the not "improper" practical ones.

In this light, let's continue to look at Gauss' investigation of the perturbations of the orbit of Ceres and Pallas, from Gauss' standpoint.

As Kepler, following Plato and Cusa, demonstrated, the solar system is a harmonically-organized process. For the mind to grasp that harmony, one must be able to "hear" the polyphony of interactions among all the heavenly bodies. Kepler presented that polyphony to us, in the form of his three principles of planetary motion, and the musical relationships of the planet's extreme angular velocities.

If, however, when you think of Kepler's solar system, you think of each planet, separately moving about the Sun, you are already on the slippery slope into the Sarpi/Newton abyss. It is but a short step, in this mode of thinking, to translating Kepler's principles, by algebraic manipulation, into a seemingly more practical system of formal equations such as Newton's inverse square law. Will you fall into the trap, that Kepler's principles are of "elegance and power," but Newton's laws say the same thing in a way more suitable for calculation as, Cal Tech's Richard Feynman, who, in his famous "lost lecture," committed the sophistry of pretending to prove "geometrically" that the planets obey Kepler's principles as a consequence of the inverse square law?

Nothing strikes terror into the mind of the formalist, more than the elegance and power of the Socratic method, just as Gauss and Kepler to this day strike terror in the minds of Marsden, Taff, and Feynman. That terror grows stronger when confronted with the physical evidence that the planets don't obey the formalist's rules. If Kepler's and Gauss' work on the larger motions of the planetary orbits were not enough, the whole Newtonian edifice, and the mental states of its adherents is completely thrown into disarray, when the tiny perturbations of the planetary orbits are considered. As Taff admits in the above mentioned text, "In my view, the problem is that when you really need perturbation theory (in celestial mechanics) it does not work very well (as applied in the past). The existing applications of it do not explain the Kirkwood gaps in the asteroid belt. They do not explain the fascinating discoveries in the Saturnian rings/moon system. They do not predict physically correct rates of change of the orbital elements of a satellite revolving about an oblate primary. Some of these failures may be attributable to faulty mathematics, some to the difference between a trajectory and a field theory. The reader is advised that the above is a minority opinion (and taken slightly out of context)."

Taff's uneasiness, once again, comes from his obsessive devotion that has lead him, like the Harvard professor, to turn away from methods of power and elegance, for the more practical ones.

The problem is straight forward. All the heavenly bodies that move about the Sun are moving in Keplerian orbits, and, these orbits, as Kepler demonstrated in the Harmonies, are interacting with each other. The effect of these inter-orbital interactions, is measurable in terms of small variations in the orbital elements that characterize the size, shape, inclination, and position, of the orbits. In other words, the planetary orbits are not fixed tracks in the sky, but regions of motion, in which the planets move, similar to the regions of ambiguity that characterize the tones of the well-tempered system of polyphony. The curves traced out by these motions, therefore, are not the simple curves you draw on a piece of paper, but much more complicated. (The reader can make a representation of this, by drawing a circle with a compass and slowly making the radius slightly larger and smaller as you rotate the compass around the center.) This representation is merely the footprint of the planet's motion. Now think of how this type of motion would be reflected in changes in the orbital elements.

The measurable variations of these orbits occur in very long cycles, but, these cycles are very significant. For example, the precession of the equinoxes is one effect of these perturbations. A longer 100,000 year cycle in the eccentricity of the Earth's orbit, resulting from these orbital interactions, accounts for the cycle of ice ages.

The scientific question posed, is, what is the harmonic relationships that are reflected in these inter-orbital interactions? In the larger planets, as Gauss says, these cycles are so small and slow as to be nearly insensible, over short intervals of time. But, in the orbits of the smaller planets like Ceres and Pallas, these cycles are measurable during shorter time intervals, and in them, the underlying harmonic ordering principle of all the inter-orbital interactions are potentially discoverable.

How do we measure the interactions between orbits?

In an 1819 paper, "The Determination of the Attraction of an Elliptical Ring," Gauss discusses the following principle of physical astronomy:

Changes in the elements of a planet's orbit, that are due to the interaction with another planet, are independent of the position of the disturbing planet. That is, the long range changes in the size, shape, inclination and position of a planet's orbit, that arise from the interaction with another planet, arise from the interaction of the two orbits. Furthermore, Gauss says, that if the total time it takes for both planets to make one complete revolution around the Sun, are incommensurable, then the effect of the disturbing planet on the elements of the disturbed planet, can be calculated using the following metaphor:

Think of the mass of the disturbing planet as spread out in a ring of uniform, but infinitely small thickness, such that the mass is distributed in this ring, proportional to the speed of the planet. That is, since the planet spends more time in the parts of the orbit farther from the Sun, and less time in the parts of the orbit closer to the Sun, and, according to Kepler's area principle, equal times sweep out equal areas, any piece of the elliptical ring, in which the planet spends equal times, will have equal amounts of mass. Or, in other words, the mass of the planet should be thought of spread out non-uniformly, in proportion to, the non-uniform motion of the planet.

The problem Gauss then solved, is how to determine the effect of this elliptical ring, on any other position in space. This lead Gauss to a surprising application of the arithmetic- geometric mean and a deeper insight into modular functions.

For now, however, it were useful to turn again to Friedrich Schiller, whose poem, "The Dance" presents to us the appropriate conceptions for the task.

See how with hovering steps the couple in wavelike motion
Rotates, the foot as with wings hardly is touching the floor.
See I shadows in flight, set free from the weight of the body?
Elves in the moonlight there weaving their vapor-like dance?
As by zephyr 'twere rocked, the nimble smoke in the air flows,
As so gently the skiff pitches on silvery tide,
Hops the intelligent foot to melodic wave of the measure,
Sweet sighing tone of the strings lifts the ethereal limbs.
Now, as would they with might traverse through the chain of the dances,
Swings there a valorous pair right through the thickest of ranks.
Quickly before them rises the path, which vanishes after,
As if a magical hand opens and closes the way.
See! Now vanished from view, in turbulent whirl of confusion
Plunge the elegant form of this permutable world.
No, it hovers rejoicing above, the knot disentangles,
Only with e'er-changing charm rule does establish itself.
Ever destroyed, creation rotating begets itself ever,
And an unspoken law guides the transformative play.
Say, how's it done, that restless renews the swaying formations
And that calmness endures even in moveable form?
Is each a ruler, free, to his inner heart only responding
And in hastening course finds his own singular path?
Wish you to know it? It is the mighty Godhead euphonic
Who into sociable dance settles the frolicking leap,
Who, like Nemesis fair, on the golden rein of the rhythm
Guides the raging desire and the uncivilized tames.
And do the cosmos' harmonies rustle you to no purpose,
Are you not touched by the stream of this exalted refrain,
Not by the spirited pulse, that beats to you from all existence,
Not by the whirl of the dance, which through eternal expanse
Swings illustrious suns in boldly spiraling pathways?
That which you honor in play--measure--in business you flee.

Translation by Marianna Wertz

by Bruce Director

When Kepler discovered the ordering principle of planetary orbits -- that the non-uniform motion of a planet can be measured by the area swept out by a line connecting the planet and the sun, such that those areas are equal for equal intervals of time elapsed -- he immediately confronted the problem of mathematical physics identified by Nicholas of Cusa in "On Learned Ignorance."

If you look on p 28 of the Fidelio article on the determination of the orbit of Ceres, you can see in Fig. 5.5 that the area swept out by the planet's motion from P1 to P2 (the curvilinear area P1-A-P2 shaded in white) is measured by the length of the line P1-N, which is the sine of the angle P1-B-P2. As the quote from Kepler on that page indicates, the ratio between the arc and the sine are infinite.

Kepler called for the discovery of the calculus to solve this problem. Leibniz made the initial discovery, and Gauss, extending the discoveries of Leibniz and their Greek predecessors, took it to a new dimension with his investigation of the hypergeometric function and the arithmetic-geometric mean.

Look again at the problem in such a way as to bring out the underlying characteristic of action. Referring again to Fig. 5.5 on p 28, think of the type of change of the right triangle P2-N-B, as the planet moves counter-clockwise around the circle, from P1. The hypotenuse of the triangle, P2-B remains constant, but the lengths of the legs change, with each change in the angle, P1-B-P2. If we consider the hypotenuse to be 1, then for any given angle, we can calculate the lengths B-N, (cosine) and the P2-N (sine), by the methods Euclid used to divide the circle.

For example, dividing the circle into 4 parts, would create angles for P1-B-P2 of 0, 90, 180 and 360 degrees. The magnitudes of the lengths of B-N (cosine) would be 1,0,1,0 respectively and the magnitudes of the lengths P2-N (sines) would be 0,1,0,1 respectively. (I have left out considerations of direction for the moment.)

Now, dividing the circle again into four more parts, adds the angles, 45, 135, 225, and 315 to the previous four angles. Now, to calculate the corresponding lengths for B-N (cosine) and P2-N (sine) we can use the Pythagorean theorem. Since by definition the angle P1-B-P2 is 45 and the angle P1-N-B is 90 degrees, the remaining angle in triangle P2-B-N is also 45 degrees making that triangle isosceles. Since the hypotenuse P2-B is 1, the lengths of the sides, will each be equal to 1/2 times the square root of 2.

To continue this process, we are at first limited to those divisions that can be accomplished by circular action itself, the so-called straight edge and compass constructions. From Euclid's time until Gauss, those divisions were limited to 3, 4, 5 and certain combinations and multiples of those divisions. Greek geometers, such as Hippias, discovered that using other curves, such as the quadratrix, the circle could be divided in other ways. In 1794, Gauss discovered a heretofore undiscovered ordering principle of circular action, by which the circle could be divided into 17, 257, and other parts.

Furthermore, the sine and cosine of 45 degrees, 1/2 the square root of 2, while constructable with straight edge and compass, is nevertheless and irrational magnitude. For the sines and cosines of other angles, non-constructable irrational magnitudes arise.

Something else can now be brought into view; the underlying non-linearity of the transcendental relationship between the circular arc and the straight line. When we divided the circle into 8 parts, the angles were divided equally. That is, 45 degrees is exactly half of 90 degrees. But, when we calculated the lengths of the sine and cosine, they did not change by 1/2. In fact, the sine (length P2-N) more than doubled, from 0 to 1/2 times the square root of 2, and the cosine (length B-N) decreased by less than half, from 1 to 1/2 times the square root of 2. If we were to calculate the sine and cosine for other angles, we would see that the sine and the cosine change non-uniformly as the angle changes uniformly.

This should surprise you. The circle is a curve of constant curvature, yet the transcendental relationship between the arc and the straight lines is non-constant! This is nub of the problem that Kepler confronted. To solve it, one would have to get at the underlying non-linear relationship between arcs and straight lines.

This is a classic example of inversion. If we change the angle by a given interval, the sine and cosine change by a different interval. As in the above example, if the angle changes from 0 to 45 degrees, the sine changes from 0 to 1/2 times the square root of 2. But, if the angle changes from 45 to 90 degres, a uniform change in angle, the sine changes by a smaller interval.

So in this direction we ask, "Given a change in angle, what is the resulting sine and cosine?" But the inverse question, "Given a change in sine, how much does the corresponding angle change?" The determination of the first question is difficult, the second, (because the sines are incommensurable magnitudes) seems impossible, unless, we can discover some unseen relationship that governs this non-linear process.

Now, in the case of the elliptical orbits, the problem seems even more complicated, because the ellipse is a curve of constantly changing curvature. In an ellipse, not only do the legs of the triangle P2-N and N-B change non-linearly, but also the hypotenuse P2-B. (For this we refer to figure 6.4 on page 33 of the Fidelio.) Using the principles of conical proportions discovered by Appolonius of Perga (A contemporary and associate of Eratosthenes, Aristarchus and Archimedes), Kepler avoids the problem of calculating the elliptical sector by calculating the proportional circular one. (Q-A-N in figure 6.4 This is done by the same method as illustrated on page 28.)

In the determination of the attraction of the elliptical ring, Gauss, instead of trying to solve the difficult problem, took on the seemingly impossible one, finding, at least to a certain degree, the unseen relationship that governs the non- linearity between the angle and the sine and cosine.

For this, Gauss began with a sort of inversion of Kepler's eccentric anomaly. If we think of the ellipse in figure 6.4 as the elliptical ring of the disturbing planet, the disturbed planet would lie along line B-P2 (not drawn in the figure). In other words, instead of thinking of P2 as the position of the planet in motion, we should think of a planet moving inside the ring. P2's position is the "projection" of the motion of the disturbed planet on the elliptical ring. The position of P2, therefore changes with the motion of the moving planet. Then, Gauss "projected" P2's position to Q. This enables Gauss to measure the non-uniformly changing length B-P2, as a function of angle E (the eccentric anomaly.)

(What is a work here, is the method of interconnected colligating cycles, in which we are interested in the way changes in one cycle are conncect to changes in other cycles. It is the type of changes we are investigating.)

Now applying the principle of inversion, Gauss seeks to determine the characteristic of change between the straight line B-P2 and the angle E. Instead of trying to calculate the actual change in length of B-P2 with respect to any particular change of angle E, Gauss seeks to determine the characteristic form of the change, as E sweeps around one entire revolution.

Beginning with Appolonius' properties of the ellipse, Gauss shows that the relationship of B-P2 to angle E, is a function of the semi-axes of the ellipse. Specifically, if the semi-major axes of the ellipse is A and the semi-minor axis is B, then P2's position is determined by A times B-N (cosine of E) and B times Q-N (sine of E). Another way to think of this, is that the ellipse in figure 6.4 is a contraction of the circle, and that P2's position is formed by contracting length Q-N by a factor of B/A. (For a discussion of the basic relationships of the ellipse see the Appendix to the Fidelio article.)

Then Gauss determines that the characteristic change of B-P2 with respect to E, is invariant under the following transformation:

If A is the semi-major axis and B is the semi-minor axis, of the ellipse, calculate the arithmetic mean (A+B/2) of these magnitudes, and call that A'. Then calculate the geometric mean (the square root of AxB) call that B'. Then construct a new ellipse with semi-major axis A' and semi-minor axis B'. Then repeat this process. This construction, leads to a discontinuous sequence of ellipses, the semi-axes of each ellipse, are the arithmetic and geometric means of the previous one. This series converges rapidly on a circle, whose radius is the arithmetic- geometric mean.

(A graphic representation of this process can be seen on page 51 and 52 of "So, You Wish to Learn All About Economics?")

What Gauss demonstrated, was, that the characteristic change of the inverse of the length of the line B-P2 with respect to angle E was the same for each ellipse in this series. And, if calculated for an entire circuit around the ellipse, is equal to 1/arithmetic-geometric mean of the semi-axes of that ellipse.

Once again, Gauss has presented us with a paradoxical solution, for the arithmetic-geometric mean, is itself a kind of transcendental and so we are measuring one transcendental by another. Gauss went on to discover that these processes are related as part of a whole group of transcendental processes subsumed under his conception of hyper-geometric functions.

In the announcement of his study on the hyper-geometric series, Gauss said, "The logarithmic and circular functions, as the simplest manner of transcendental functions, are those, with which analysis has been mostly occupied. They earn this nobility because of their continuous hold on almost all mathematical investigations, theoretical and practical, and because of their almost inexhaustible richness of interesting truths that their theory presents..." Gauss goes on to discuss the other higher forms of transcendental functions saying, "The transcendental functions have their source at all times, often reclining or hiding, in the infinite ..." These hidden sources, Gauss explores in the domain of hyper-geometry.