Gauss' Discovery of the Hypergeometric Nature of Physics

by Merv Fansler

1 Introduction

An ever-present question relentlessly confronts the reader of Gauss' Summarische Uebersicht: What was unique, principally, in Gauss' approach to the problem of determining the orbit of Ceres, which advantaged him to succeed where all others had failed? This writing does not purport to offer an explicit answer to this query; rather, here will proceed the unraveling of a particular hypothesis, which arose in the struggle to resolve this question. The least which might be offered at this point, however, is that, by way of this present investigation, one should begin to see why the answer to this question remains so persistently elusive. In addition to this, it is hoped that this report might aid in defining the specific quality of the answer of which we are in search. To these ends, we here take up a subject, which, though never explicitly mentioned by Gauss in his published works on astronomical subjects, first presented itself to him in its full magnificence while he struggled with solving one of the most difficult problems in all astronomy.¹ That subject: hypergeometric series.

2 Post Hoc

The first public mention of hypergeometric series by Gauss appeared in his 1799 doctoral dissertation, Demonstratio nova theorematis.... The first part of this work consisted of a series of penetrating critical analyses and refutations of the previous failed attempts to prove the Fundamental Theorem of Algebra which had been given by some of the idolized contemporary mathematicians. In fact, Gauss' insight into the impostures of EULER, LAGRANGE, and D'ALEMBERT, on this matter, is so incisive that, upon first reading the paper, one is imparted a prescience that there must be some other vantage point from which he is approaching the problem. Once the second half of the paper is reached, this foreknowledge is completely confirmed.

In the second part of the paper, Gauss shares his vantage point with the reader, revealing that his insight was not a matter of a superhuman acumen, but rather was one of method. He, unlike his contemporaries, was willing to seek out, rather than conceal, the physical implications of

. In doing so, he enabled himself to conceive of a higher domain of transcendental functions which was capable of generating all the relationships found in algebraic functions.

[graphic of gauss surfaces ]

Furthermore, from a comparison of a Cartesian representation of an algebraic function, to the one which Gauss discovers, one can begin to see why Gauss had such an `as-if-from-above' view of everything.

[insert graphic of gauss surface -> cartesian]

This first discussion of hypergeometric series appears in his refutation of D'ALEMBERT's alleged proof of the Fundamental Theorem of Algebra. Summarily, D'ALEMBERT puts forth the argument that NEWTON's method for achieving a numerical solution for roots of an algebraic equation using converging infinite series can be applied not only in cases of finding real roots, but also in cases where the roots take on the form a+b

. Gauss points out that D'ALEMBERT ignores the possibility that the infinite series, which might emerge in such cases, could be of the hypergeometric form, in which instance the series would become divergent and absolutely useless.

Furthermore, not only did Gauss draw attention to D'ALEMBERT's lack of consideration of such series, he also takes the opportunity to reprove EULER for similar presumptions. Namely, in a footnote to this section, Gauss reprimands EULER for using hypergeometric series in his calculus textbook² with the assumption that such series would converge.³ By pointing out the fact that the series in question were actually divergent, the validity of EULER's conclusions were consequently discredited.

Yet, typical of Gauss, his aim was never merely to tear down the work of others, nor to promote the authority of his own work, but instead to restitute a method of investigation that would open the field for others to independently explore. Consequently, one finds in his dissertation an abundance of questions, each of which represent a rich pathway awaiting a daring explorer. In this spirit, at the conclusion of his reprimand of EULER, he indicates a door beckoning to be entered:

This has, as far as I know, been noticed by no one until now. Thus it is exceedingly desirable to clearly and rigorously demonstrate why such series, which initially converge very strongly, then ever weaker and weaker, and finally diverge more and more, nevertheless yield nearly the exact sum, only in the event that not too many terms are taken; and in how far such a sum may, with reliability, be taken as correct.
- Gauss, Demonstratio nova theorematis

Gauss would not again publicly mention hypergeometric series for 13 years, yet when he did, he would do so in no perfunctory way. On January 30th, 1812, Gauss presented an essay to the Royal Society of the Sciences at Göttingen in which he unveiled his work on transcendental functions, demonstrating that not only could many simple and even higher transcendental functions be represented by hypergeometric series, but, furthermore, when taking on such a form, new interrelationships amongst seemingly unrelated transcendental functions abundantly present themselves.

Such discoveries, however, were perhaps not the most intriguing aspect of his presentation. A more surprising revelation was pointed out by Gauss: in the methods he had employed in his Theoria Motus, 1809, to develop what he considered to be the most convenient method for solving a problem in orbital determination, he had already been tacitly employing hypergeometric series!

In light of this revelation, the following questions arise: How far along had he developed his methods at the point at which he authored the Theoria Motus? Could his presentation in the Theoria Motus reflect a derivation other than what he had actually carried out? When does he first begin examining the astronomical problem in the way he does-might it go back to his determination of the Ceres orbit? If so, given the nature of the available Ceres observations, could his method of hypergeometric series have provided a critical difference in his determination over the attempts of others? Might there have been something which confronted him in his astronomical investigations, which prompted his development of such a method? Or, had he developed his method in earlier investigations, enabling him to apply it to astronomy?

To commence our search for answers to these questions, let us first familiarize ourselves with the problem that Gauss was confronted with.

3 Kepler Problem, Yet Again

Let us retrace our steps through the problem of the determination of an orbit from three geocentric positions and their corresponding times. Ironically, the first question that a child might ask about the celestial object-How far away is that?-is also the first question that must be answered. What means do we have of measuring such a distance? No matter how high we climb, the object seems not to change its apparent distance from us. No immediate terrestrial metrics available to us seem to size up to the problem. Thus, the initial challenge confronting the inquirer is one of determining what metric is intrinsic to the nature of the action they behold. Fortunately we are not bound by the foot, for Kepler had proffered us his feat.

With Kepler then, as one contemplates an object of the Solar System, one knows with certainty that that object expresses all the harmonic principles which, as Kepler discovered, bound all the parts into a unity. Though we may not know how far away a planet is, we know that that distance is bounded by an entire orbit, wherein that orbit itself is bounded by the harmonic characteristics of a solar system. From Kepler's Astronomia nova, we know that the distance of a planet to the Sun is a quantum of the action of sweeping out equal areas in equal times. It is further known, from Kepler's Harmonice Mundi, that the two quanta, of the Earth's distance to the Sun and a planet's distance to the Sun, are related to each other by the respective major axes or parameters. Consequently, it is only through the knowledge of these harmonic metrics that we are able to ascend to measure the distance of a planet from the Earth.

However great this ascent might be, it is not the only conceptual mountain that must be scaled. Once we obtain distances of the planet to the Earth, there is still much to be accomplished. Working back from a geocentric perspective to a heliocentric one is evident enough as a matter of spherical trigonometry. Once that is achieved, the first two elements, the inclination of the orbital plane to the ecliptic and the longitude of the node, readily present themselves to the resourceful pursuer. After this though, the hasty and the sure-paced alike will be halted in their paths, for once again an age-old problem begs them pay their obeisance and dedicate some solemn moments of their pilgrimage to comprehending the incomprehensible incomprehensibly.

3.1 Gauss' Homage

Gauss provides the reader of his Theoria Motus an introduction to this age-old problem in § 84 as follows:

Since it is possible to determine the whole orbit by two radii vectors given in magnitude and position together with one element of the orbit, the time also in which the heavenly body moves from one radius vector to another, may be determined... Hence, inversely, it is apparent that two radii vectors given in magnitude and position, together with the time in which the heavenly body describes the intermediate space, determine the whole orbit. But this problem, to be considered among the most important in the theory of the motions of the heavenly bodies, is not so easily solved, since the expression of the time in terms of the elements is transcendental, and, moreover, very complicated. It is so much the more worthy of being carefully investigated...

Taking Gauss' advice then, it is this problem that we will henceforth dedicate ourselves to resolve, in hopes that we might attain what Gauss regarded as so worthy of our effort. Before we directly take on the "very complicated" challenge, let us first familiarize ourselves with the problem at hand by demonstrating to ourselves that the first part of Gauss' statement really is possible. That is, let us first take up the problem of determining an orbit beginning from the case where we have already obtained the magnitude of two of the radii vectors together with one of the elements. After having accomplished this, we shall then attempt to invert the process.

To begin, an example will be most suitable for our purposes. Let us take the case directly from Gauss' calculation of the orbit of Ceres, the elements of which were presented by VON ZACH in the December issue of the Monatliche Correspondenz.⁴ For the determination, Gauss used the three observations

from PIAZZI's 1801 data. If we take for the calculation the radii vectors from the outer two observations, then we should arrive at a calculated time elapsed of 40.8949120 days.

Let the radii for the first and third observations be r = 2.7337947, r'= 2.7685475, and the angle between them be q = 8° 49'24".09 or, expressed in parts of the radius, q = 0.1539967. Let us take the semi-parameter as our given element, namely p = 2.7451305.

If one expresses the radius in terms of the true anomaly the following relationship is found:

where e and v denote the eccentricity and true anomaly, respectively.⁵

Separating our known quantities, we will have the two relations:

What can we draw from these relationships? For one, since we know that the eccentricity must be positive, the signs of these cosines, together with the presumption that the observations occur at successive times, give us insight into the qualitative positions of these observations with respect to the perihelion. Namely, since the cosine is positive, and the distance to the Sun is less than the semi-parameter, for the first observation, we can conclude that the observation must have occurred when the planet was nearer to the perihelion than the aphelion. Further, since the next observation, which is only a little more than 8° beyond the first, has a negative cosine, and has a distance from the Sun greater than the semi-parameter, the planet must have gone past the quadrant in the direction of the aphelion. Thus, we already know something qualitative about these positions with respect to the entire orbit.

We still do not quantitatively know the value of either e or the true anomalies, so we must look to see if these relationships formed from our equations might be derived from something we do know. Re-examining the equations,

e cos v = 0.0041465 and e cos v' = -0.0084582,

one may notice that it is possible to arrange these relationships on a circle with radius = e and angles = v, = v'.

If we set e cos v = x and e cos v' = y, then one can prove the relation:⁶

e sin(v¢-v) =

x² +y² -2 x y cos(v¢-v)

Hereafter the eccentricity will be easily calculated, namely

where q = v¢ - v.

Now, knowing the eccentricity, our equations will yield us values of the respective true anomalies for those positions, v = 87° 6' 0".18 and v'= 95° 55'24".28, or v = 1.5201827 and v'= 1.6741794, in parts of the radius. The semi-parameter can also be converted into the semi-major axis, a, since p = a cos² f, where f denotes the angle whose sine is equal to the eccentricity, or cos f = Ö{(1 - e²)}. Thus, a = 2.7636956.

Drawing upon the relationship between the true anomaly to the eccentric anomaly, a cos f sin E = r sin v, we find E = 82° 24' 52".30 and E'= 91° 13'38".48, or E = 1.4384049 and E'= 1.5922177. The corresponding mean anomalies in turn are M = 77° 45' 34".70 and M'= 86° 31'56".82, or M = 1.3571617 and M'= 1.5102762. Hence the mean motion amounts to DM = 8° 46'22".12 or DM = 0.1531145.

We have the relation:

Multiplying this by the number of days in a sidereal year, 365.2563835, we obtain 40.8949302 days elapsed. If we compare this calculated time elapsed to the above given observed time elapsed, it will provide us a means of checking the accuracy of our calculated elements. Above we found that the observed time elapsed was 40.8949120. Thus the difference between our calculated time elapsed and our observed time is 0.0000112 days, or a little less than a second - quite an acceptable degree of accuracy!⁷

3.2 Inversion

That seemed to be not as difficult as one might have expected. Yet, if anyone remembers Chapter 60 of KEPLER's Astronomia nova, though calculating the mean anomaly from the eccentric anomaly presented little difficulty, inverting that calculation proved an insuperable task. The question now: Can we invert the process which was just completed? That is, given the two radii vectors, the angle between them, and the time elapsed, can we derive a value for the semi-parameter, eccentricity, or position of the perihelion?

If we attempted a simple stepwise inversion, what would be our first step? In the opposite direction, the last step we made involved converting the mean motion into a number of days, such that our first step in this direction should be converting our number of days into a mean motion. Yet if we look at the equation,

one is confronted with the problem that the value of the semi-axis major is as yet unknown. Further, knowing the semi-axis major involves knowledge of the parameter and the eccentricity, neither of which we have. Thus, one must try to express the major-axis in terms of our known values.

Supposing that this is possible, we are faced with yet another problem: once we obtain the mean motion, we are wont to proceed to the eccentric anomalies. Here is where the Kepler Problem arises, yet now it takes on an added difficulty. Since we do not know either of the mean positions independently, but only their difference, we must rather take on the more difficult task of finding the difference in the eccentric anomalies.

DM = (E¢ - e sin E') - (E - e sin E)

Now the reader might, appropriately, become wary of taking this path, for even in the original Kepler Problem, of finding a single eccentric anomaly from a single mean anomaly, this could not be solved explicitly, but could only be approximated by an iterative process. Now, instead of merely solving the Kepler Problem for a single time, we must pursue a solution to the problem for two times, both of which are unknown to us, although we know their difference.

Here something quite profound begins to suggest itself as underlying the problem being confronted. For at this moment, the difference between a simple geometric elliptical motion and the efficient principled action of the physical elliptical motion found in the motion of planets presents itself as seemingly irreconcilable. Though the expression of this physical action as a whole may clothe itself in what many consider to be a simple geometric figure, the temporal unfolding of that action wholly transcends any attempted identification of it with its empirical appearance. Ironically, KEPLER was successful in identifying the nature of such action (and even communicating it to his fellow man), as his Astronomia nova attests, yet, fundamentally his comprehension of it could still only be stated in the form of a paradox.

One might object, "But do we not know the laws of planetary motion?" "Have these not been precisely defined?" "How could Kepler claim to discover a 'principle', yet not be capable of expressing it mathematically?" "How can one be considered to know something, if all they really know is that they do not know it?"

For now, we will leave it to the reader to attempt this pathway on their own (though we will be returning to this trail when we directly take up Gauss' approach).

3.3 Another Route

At this point we will make an attempt to circumvent these difficulties by investigating the applicability of an approximation technique. The reader should already have encountered one approximation technique for this problem, which will not be treated here.⁸ Let us introduce another approach for pedagogical purposes.

After KEPLER had devastatingly demonstrated the futility of adhering to mere geometric modeling for comprehending the motions of the planets in his Astronomia nova, he proceeded to introduce a new mode of investigation to astronomy: hypothesizing physical causality. With this concept of the knowability of physical causality, he would arrive at a new hypothesis that would transform all astronomical investigations thereafter. Yet he admitted, even as early as his Mysterium cosmographicum, that the singular hypothesis which he arrived at, that the planetary motions locate their source and ordering in the Sun, was not entirely new, but rather was a reincarnation of Pythagorean knowledge.

KEPLER first begins to elaborate upon this hypothesis for the reader of his Astronomia nova in the thirty-second chapter. In order to establish the existence of such a causal relation, he investigates the interaction of a planet's angular motion, both about the center of the orbit and about the Sun, with the planet's distance to the Sun. By means of this investigation he is able to demonstrate that the effect of the planet's speeding up and slowing down along different parts of its orbit, which previously had only been treated geometrically, could be the result of a physical principle. Yet at that point, he still did not establish what figure the motion resultingly traversed, but rather only how the motion changed.

These differential relations, which he established in Chapter 32, specified that the angle about the center changes inversely proportional to the distance from the Sun, and, consequently, that the angle about the Sun changes in an inverse proportion to the square of the distance from the Sun. In the subsequent chapters of the book (33-38), KEPLER commences the search for what must be the nature of the planet and Sun, in order that such relations could exist. After this, much of the remainder of the Astronomia nova is directed toward seeking out what motion might satisfy such differential relationships-that is, what figure these differential relations generate.⁹

Although KEPLER's results had been extracted from his Astronomia nova and reduced to the form of "Laws" (i.e., elliptical motion and equal area, equal time), these differential relations were still underlying the analytic representations that had become prevalent by Gauss' day.¹⁰ Namely, it can be shown that the second of these relations, that between the angle about the Sun with the distance from the Sun, finds itself expressed as