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- Gauss' Determination of the Orbit of Ceres, Introduction, by
Bruce
Director,
*The American Almanac*, December 15, 1997.

- Gauss' Determination of the Orbit of Ceres, Part 1, by
Jonathan Tennenbaum,
*The American Almanac*, December 22, 1997.

- Gauss' Determination of the Orbit of Ceres, Part 2, by
Jonathan Tennenbaum

Part 1 -- Introduction.

January 1, 1801, the first day of a new century. In the early morning hours of that day, Guisseppi Piazzi, peering through his telescope in Palermo, discovered an object in the sky which appeared as a small dot of light in the dark night sky. He noted its position as a point on the inside side of an imaginary sphere with him at the center. On a subsequent night, he saw the same small dot of light, but this time it was in a slightly different position on the inside of the imaginary sphere. He had not seen this object before, nor was there any recorded observations of it. Over the next several days, Piazzi watched this new object, carefully noting its change in position from night to night, recording its position as the intersection of two circles on the imaginary sphere. One set of circles was thought of as ascending from the horizon overhead and then descending. The other set of circles was parallel to the horizon. Each position could be recorded, therefore, by two angular measurements, one in each direction. In this way, the observations then known only to Piazzi could be communicated to others. On each clear night over the next 42 days, Piazzi noted the positions of this new object on the imaginary grid on the celestial sphere. Each night, the object appeared at a slightly different position. Each night, the new object appeared later and later in the evening, until on February 11, 1801, the object did not appear until after the sun had risen. What had Piazzi discovered? Was it a planet, a star, a comet, or something else which didn't have a name? (At first, Piazzi thought he had discovered a small comet with no tail. Later, he and others speculated it was a planet between Mars and Jupiter.) And now that it had disappeared, what was its trajectory? When and where could it be seen again? If it were orbiting the sun, how could it's trajectory be determined from these few observations made from the earth, which itself was moving around the sun? Had Piazzi observed the object while it was approaching the sun, or was it moving away from the sun? Was it moving away from the earth or towards it, when these observations were made? Since all the observations appeared as changes in positions on the inside of the imaginary celestial sphere, what motion were these changes in position a reflection of? What would these changes in position be if Piazzi had observed them from the sun? Or, a point outside the solar system itself? A God's-eye view? It was six months before Piazzi's observations were published in the main German-language journal of astronomy, von Zach's Monatliche Corespondenz, but news of his discovery had already spread to the leading astronomers of Europe, who searched the sky in vain for the object. Unless an accurate determination of the object's trajectory was made, re-discovery was uncertain. There was no direct precedent that could be drawn on to solve this crisis. The only previous experience anyone had with determining the trajectory of a new object was the 1781 discovery of the planet Uranus by William Herschel. In that case, astronomers were able to observe Uranus' position on many different nightse, recording numerous changes of position of the planet with respect to the Earth. With these observations, the mathematicians simply asked, "On what curve is this planet travelling, that would produce these observations?" If one curve didn't produce the desired mathematical result, another was tried. As Gauss described it in the Preface to his Theoria Motus, "As soon as it was ascertained that the motion of the new planet, discovered in 1781, could not be reconciled with the parabolic hypothesis, astronomers undertook to adapt a circular orbit to it, which is a matter of simple and very easy calculation. By a happy accident, the orbit of this planet had but a small eccentricity, in consequence of which the elements resulting from the circular hypothesis sufficed at least for an approximation on which could be based the determination of the elliptic elements. There was a concurrence of several other very favorable circumstances. For, the slow motion of the planet, and the very small inclination of the orbit to the plane of the ecliptic, not only rendered the calculations much more simple, and allowed the use of special methods not suited to other cases; but they removed the apprehension, lest the planet, lost in the rays of the sun, should subsequently elude the search of observers, (an apprehension which some astronomers might have felt, especially if its light had been less brilliant); so that the more accurate determination of the orbit might be safely deferred, until a selection could be made from observations more frequent and more remote, such seemed best fitted for the end in view." The false belief that a large number of observations, as far apart from one another as possible, was required to determine the orbit of a heavenly body, is a fault of the linearization in the small mathematics of Euler-Newton-Sarpi, not a limitation of nature, or the human mind. If the universe becomes more linear in the small, the closer your observations are to one another, the more indeterminate their relationship. This delusion can be maintained, in this case, only if the problem of determining the orbit of an unknown planet is treated as a purely mathematical one. For example, think of three dots on a plane. On how many different curves could these dots lie? Now add more dots. The more dots, covering a greater part of the curve, the more precise determination of the curve. A small change of the position of the dots, can mean a great change in the shape of the curve. The less dots and the closer together they are, the less precise is the mathematical determination of the curve. If this false mathematics is imposed on the universe, a great number of observations are required to determine an orbit of a planet. But the changes of observed positions of an object in the night sky, are not dots on a piece of paper. These changes in position are a reflection of physical action, which is self-similar in every interval of that action, as Cusa, Kepler, and Leibniz knew. Consequently, every small interval of action will reflect the larger process. Thus, the smaller the interval of action investigated, the more accurate the determination of the orbit. This distinction will become more clear, as, over the next several weeks, we work through Gauss' determination of the orbit of Ceres. It was only an accident, that the problem of the determination of the orbit of Uranus, could be solved without challenging the falsehood of linearization in the small. But such accidental success of a wrong method, was shattered by the problem presented by Piazzi's discovery. The universe was demonstrating Euler was a fool. (Years later, Gauss would calculate in one hour, the trajectory of a comet which had taken Euler three days, a labor in which Euler lost the sight of one eye. "I would probably have become blind also, if I had been willing to keep on calculating in this manner for three days," Gauss said of Euler.) It was September of 1801 before Piazzi's observations reached the 24-year old Gauss, but Gauss had already anticipated the problem, and ridiculed other mathematicians for not considering it, "since it assuredly commended itself to mathematicians by its difficulty and elegance, even if its great utility in practice were not apparent." Because others assumed this problem was unsolvable, and were deluded by the accidental success of the wrong method, they refused to believe that circumstances would arise necessitating its solution. Gauss, on the other hand, considered the solution, before the necessity presented itself, knowing, based on his study of Kepler and Leibniz, that such a necessity would certainly arise. Before working through the crucial conceptions at issue in Gauss' determination of the orbit of Ceres, we suggest the reader perform the following constructions to familiarize yourself with some of the basic geometric relationships of conic sections. Take a piece of wax paper and on it draw a circle. (The best way to do this is by tracing the edge of plate with a marking pen.) Then put a dot at the center of the circle. Now fold the circumference onto the point at the center and make a crease. Unfold the paper and make a new fold, bringing another point on the circumference to the point at the center. Make another crease. Repeat this process around the entire circumference (approximately 25 times). At the end of this process, you will see a circle enveloped by the creases in the wax paper. Now take another piece of wax paper and do the same thing, but this time put the point a little away from the center. At the end of this process, the creases will envelope an ellipse, with the dot being one focus. Repeat this construction several times, each time moving the point a little farther away from the center of the circle. Then try it with the point outside the circle. Then make the same construction, using a line and a point. In this way, you can construct all the conic sections as envelopes of lines. Now think of the different curvatures involved in each conic section, and the relationship of that curvature to the position of the dot (focus). To see this more clearly, do the following. In each of the constructions, draw a straight line from the focus to the curve. How does this the length of this line change, as it rotates around the focus? How is this change different in each curve?

Last week, we journeyed back to the turbulent year of 1801, to share in the excitement of the great challenge which Piazzi's observation of an unknown planet placed in front of the scientists of his time, and which only the 24-year-old Carl Friedrich Gauss was able to meet. What did Gauss do, which other astronomers and mathematicians of his time did not, and which led those others to make widely erroneous forecasts on the path of the new planet? Perhaps we shall have to consult Gauss' great teacher, Johannes Kepler, to give us some clues to this mystery. Gauss first of all adopted Kepler's crucial hypothesis, that the {motion of a celestial object is determined solely by its orbit}, according to the intelligible principles demonstrated by Kepler to govern all known motions in the solar system. In the Keplerian determination of orbital motion, no information is required concerning mass, velocity or any other details of the orbiting object itself. Moreover, as Gauss demonstrated, and we shall rediscover for ourselves, the orbit and the orbital motion in its totality, can be adduced from nothing more than the internal "curvature" of any portion of the orbit, however small. Think this over carefully. Here the science of Kepler, Gauss, and Riemann dinstinguishes itself {absolutely} from that of Galileo, Newton, Laplace etc. Orbits and changes of orbit (which in turn are subsumed by higher-order orbits) are {ontological primary}. The relation of the Keplerian orbit, as a relatively "timeless" existence, to the array of successive positions of the orbiting body, is like that of an hypothesis to its array of theorems. In truth, we can say it is the orbit which "moves" the planet, not the planet which creates the orbit by its motion! If we interfere with the motion of an orbiting object, then we are doing work against the orbit as a whole. The result is to change the orbit; and this, in turn, causes the change in the visible motion of the object, which we ascribe to our efforts. That, and not the bestial "pushing and pulling" of Sarpian-Newtonian physics, is the way our universe works. Any competent astronaut, in order to successfully pilot a rendezvous in space, must have a sensuous grasp of these matters. Gauss' entire method rests upon it. Gauss adopted an additional, secondary hypothesis, likewise derived from Kepler: At least to a {very high degree of precision}, the orbit of any object which does not pass extremely close to some other body in our solar system (moons are excluded, for example), has the form of a simple conic section (a circle, ellipse, parabola or hyperbola) with focal point at the center of the Sun. Under such conditions, the motion of the celestial object is {entirely determined} by a set of five parameters, which specify the form and position of the orbit in space, and which became known among astronomers as the "elements of the orbit." Once the "elements" of an orbit are specified, and {for as long as the object remains in the specified orbit}, its motion is entirely determined {for all past, present and future times}! Gauss demonstrated, in fact, how the "elements" of any orbit, and thereby the orbital motion itself in its totality, can be adduced from nothing more than the curvature of any "arbitrarily small" portion of the orbit; and how the latter can in turn be be adduced -- in an eminently practical way -- from the "intervals" defined by only three good, closely-spaced observations of apparent positions as seen from the Earth! The "elements" of a simple Keplerian orbit consist of the following: 1) Two parameters, determining the position of the {plane} of the object's orbit relative to the Earth's orbit (the so-called ecliptic). Since the Sun is the common focal point of both orbits, the two orbital planes intersect in a line, called the {line of nodes}. The relative position of the two planes is uniquely determined, once we prescribe their angle of inclination to each other (i.e. the angle between the planes) and the angle made by the line of nodes with the major axis of the Earth's orbit. 2) Two parameters, specifying the {shape} and {overall scale} of the object's Keplerian orbit. It is not necessary to go into this in detail now, but the chiefly-employed parameters are: (i) the relative scale of the orbit as specified (for example) by its width when cut perpendicular to its major axis through the focus (i.e. center of the Sun); (ii) a parameter of shape known as the "eccentricity", which we shall examine later, but whose value is 0 for circular orbits, between 0 and 1 for elliptical orbits, exactly 1 for parabolic orbits and greater than 1 for hyperbolic orbits. Instead of the eccentricity, one can also use the perihelial distance, i.e., the shortest distance from the orbit to the center of the Sun, or its ratio to the width parameter; (iii) one parameter specifying the relative "tilt" of the main axis of the object's orbit within its orbital plane. For this purpose, we can take the angle between the major axis of the object's orbit and the line of nodes. As I said, the entire motion of the orbiting body is determined by these elements of the orbit alone. If you have mastered Kepler's principles, you can compute the object's precise position at any future or past time. All that you must know, in addition to the five parameters just described, is a single time when the planet was (or will be) in some particular locus in the orbit, such as the perihelial position. (Sometimes astronomers include the time of last perihelion-crossing among the "elements.") Now, let us go back to Fall 1801, as Gauss pondered over the problem, how to determine the orbit of the unknown object observed by Piazzi, from nothing but a handful of observations made in the weeks before it disappeared in the morning glare of the Sun. The first point to realize, of course, is that the tiny arc of a few degrees, which Piazzi's object appeared to describe against the background of the stars, was not the real path of the object in space. Rather, the positions recorded by Piazzi were the result of a rather complicated combination of motions. Indeed, the observed motion of any celestial object, as seen from the Earth, is compounded {chiefly} from the following three processes or degrees of action: 1. The rotation of the Earth on its axis (uniform rotation, period one day). 2. The motion of the Earth in its known Keplerian orbit around the Sun (nonuniform motion, period one year). 3. The motion of the planet in the unknown Keplerian orbit (nonuniform motion, period unknown, or nonexistent in case of a parabolic or hyperbolic orbit). Thus, when we observe the planet, what we see is a kind of "blend" of all of these motions, mixed or "multiplied" together in a complex manner. Within any interval of time, however short, all three degrees of action are operating {together} to produce the apparent positions of the object. As it turns out, there is no simple way to "separate out" the three degrees of motion from the observations, because (as we shall see) the exact way the three motions are combined, depends on the parameters of the unknown orbit, which is exactly what we are seeking! So, {from a deductive standpoint}, we would seem to be caught in a hopeless vicious circle. I shall get back to this point later. Although the main features of the apparent motion are produced by the "triple product" of two elliptical motion and one circular motion, as just mentioned, several other processes are also operating, which have a comparatively slight, but nevertheless distinctly measurable effect on the apparent motions. In particular, for his {precise} forecast, Gauss had to take into account the following known effects: 4. The 25,700-year precession of the equinoxes, which reflects a slow shift in the Earth's axis of rotation during the period of observation. The angular change of the Earth's axis in the course of a single year, causes a shift in the apparent positions of observed objects of the order of tens of seconds of arc (depending on their inclination to the so-called celestial equator), which is much larger than the margin of precision which Gauss required. (In Gauss' time astronomers routinely measured the apparent positions of objects in the sky to an accuracy of one second of arc, which corresponds to a 1,296,000th part of a full circle. Recall the standard angular measure: one full circle = 360 degrees, one degree = 60 minutes of arc, one minute of arc = 60 seconds of arc. Gauss is always working with parts-per-million accuracy, or better.) 5. The so-called nutation, which is a smaller periodic shift in the Earth's axis, superimposed on the 25,700-year precession, and chiefly connected with the orbit of the Moon. 6. A slight shift of the apparent direction of a distant star or planet relative to the "true" one, called "aberration," due to the compound effect of the finite velocity of light and the velocity of the observer. The apparent positions of stars and planets, as seen from the Earth, are also significantly modified by the diffraction of light in the atmosphere, which bends the rays from the observed object, and shifts its apparent position to a greater or lesser degree, depending on its angle above the horizon. Gauss assumed that Piazzi, as an experienced astronomer, had already made the necessary corrections for diffraction in the reported observations. Nevertheless, Gauss naturally had to allow for a certain margin of error in Piazzi's observations, arising from the imprecision of optical instruments, in the determination of time, and other causes. Finally, in addition to the exact times and observed positions of the object in the sky, Gauss also had to know the exact geographical position of the Piazzi's observatory on the surface of the Earth. Let us assume, for the moment, that the complications introduced by the effects 4,5 and 6 above are of a relatively technical nature and do not touch upon what Gauss called "the nerve of my method." Focus first, on obtaining some insight, into the way the three main degrees of action (1), (2), (3) combine to yield the observed positions. For exploratory purposes, do something like the following experiment, which requires only a large room and tables. Set up one object to represent the Sun, and arrange three other objects to represent three successive positions of the Earth in its orbit around the Sun. This can be done in many variations, but a reasonable first selection of the "Earth" positions would be to place them on a circle of about two meters radius around the "Sun", and about 23 centimeters apart -- corresponding, let us say, to the positions on the Sundays of three successive weeks. Now arrange another three objects at a greater distance from the "Sun", for example 5 meters, and separated from each other by, say 6 and 7 centimeters. These positions need not be exactly on a circle, but only very roughly so. They represent hypothetical positions of Piazzi's object on the same three successive Sundays of observation. For the purpose of the sightings we now wish to make, the best choice of "celestial objects" is to use small, bright-colored spheres or beads of diameter 1 cm or less, mounted at the end of thin wooden sticks which are fixed to wooden disks or other objects, the latter serving as bases placed on the table. Now, sight from each of the Earth positions to the corresponding hypothetical positions of Piazzi's object, and beyond these to a blackboard or posters hung from an opposing wall. Imagine that wall to represent part of the "sphere of fixed stars." Mark the positions on the wall which lie on the lines of sight between the three pairs of positions of the Earth and Piazzi's object. Those three marks on the wall, represent the "data" of three of Piazzi's observations, in terms of the object's apparent position relative to the background of stars, assuming the observations were made on successive Sundays. Experimenting with different relative positions of the two in their orbits, we can see how the phenomenon of apparent retrograde motion and "looping" can come about (in fact, Piazzi observed a retrograde motion). Experiment also with different arrangements of the spheres representing Piazzi's object, as might correspond to different orbits. From this kind of exploration, we are struck with an enormous apparent ambiguity in the observations. What Piazzi saw in his telescope was only a very faint point of light, hardly distinguishable from a distant star except by its peculiar motion from day to day. On the face of things, there would seem to be no way to know exactly how far away the object might be, nor in what exact direction it might be moving in space. Indeed, all we really have are three straight lines-of-sight, running from each of the three positions of the Earth to the corresponding marks on the wall. For all we know, each of the three positions of Piazzi's object might be located anywhere along the corresponding line-of-sight! We do know the {time intervals} between the positions we are looking at (in this case a period of one week), but how can that help us? Those times, in and of themselves, do not even tell us how fast the object is really moving, since it might be closer or farther away, and moving more or less toward or away from us. Try as we will, there seems to be no way to determine the positions in space from the observations in a deductive fashion. But haven't we forgot what Kepler taught us, about the primacy of the orbit over the motions and positions? Gauss didn't forget, and we shall discover his solution, in the coming installments of this series. ADDENDUM FROM JONATHAN TENNENBAUM ON PEDAGOGICAL DISCUSSION: Lyn beat me to it, but it still might be worth adding the following corrective which did not make into the Saturday briefing for technical reasons: Lest readers be misled by some hasty formulations in my pedagogical discussion above, I want to emphasize the following: The Keplerian orbits, referred to there, naturally do not exist as independent entities. Rather, what was said about the primacy of the orbits over the planets and their motions, applies even more to the Keplerian harmonic ordering of the solar systen as a whole. By virtue of the "strong forces" of that ordering, certain orbits are permitted, others rejected, and the curvature of any orbit in the small reflects the existence of all other orbits. It was exactly toward the goal of elucidating those harmonic "strong forces," that Gauss directed his work on the orbital determinations of Ceres. [jbt]

"In investigations such as we are now pursuing, it should not be so much asked `what has occurred,' as `what has occurred that has never occurred before.'" -- C. Auguste Dupin, in Poe's "The Murders in Rue Morgue" (1). With Dupin's words in mind, let us return to the dilemma, in which we had entangled ourselves in last week's discussion. That dilemma was connected with the fact, that what Piazzi observed as the motion of the unknown object against the stars, was neither the object's actual path in space, nor even a simple projection of that path on the "celestial sphere" of the observer, but rather the result of the motion of the object and that of the Earth "mixed" together. Thanks to the efforts of Kepler and his followers, the determination of the orbit of the Earth, subsuming its distance and position relative to the Sun at any given day of the year, was quite precisely known by Gauss' time. Accordingly, we can formulate the challenge posed by Piazzi's observations in the following way: We can determine a precise set of positions of the Earth in space at the precise times of Piazzi's observations, and from that the exact position which Piazzi's observatory in Palermo occupied in space at each of those precise times, as the calculable result of the Earth's motion in its orbit together with its rotation on its axis. From each of the positions of Palermo, draw a straight "line of sight" in the precise direction in which Piazzi saw the object at that moment (i.e., the presumed direction of the light ray arriving at Piazzi's telescope from the object, assuming the ray to deviate only imperceptibly from a straight line). Lacking "hard" information about the size, distance, and velocity of the object, all we can say with certainty, about the actual positions of the unknown object at the given times, is that each position lies somewhere along the corresponding straight line. What shall we do? In the face of such an apparent degree of ambiguity, those who would attempt to immediately "curve-fit" an orbit, will be thrown into complete disarray. For, there are no well-defined positions on which to "fit" an orbit! This shock should prompt us to turn on our brains (which are switched off during any fit of curve-fitting): Don't we know something more, which could help us? After all, Kepler taught, that the geometrical forms of the orbits are (to within a very high degree of precision, at least) plane conic sections, having a common focus at the center of the Sun. Kepler also provided a crucial, additional set of constraints, which determine the precise motion in any given orbit, once the "elements" of the orbit (described in last week's discussion) have been determined. Now, unfortunately, Piazzi's observations don't even tell us what plane the orbit of Piazzi's object lies in. The fact that the observed positions did not lie exactly on the ecliptic circle (the circle of the Sun's apparent motion against the stars), meant that in any case the required orbital plane is not identical to the plane of the Earth's orbit. That left open an infinity of possible orientations for the plane in question. How do we find the right one? Take an arbitrary plane through the Sun. Generally speaking, the lines-of-sight of Piazzi's observations will intersect that plane in as many points, each of which is a candidate for the position of the object at the given time. Next, try to construct a conic section, with a focus at the Sun, which goes through those points or at least fits them as closely as possible. (Alas! We are back to curve-fitting!) Finally -- and this is the substantial new feature -- check whether the time intervals defined by a Keplerian motion along the given conic sections between the given points, agree with the actual time intervals of Piazzi's observations. If they don't fit, which will be nearly always, then we reject the orbit. For example, if the intersection-points are very far away from the Sun, then Kepler's constraints (which we shall examine carefully in the next discussion) would imply a very slow motion in the corresponding orbit; outside a certain distance, the corresponding time-intervals would become larger than the times between Piazzi's actual observations. Conversely, if points are very close to the Sun, the motion would be too fast to agree with Piazzi's times. The consideration of time-intervals thus helps to limit the range of "trial-and-error" search somewhat, but the domain of apparent possibilities still remains monstrously large. With the unique exception of Carl Gauss, astronomers felt themselves forced to make ad hoc assumptions and guesses in order to radically reduce the range of possibilities, thereby reducing the amount of trial-and-error to a minimum. Thus, upon receiving the first sets of data from Piazzi, the astronomer Olbers and others decided to start with the working assumption, that the sought-for orbit was very nearly circular. The case of a hypothetical, perfectly circular orbit, the motion becomes particularly simple; and indeed, Kepler's third constraint (usually referred to as his "Third Law") determines a specific rate of uniform motion along the circle, as soon as the radius of the circular orbit is known. According to that third constraint, the square of periodic time in any closed orbit -- i.e., a circular or elliptical one -- as measured in years, is equal to the cube of the orbit's major axis, as measured in units of the major axis of the Earth's orbit. Next, Olbers took two of Piazzi's observations, and calculated the radius a circular orbit would have to have, in order to fit those two observations. In fact, it is easy to see how to do that in principle: Imagine a sphere of variable radius R, centered at the Sun. For each choice of R, that sphere will intersect the lines-of-sight of the observations in two points, P and Q. Assuming the planet were actually moving on a circular orbit of radius R, the points P and Q would be the corresponding positions at the times of the two observations, and the orbit would be the great circle on the sphere passing through those two points. On the other hand, Kepler's constraints tell us exactly how large the arc is, which any planet would traverse, during the time interval between the two observations, if its orbit were a circle of radius R. Now compare the arc determined from Kepler's constraint, with the actual arc between P and Q, as the radius R varies, and locate the value or value of R, for which the two become coincident. That determination can easily be translated into a mathematical equation, whose numerical solution is not difficult to work out. Having found a circular orbit fitting two observations in that way, Olbers then used the comparison with other observations to "correct" the original orbit. While Olbers, Piazzi himself and some other astronomers stuck with the circular orbit hypothesis, another group of astronomers, including Burckhardt and others, seeking better agreement with the whole series of observations, modified their original circular approximations to slightly eccentric ellipses. Parabolic orbits were also considered. Toward the end of the Summer of 1801, astronomers all over Europe began to search for the object Piazzi had seen in January-February, and which (according to the estimated orbits) should have moved far enough away from the Sun to once again be visible in the night sky. The search was guided by forecasts of the object's day-to-day position, derived from the approximate orbits that Olbers, Piazzi, Burckhardt and numerous others had constructed using Piazzi's original observations. The search was in vain! For five months, exhaustive and increasingly desparate searches by Europe's foremost observers, failed to turn up any sign of the object. Finally, in early December, the astronomcal publication "Zachs monatliche Correspondenz" published the elements of the orbit calculated by Gauss, which was substantially different from all the others. According to Gauss' analysis, the object would be located more than 6 degrees further to the South from the forecasted positions of Olbers, et al. -- an enormous angle, in astronomical terms! Shortly thereafter the object was indeed found, by Zach and then by Olbers, very close to the position calculated from Gauss' orbit. Olbers called it Ceres, following Gauss' own proposal, Ceres Fernandea. Characteristically, Gauss' method used no trial-and-error at all! Without making any assumptions on the particular form of the orbit, and using only three well-chosen observations, Gauss was able to construct a good first approximation to the orbit immediately, and then perfect it without further observations to a high precision, making the rediscovery of Piazzi's object possible. "Time-reversal" lies at the center of Gauss' method. We have to treat a set of observations (including the times as well as the apparent positions) as being the equivalent of a set of harmonic intervals. Even though the observations are "jumbled up" by the effects of projection along lines-of-sight and motion of the Earth, we must start from the standpoint, that the underlying curvature, determining an entire orbit from any arbitrarily small segment, is somehow lawfully expressed in such an array of intervals. To determine the orbit of Piazzi's object, we must be able to identify the specific, tell-tale characteristics which reveal the whole orbit, so-to-speak, from "between the intervals" of the observations and distinguish it from all other orbits. This requires that we conceptualize the higher curvature underlying the entire manifold of Keplerian orbits, taken as a whole. As we shall see, however, the higher curvature required, cannot be expressed by the sorts of mathematical functions, that existed prior to Gauss's work. Shed some light on these matters, by the following elementary experimental-geometrical investigation. (These should be done with a certain degree of care and precision, not in a sloppy, "Boomer" fashion! Without assimilating a sensuous notion of physical precision of determination within lawful relationships, readers will not be able to "get inside" Gauss' thinking on this and other topics.) Construct an ellipse, of the form of Mars' orbit, in the following way: Place two thin nails in a large board, on which poster paper has been attached, at a distance of 5.6 centimeters from each other. Attach the ends of a piece of strong, inelastic, but thin and flexible string to each of the nails in such a way, that the total length between the attached ends is 60 centimeters. Now, stretching the string taut with a suitable drawing instrument, and moving it subject to that constraint, generate an ellipse as the manifold of loci, the sum of whose distances to the two nails is 60 centimeters. Now, remove the two nails and draw a small circle, with a radius about 1 millimeter, around one of the holes to designate the Sun. This kind of construction of ellipses should be familiar to everyone, but probably few have ever carried it out with the actual dimensions of a planetary orbit. Observe that the circumference generated, is hardly distinguishable, by the naked eye, from a circle. Indeed, mark the midpoint of the ellipse (which will be the point mid-way between the foci), and compare the distances from various points on the circumference, to the center. You will find a maximum discrepancy of only about one millimeter (more precisely 1.3 mm), between the maximum distance (attained by the points at the two ends of the axis connecting the two foci) and the minimum, attained at the endpoints of the perpendicular axis. Thus, the deviation from a perfect circle is only on the order of 4 parts in a thousand. How was Kepler able to detect and demonstrate the non-circular shape of Mars' orbit, given such a minute deviation, and correctly ascertain the precise nature of the non-circular form, on the basis of the technology available at his time? Observe, however, that the distances to the Sun (the marked focus) change very substantially, as we move along the ellipse. Now, choose two points P and P' anywhere along the circumference of the ellipse, 2 centimeters apart. The interval between them would correspond to successive positions of Mars at times about 7 days apart (actually, up to about 10% more or less than that, depending on exactly where P and P' lie, relative to the perihelion and aphelion positions). Draw radial lines from each of P, P' to the Sun, and label the corresponding lengths R, R'. Now, consider what is contained in the curvilinear triangle formed by those two radial line segments and the small arc of Mars' trajectory, between P to P'. Compare that arc with that of analogous arcs at other positions on the orbit, and consider the following propositions: Apart from the symmetrical positions relative to the two axes of the ellipse, no two such arcs are exactly superimposable in any of their parts. Were we to change the parameters of the ellipse, for example the separation distance between the foci, by any amount however small, then none of the arcs on the new ellipse would be superimposible with any of those on the first, in any of their parts! Thus, each arc is uniquely characteristic of the ellipse of which it is a part. The same is true among all species of Keplerian orbits. Consider what means might be devised, to reconstruct the whole orbit from any one such arc. By what means one might determine, from a small portion of a planetary trajectory, whether it belongs to a parabolic, hyperbolic, or elliptical orbit? Now, compare the orbital arc between P and P' with the straight line joining P and P'. Together they bound a tiny, virtually "infinitesimal" area. Evidently, the unique characteristic of the particular elliptical orbit must be reflected somehow in the specific manner in which that arc differs from the line, as reflected in that "infinitesimal" area. Finally, add at third point P'', and consider the curvilinear triangles corresponding to each of the three pairs P-P', P'-P'' and P-P'', together with the corresponding rectilinear triangles and "infinitesimal" areas which compose them. The harmonic, mutual relations among these and the corresponding time intervals, lie at the heart of Gauss's method, which is the exact opposite of "linearity in the small"! ------------------------------------------------- (1) I would presume, that the quotes from C. Auguste Dupin in Poe's stories are not mere inventions of Poe, but directly reflect intensive discussions on scientific and intelligence methods, which Poe himself carried on in Paris with Dupin personally and other representatives of the French side of Gauss' "American conspiracy" in the early 1830s. It would hardly be surprising, if the discussions of intelligence method focussed on Gauss' determination of Ceres' orbit as a paradigm "detective story." Consider, for example, the following quotes from "The Murders in the Rue Morgue" and "The Purloined Letter," with included unmistakable reference to the destructive influence of LaPlace et al. "The mental features discoursed of as the analytical, are, in themselves, but little susceptible to analysis. We appreciate them only in their effects. We know of them, among other things, that they are always to their possessor, when inordinately possessed, a source of the liveliest enjoyment. As the strong man exults in his physical ability, delighting in such exercises as call his muscles into action, so glories the analyst in that moral activity which disentangles.... His results, brought about by the very soul and essence of method, have, in truth, the whole air of intuition. The faculty of re-solution is possibly much invigorated by mathematical study, and especially by that highest branch of it which, unjustly, and merely on account of its retrograde operations, has been called, as if par excellence, analysis. Yet to calculate is not in itself to analyze ... Analytical power should not be confounded with simple ingenuity; for while the analyst is necessarily ingenious, the ingenious man is often remarkably incapable of analysis. The constructive or combining power, by which ingenuity is usually manifested ... has been so frequently seen in those whole intellect bordered otherwise on idiocy, as to have attracted general observation among writers on morals." "Vidocq, for example, was a good guesser, and a persevering man. But, without educated thought, he erred continually by the very intensity of his investigations. He impaired his vision by holding the object too close. He might see, perhaps, one or two points with unusual clearness, but in doing so he, necessarily, lost sight of the matter as a whole ..." "They have no variation of principle in their investigations; at best, when urged by some unusual emergency -- by some extraordinary reward -- they extend or exaggerate their old modes of practice, without touching their principles. WHat, for example, in the case of D--, has been done to vary the principle of action? What is all this boring, and probing, and sounding, and scrutinizing with the microscope, and dividing the surface into registered square inches -- what is it all but an exaggeration of the application of the one principle or set of principles of search, which are based on the one set of notions regarding human ingenuity...?" "`The Minister ... has written learnedly on the Differential Calculus. He is a mere mathematician, and no poet. "`You are mistaken; I know him well; he is both. As poet and mathematician, he would reason well; as mere mathematician, he could not have reasoned at all ..." "`You surprise me,' I said, `by these opinions, which have been contradicted by the voice of the world. You do no mean to set at naught the well-digested idea of centuries. The mathematical reason has long been regarded as the reason par excellence.'... "`... The mathematicians, I grant you, have done their best to promulgate the popular error to which you allude, and which is none the less as error for its promulgation as truth. With an art worthy a better cause, for example, they have insinuated the term `analysis' into application to algebra. The French are originators of this particular deception ...' "`You have a quarrel on hand, I see,' said I, `with some of the algebraists of Paris; but proceed.'"

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