How Gauss Determined the Orbit of Ceres

by Bruce Director and Jonathan Tennenbaum

Printed in The American Almanac, December 15 and 22, 1997.


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Contents


How Gauss Determined the Orbit of Ceres
Part 1 -- Introduction.

by Bruce Director

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        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?

HOW GAUSS DETERMINED THE ORBIT OF CERES, PART 2

by Jonathan Tennenbaum

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        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]

Gauss' Determination of Ceres Part 3

by Jonathan Tennenbaum

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        "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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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.


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