Always ask before diving into what may seem like a sea of
mathematics: “what are the thoughts that generate these footprints?” If you follow your investigation with that in
mind, you will be able to build a raft you can always come back to when having
swam too far out into overwhelming waves of equations. Rest assured, any genius like Gauss did not
start out with equations before tackling a physical problem like finding the orbit
of a planet.
If you’ve studied Johannes Kepler’s works, the New Astronomy
and the Harmony of the World, with the help of the websites provided by the LaRouche Youth Movement and the
scientific journal Dynamis,
then you may be a little uncomfortable with Carl Friedrich Gauss’ approach,
which approximates Kepler’s candor only
in the preface of his book on his discovery of the orbit of the asteroid Ceres.
"The Methods first employed have undergone so many and
such great changes, that scarcely any trace of resemblance remains between the
method in which the orbit of Ceres was first computed, and the form given in
-Carl Friedrich Gauss
If you have not studied Kepler in depth, do so before
Since everyone who is reading this has had at least some
exposure to how Kepler created mathematical physics and shaped the foundations
for science to come, we’ll begin with where Kepler left off.
After Kepler had released his amazing discovery, the
astronomical community was stunned. Scientists were torn. Using the
equant was untruthful, but the eccentric anomaly which was so easily calculated
from the equant could not be calculated from the mean anomaly of the ellipse.
Venetians like Galileo tried desperately to bury Kepler’s works.[i] This tension would only be resolved decades
later by Gottfried Leibniz, whose own discoveries only survived to reach Gauss
because of the unrelenting counteroffensive which Abraham Kästner and his circles led.
Unlike the dialogue Kepler has with that soul engaged in the
reliving of his discovery, a dialogue with the Gauss who is steeped in a
terrifying political climate,[ii] requires some tools which you already have the basis for, but may require some preliminary
investigation to make sure they are at your fingertips as we know they were for
Gauss during his investigations. Here,
we will only scratch the surface.
For this first installment, we will be accompanying Kepler,
Leibniz and Kästner through an investigation of the “conic sections” part of
Gauss’ Theory of the Motion of the
Heavenly Bodies Moving About the Sun in Conic Sections.
The first conic section to be considered is the circle,
which Kepler has already proven to be a figure which does not correspond to a
species that acts in a continuously changing way, in a universe which always
tends necessarily towards higher states of change. Its significance will come to light only once
we look at its place among the other conic cuts.
On the other hand, getting into the substance of a treatment
of the ellipse will require a bit of review.
To see if this cut is the same figure as what Kepler calls the ellipse, review
the properties of his construction. Does it have the same characteristics? After all, it may not seem so obvious,
because even visually, it seems as if the thinner side of the cone should
produce a more curved extreme.
construction of it, remember that the distances from the Sun are simply the
diametral distances through the center (see Sufficient Harmony: From
Kepler to Gauss) which are then placed at the
sun or the “center of the world” to more accurately represent the generating
source of the species. The diametral
distances are swung to the sine line dropped perpendicularly from the eccentric
position to the line of apsides.
This figure actually looks very similar to three other
hypotheses which Kepler discarded on the way here: the oval, the “middle road”
and the “puffed cheek.”[iii] What is so unique about this one, that it
could serve as an expression or manifestation of the curvature of space, and
allow us to both "look” for proportions in mean distances and “listen” for
harmonies in extreme motions, a combination that was necessary for us to
“sense” the harmony or the shape of space, thus giving Kepler the ability to
tune the constantly changing universe?
Kepler states that there is a direct constant proportion
between the perpendicular length dropped from the circle to the line of apsides
and that dropped from the ellipse. Can
you prove this using his construction? As with every investigation, start with what you know. For example, take the case at quadrature from
the center. What is the diametral
distance at quadrature? What is the
ratio in this case?
Here, a is the
diametral distance at quadrature, which is, in this case, also the semimajor
axis, while e is the ratio of the eccentricity to the major axis.
What is the relationship of the minor axis to the
eccentric’s eccentricity? What are these
relationships with respect to the aphelion and perihelion?[iv]
Take one more specific case at quadrature from the sun.
Does this hold the same proportion as found earlier? Find the relationship between this length (k) to the minor axis, in terms of the
aphelion and perihelion.
This length is called the half-parameter, whose significance
we shall return to shortly.
The plane construction of the ellipse can be thought of from
the standpoint of the definition originally given by Kepler in his Optics:
"The lines drawn from these points [in yellow] touching
the section, to their points of tangency, form angles equal to those that are
made when the opposite points are joined with these same points of
tangency. For the sake of light, and
with an eye turned towards mechanics, we shall call these points ‘foci.’"
From Kepler’s construction, the two diametral
distances will always add up to the length of the diameter. This is also the least path taken by light
were one to shine a light from one of the foci.
In the same chapter of the New Astronomy in which Kepler
claims the constant proportion property of the circle to the ellipse,[v] he also claims that the areas of the two have the same proportion!
The area of the circle sector is easy enough to find, but
what about the corresponding sector of the ellipse? If you have an itching hypothesis, try it
out, otherwise, we will come back to this question later, after having looked
at the cut of the cone, commonly called an ellipse.
Begin with a simple case, namely, the cone of zero
pitch. How does the cut grow as the tilt
Now try the same investigation on another unique cone, the
right cone, which gives circles with radii equal to the height up to those
corresponding circles from the apex.
Does this figure maintain the same relationships as before
in the Kepler construction? Or, even to
the figure in the cylinder? If so find
the focus and its placement in relation to the tilt of the cut. This
investigation will be a good practice in projective geometry before we get into
the spherical trigonometry. For example,
what does someone who is looking level to the cone, parallel to the axis
see? What about from the bottom or top?
What does your cut look like projected down to the plane at
the apex? Does the projected figure have
the same properties? Is the parameter,
or the eccentricity the same? If not,
what would you get if the parameter remained fixed and everything else were
allowed to vary? This will become
important later on when we investigate the hyperbola and parabola.
In chapter 60 of the New Astronomy, Kepler claims that the
areas of the circular sector and the corresponding ellipse sector have the same
proportion as the lines perpendicular to the line of apsides did. Is this true? Remember that the area of the circular sector measures the time it takes
to traverse the corresponding arc on the ellipse, a property we may need to
use, for example: when given an amount of time (for example, 6-10 months), and
asked to determine where a heavenly body (like an asteroid) will be once the
whole orbit is determined. In other
words, from a given initial position and a duration, how much arc will have
"Nowhere in the annals of astronomy do we meet with so
great an opportunity, and a greater one could hardly be imagined, for showing
most strikingly, the value of this problem, than in this crisis and urgent
necessity, when all hope of discovering in the heavens this planetary atom,
among innumerable small stars after nearly a year, rested solely upon a
sufficient approximate knowledge of its orbit to be based upon a very few
Now, before going on to the other conic sections and their
projections, let’s take a very preliminary look at another type of projection
that will be extremely useful and will require much more investigation. Let’s look at this spherical projection.
What are the characteristics of this projection and its
relation to the circle? Compare this to
earlier when we projected the ellipse to a circle in the cylinder. Which relations hold, and which don’t? You may have realized that if you tried to
draw this figure, you had to begin with a projection of a tilted circle. What allows you to recognize it as a circle? In fact ironically in the cone, the
representation of the ellipse looks more like a circle than the representations
of the circles do. What is the relation
of the area projected down? You will
have to become very familiar with these types of inversions, for example,
projecting a hypothesis onto the celestial sphere and back.[vi]
As we continue, avoid ending up like
Newton, by keep in mind that none of these
figures are primary or self evident.
"237. Scholium. Kepler found from the observations,
that the planets go in ellipses around the sun, which lies at the focus of
these ellipses (Astr. 247). Regarding this,
Newton showed that this would happen if the
planet were driven or pulled around the sun by a force which varied inversely
as the square of the distance (Princ. L. I. Pr. II). I consider the proof
thereof to be inadequate...."
In his textbook, Kästner showed that the inverse square law
could be entirely derived from properties of the ellipse, without going into
what gravity is at all! Therefore,
simply stating that the effect could be calculated by this formula in no way
shows anything about gravity, much less gives an explanation why there would be
a force acting at a distance along straight lines through a vacuum by some
property inherent in bodies.
"239.... The force is therefore inversely as the square
of the distance.
proved 239, but did not clearly show, that inverted, if the force was so
constituted, it would produce a conic section. Since, on page 17 of Book 1, he
already assumed it would make a conic section, and only investigated which one
it would be. This latter is much easier
than the former... and requires only to differentiate, where the former
In fact Leibniz goes a little further. He shows that the inverse square law could be
derived from several different theories which are completely opposed:
"I have considered Newton’s book, …I do not understand
how he conceives gravity…When I worked out my arguments about harmonic
circulation, that is to say, [circulatory motion in which speed is] inversely
proportional to distances, and encountered Kepler’s rule (of times proportional
to areas), I perceived the excellent advantage of this kind of circulation: it
alone is able to conserve itself in a medium that also circulates, and to bring
into lasting accord the motion of a solid body and that of the ambient
fluid. This was the physical explanation
which I once claimed to give for this circulation, bodies having been
determined in this way the better to be harmonized with one another. For harmonic circulation alone has the
property that the body circulating in this way keeps exactly the force of its
direction or previous impression as if it moved in a vacuum merely by its own
impetuosity together with gravity...It surprises me that Newton did not think
to give some explanation of the law of gravitation, to which I was also led by
-Leibniz to Huygens
Leibniz shows later that, the same equation is arrived at if
you considered gravity to be an attractive force with rays like light or if you
assume space to be a plenum, instead of a void, and the circulation of the sun
to cause centrifugal force, driving faster matter away and thereby creating a
situation where slower matter would fall as if being pulled toward the sun!
Kepler points out the uniqueness of the parabola in his Optics:
"The most obtuse of all hyperbolas is a straight line;
the most acute, a parabola. Likewise,
the most acute of all ellipses is a parabola, the most obtuse, a circle. Thus the parabola has on one side two things
infinite in nature - the hyperbola and the straight line - and on the other
side two things that are finite and return to themselves - the ellipse and the
However, Gauss rejects the parabola from the start,
beginning his work with an attack on all the people who assumed a parabolic
orbit for comets. He then goes on to
attack those few would admit that the orbits were in fact not parabolic, but
would then proceed to use the parabola as their first approximation, before
adding a few adjustments.
"There existed, in point of fact, no sufficient reason
why it should be taken for granted that the paths of comets are exactly
parabolic: on the contrary, it must be regarded as in the highest degree
improbable that nature should ever have favored such a hypothesis."
Nevertheless, the difficulties encountered in carrying out
the same investigations as above for the parabola will not be unfruitful. For example, for the parabola and for the
hyperbola, find the foci, or where the sun would be for a non-periodic
comet. What is the eccentricity? How about the major and minor axis! Now, what about the parameter k? Is that within sight? [See
Briefly on the Parameter
All the conic sections can be traversed by either
maintaining the same parameter and
increasing the eccentricity, thereby creating a family of related conic sections , which are infinite in
number, or vice versa, glimpsing at a
layer of several families. Kepler uses
still another which is the most useful for astronomy. He maintains the perihelion (or sagitta, as Kepler calls it) while
varying the parameter. Thus, if only the
perihelion, and the parameter were obtained or their ratios), the entire orbit
can be fully determined! The question of
how the parameter is determined will be the subject of another installment.
The parameter can be
changed by moving the mouse up and down, while the eccentricity can be changed
by moving the mouse left and right. Although only part of the hyperbola and parabola are seen, all of it can
be known. For the circle, the sagitta to parameter ratio is one, less
than one for the ellipse, 1/2 for the parabola, less than 1/2 for the hyperbola, and
0 for the line.
Remember, the constant relationship of area, or arc swept
out, to the time it takes to sweep out that area, is the first formulation of
the whole acting in the part and the part in the whole within each orbit. The orbits are, in turn, related to each
other as a whole by the sesquialterate relationship [See Box 4] of periodic
times of vs. mean distances of each orbit. But, what happens if part of the whole is at infinity? But aren’t these laws universal? Are non-periodic comets unlawful?
"I make use of a hypothesis with appears reasonable to
me, that is the same amount of power in each orbit or concentric circular
circumference of this circulating matter. This also means that they counterbalance each other best and that each
orbit conserves its own power.
"Now I measure power or force by the quantity of
effect. For example, the force needed to
raise a pound one foot is one fourth the force capable of raising one pound
four feet, for which twice the speed is required. Whence it follow that the forces are
proportional to the squares of the speeds.
"Let us consider, for example, two orbits or concentric
circumferences. Since the circumferences
are proportional to the rays or distances from the center, the quantities of
matter in each fluid orbit are also proportional.
"Now if the powers of the two orbits are equal, the
squares of their velocities must be inversely proportional to their quantity of
matter, and consequently to their distances. Or in other words, the velocities of the orbits should be inversely
proportional to the square roots of the distances from the center.
"From this, two important corollaries follow, both
verified by observation. The first is
that the squares of periodic times are directly proportional to the cubes of
distances. For the periodic times are
directly proportional to the orbits or distances and inversely proportional to
the velocities; and velocities are inversely proportional to the distances and
to the square roots of the distances; therefore, periodic times are directly
proportional to the distance and to the square root of the distance, which is
to say that the squares of the periodic times are as the cubes of the
-Leibniz letter to
Leibniz employs his life's work on motion, from his
development of the concept of dynamics, to show that his principle of
conservation of vis viva, combined
with the relationship derived by Kepler, of the instantaneous speed of a
heavenly body maintaining an inverse relation to its distance from the source
of immaterial species, results in Kepler's third law, which Kepler had arrived
at by observation. From the two, the
velocity of the fluid matter is found to be both proportional to the square
root of the distance and directly proportional to the distance as well. How could this be?[vii]
Here is Kästner’s construction of the hyperbola.
"Whereas before, the ellipse could be generated by a simple
mapping of one extension by a constant proportion, this time, the circle is
still being mapped by constant proportion, but to somewhere outside of
itself. Or, as Kästner puts it,
"just replace c [the minor axis of the ellipse] with c times the square root of -1."
Finally, in Gauss treatment, the equations he derives for
the hyperbola are stated matter of fact and are the same as the ellipse, except
for one small modification. They're all
The Curved and the
"Opposite limits [of the conic projections] are the
circle and the straight line: The former is pure curvedness, the latter pure
straightness. The hyperbola, parabola
and ellipse are placed in between, and participate in the straight and the
curved, the parabola equally, the hyperbola in more of the straightness, and
the ellipse in more of the curvedness."
This characteristic, of action bounded by the curved and the
straight, was further developed by Leibniz in his investigation of the
exponential and the catenary curve.
In the problem of solving the catenary curve, Leibniz had
encountered an "infinite" similar to the one Kepler had run into and
challenged posterity with. The
transcendental character of the catenary, arising from the relationship of the
action and the field of action (see Experimental Metaphysics: On the Subject of
Leibniz's Captive - Dynamis October 2006) exposed the Kepler paradox. It was this development of the Kepler problem
which finally led Gauss to present a geometrical representation of the complex
domain in his 1799 doctoral dissertation.
Now the question to us is: what does the nature of these
conic sections, which all planetary orbits trace, tell us about the nature of
The catenary (in
blue) can be expressed as the arithmetic mean of two geometric curves (in red).
The catenary, or hanging chain, derives its transcendental
nature from its embodiment of two qualitatively different actions: one being
the tension from that which holds the chain apart (i.e. horizontal tension),
and the other being the gravitational field that the chain exists in. The former is a constant throughout the
chain, while the latter expresses itself differently at every point depending
on how much chain is below that point for gravity to act on. Though qualitatively different, the catenary
expresses the two simultaneously. However, since the horizontal tension is constant throughout the chain,
changes in the curvature of the chain are dominated by the vertical tension,
i.e. the gravitational field, so that the chain can be used as a yard stick for
How do the characteristics of the catenary express
themselves in the complex domain?
The complex grid is
mapped according to self-similar growth
of one orthogonal set with respect to a uniform growth in the other. Vertical lines are transformed into pure
circular motion and horizontal lines are transformed into pure divergence.
The exponentials now constitute a whole range of actions,
which are bounded by the circle and the line, or maximum and minimum curvature,
What becomes of the catenary?
To look at this in another light, imagine an infinitely
large chain, or what would be equivalent,
imagine the ground underneath the catenary to be a small part of an a very
large sphere, and let that sphere shrink until it becomes very small in
comparison to the catenary. What shape
would the chain take on?