What do Cones have to do with Gravity?


by Liona Fan-Chiang

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 this work."

-Carl Friedrich Gauss

If you have not studied Kepler in depth, do so before reading on.

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 Curved

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.

Cut of Cone


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.

Box 1

From the 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.

Also refer to the Harmonies website constructions of conic sections.

Two Sectors

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 increases?

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 been traversed?

Arc Area

 

"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 observation."

- Gauss

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.

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.

Box 2

"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...."

-Kästner Anfangsgründe.

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.

"240. Scholium. Newton 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 requires integration."

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 elliptical motion."

-Leibniz to Huygens 1690

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!

The Straight

Begin an investigation of the parabola and hyperbola with a review of a thorough treatment of the two on the Harmony of the World website.

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 circle."

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 Box 3]

Box 3

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?

Box 4

"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 distances."

-Leibniz letter to Huygens 1690.

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."

-Kaestner on conic sections, Anfangsgründe

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 imaginary!

The Curved and the Straight Coincide

 

"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."

-Kepler Optics

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 gravity?

Box 5

The Catenary

Catenary

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 gravity.[viii]

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, respectively.

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?

 



[iii] See pedagogical for Ch.58 of the New Astronomy

[iv] See Harmonies site on the ellipse

[v] See pedagogical for chapter 59 of the New Astronomy

[vi] For a head start on spherical trigonometry, refer to Kästner’s spherical trigonometry

[vii] For Kästner's treatment of Leibniz's conservation of vis viva and the principle least action, in the context of the political fight with the Maupertuis circles, see his Anfangsgrunde Mechanics.

[viii] For a more complete treatment of the transcendental nature of the catenary, see [“Experimental Metaphysics-On the Subject of Leibniz's Captive”-Dynamis October 2006]