Introduction
Harmonies and Solids
Chapters 3 and 4
Ellipse
Proposition 13
Prop 13 Calculator
Chapter 5
Harmony of the World
Chapter 9
A Dissonant Harmony

©2006 LaRouche Youth L.L.P.
Chapters 3 and 4

Chapters 3 & 4

What causes change?  Or, maybe another question might be easier to start with.  How do you know that something has changed?  Your keys aren’t where you left them.  They’ve moved?  Perhaps somebody took them.  Things appear different.  Objects in space aren’t where they were.  Is that change?

Take the case of the moment of Ben Franklin's discovery of electricity . Did the popoulation change?  Did the relations among nations suddely change?  What about the course of human history? Or, take the example of the election of Franklin Roosevelt. These are particular moments of history that may seem small in themselves but, when viewed from a higher standpoint, stand out as singularities in a developing process. So, the question really is, what moves and shapes such a process?

Any student of the history of science, or the science of history, for that matter, ought to be familiar with this sort of question.  A lesson from Johannes Kepler may help demonstrate the issue at hand.

How do you know what kind of Universe you are part of?  Where do you begin to investigate such a thing?  Why not start with what you see?  Any investigation is going to require some observations.  In fact, if you haven’t already done it, before continuing reading this, I strongly encourage you to go out for a night, or better yet, several nights, and pay attention to what appears to change, and how it changes.  This is crucial to the process of reliving a discovery, especially Kepler’s.  Attempting to do astronomy, without the first-hand experience of the night time sky, can lead to the disease of Cartesianism, i.e. situating oneself in an imaginary universe with presupposed types of interactions among objects, which do not correspond to reality.  This disease can lead to irritability, frustration, rage, and worst of all, political impotence.  So, please do not run this risk. 

One of the first things that may stick out is that, at any given moment, you don’t see anything change.  The sky looks still.  As time goes by over the course of the night, you observe that a particular grouping of stars seems to have kept the same relationship to one another, but, from where you’re standing, you can see that they have moved.  Maybe they moved closer to or farther from the horizon.  Maybe the orientation changed.  But, regardless of how they’ve moved, there is no question about whether they’ve moved.  Did you see the motion?  Just as in music, you cannot literally hear the interval between two tones, so too your eyes don’t see the motion.  In both examples, you perceive the effect of some sort of motion and your mind attempts to solve this somewhat puzzling experience.  This becomes clearer when you begin to contemplate that the stars you saw move, didn’t move at all.

In order for Kepler to present to the world his discovery concerning the harmonies adorning the solar system, he spent 17 years mapping out the motions of the solar system according to the method laid out in the Commentaries on Mars. That work was devoted to uncovering what was behind the motions in the heavens.  Being a student of Nicholas of Cusa, whose work served as a bridge between the work of the ancient Greeks and the 15th Century Renaissance, Kepler recognized motion as the effect of an unseen principle.  According to Cusa

Form descends, so that it exists contractedly in possibility; that is, while possibility ascends toward actual existence, form descends, so that it limits, perfects, and terminates possibility.  And so, from the ascent and descent motion arises and conjoins the two.  This motion is the medium-of-union of possibility and actuality, since from movable possibility and a formal mover, moving arises as a medium.  Therefore, this spirit, which is called nature, is spread throughout, and contracted by, the entire universe and each of its parts.  Hence nature is the enfolding (so to speak) of all things which occur through motion. 

De Docta Ignorantia II

From the standpoint of the observer on earth, who is well acquainted with the night-time sky, you may notice a star that is not part of a particular constellation rise with that constellation one evening.  Over the next several days it seems to move through that constellation, irregularly. To try to track down the cause of this effect, lets trace out its path.  What kind of measurement are we going to take?  After doing some investigations, you may bump into some interesting ironies.  The measurements that you make are tied to your own perceptions, which in this case are visual.  So, our means of investigation and the means by which we may verify a hypothesis, are limited by the bounds of sense-perception, whereas, the principle that we are investigating is not even sensible.  Luckily, as Kepler demonstrated, the human mind, also, like the principle we are investigating, is not bounded by perception.

So, how does this object change position in the sky?  How much does it change over the course of one night?  How much of this change is due to the Earth’s motion?  (Don’t assume that Earth’s rotation is the only consideration in this).  How does it change from one night to the next?  And, let’s not forget the greater question:  what is causing these changes? The deeper that you delve into such matters, the clearer it becomes that there is not one, but several interacting motions occurring, which conspire to produce the effect of what we perceive from the Earth. 

Kepler reveals his method of unraveling these interacting motions and their effects in his Commentaries on Mars. With that intention Kepler introduces a concept that later becomes known as the infinitesimal.  Consider this: what is the smallest interval of change of the object's true position (a position that can be known but not seen) that can be measured through observation?  Since a ‘minimal arc’ cannot be observed, this smallest unit of change in the object’s orbit must be envisioned in the mind!  In that way we can approach the cause of this change that is expressed in every infinitesimal interval, or minimal arc.  Understanding that these infinitesimal changes are due to physical principles, he is able to see with his mind what the true motions must be.  From that vantage point you can then begin to investigate the question of what is making these particular motions in the heavens necessary, or even possible- a question of utmost importance for Kepler. 

Kepler establishes that the amount of time that it takes a planet to travel over an arc is determined by the amount of area that is swept out by the arc (measured from the sun).  In other words, that at every minimal arc, the speed of the planet is inversely proportional to its distance. 

Link to New Astronomy webpage on Equal Area Equal Time

As Kepler demonstrates, this area-time relationship along with the elliptical orbit, are the effects of the relationship of the planet to the sun.  But, wait a minute; this relationship does not exist on its own.  It is part of the entire solar system.  As mentioned above, in the years following the publication of the Commentaries on Mars, Kepler continued to diligently gather together the data required to map the whole solar system in the same way that he maps the orbit of Mars in that work.

As far back as the Mysterium Cosmographicum  Kepler knew that Harmony had to be embedded in the creation of the Heavens.  In Book V of the Harmonices Mundi, Kepler resumes this investigation.  After examining the periodic times and the bulks of the planets for the harmonies, Kepler looks at the distances. He first compares the greatest and smallest distances in a particular orbit, and then compares these distances in various orbits.  He concludes these examinations stating that these things are not appropriate for such an investigation of harmonies, because harmony is more intimately connected with motion.  He therefore begins again.

After the previous failed attempts, Kepler chooses to use observations of angular changes in a planet’s location over equal intervals of time, an earth day.  He takes these measurements for all of the planets at their greatest (aphelial) distance and closest (perihelial) distance from the sun.  He applies these angular changes to the find the true daily paths of the planet at these distances.  To do this place yourself, in your imagination, in the center of the planet’s orbit.

 
Here the little man in the middle is imagining himself to be at the center of the orbit. He would see some amount of arc. Multiplying that by the average distance of the planet, or the semidiameter of the orbit, projects this arc onto a circle circumscribing the ellipse. Yes, this is an approximation, but, the eccentricities of most of the planets are so small that the orbits are very close to these circles; so close, that the arithmetic and geometric means of the greater and lesser semidiameters hardly differ. But, if you want more precision, see if you can come up with a better method!

But since the true motions of the planets are the inverse of their distances, this investigation yielded the same results as the previous one.  But, of course it would.  For who could detect or take pleasure in such harmonies?  There would have to be some sensory organ present throughout the world.  Otherwise there would have to be some mind equipped with the knowledge of geometry and arithmetic, situated to thus calculate such harmonies, a thing totally unnatural.  Harmony should be enjoyed, not calculated! Therefore, Kepler dismisses these hypotheses, and finally concludes that we should “turn our eyes to the apparent daily arcs, all indeed apparent to one definite and prominent position in the world, that is to say to the actual solar body, the source of the motion of the planets.”

In order to do this, we must imagine the view from the sun.  This involves a bit of an investigation into sense perception.  How does the size of an object change as you move toward or away from it, or as it moves toward or away from you? 


Here you can see that the perceived size of an object changes as the distance between the observer and the object changes.


Starting from the principle discovered by Kepler that the true motion on a minimal arc is inversely proportional to the distance of that arc, we can find how this motion appears on the sun.  In this example, taken from Chapter 3 of book V, we will treat these daily arcs as the minimal arcs, since they are the smallest observable measure.  This means that the distance at the start and end of the arc is equal.  It also means that we can treat the sines of the arcs as the measure. Making this assumption, you can see how the apparent motion turns out to be the inverse square of the distance.

Whereas the true motion of the planet on the eccentric is inversely proportional to the distance, the apparent motion from the sun is inversely proportional to the square of the distance, as seen in the example.  However, as with all problems of sense perception, some ambiguities arise.  For example, in the middle longitudes, the difference in distance between the sun and the center is so small that Kepler can use the true motion and apparent motion almost interchangeably, with one exception.  In the case of Mercury, the eccentricity is so large that, that the mean daily arc appears about 5 minutes smaller than it actually is.


This animation demonstrates that at quadrature, 90 degrees from aphelion as measured from the center of the orbit, the change in the apparent size of the arc is minimal for small eccentricities.  Whereas at the extremes, aphelion and perihelion, the distance is either increased or diminished by the whole eccentricity, the distance of the sun from the center of the orbit.  The change of distance at quadrature is a tiny portion of the eccentricity, which for most of the planets is small enough for the change to be negligible. This is not the case, however, for Mercury.

Mercury, whose orbit is the most eccentric of all the planets, is represented in this diagram.  As in the earlier animation on the apparent size of the arcs, if we take this arc to be a minimal arc, the distances at the beginning and end of the arc are equal.  From the center of the orbit, this arc appears smaller than from the sun. The numbers used for the distances, taken from Kepler, are in proportion to the actual distances, keeping the angles the same.  Here, you can see what amounts to a small, but significant difference in distance at quadrature, caused by the large eccentricity. The triangle to the right of the ellipse shows these distances. The rest of the diagram shows the steps taken to find the difference in the apparent size of the arc. The total effect of this eccentricity produces a difference in 5’32” of arc in the apparent mean motion.  Using this apparent mean motion changes the value of the apparent extreme motions.  The true mean motion results in a daily apparent arc of 167” at aphelion, and 393” at perihelion.  Using the corrected value of the apparent mean motion reduces values to 164” and 384”, respectively. 

Kepler, taking this data, compares the apparent motion of each planet at aphelion with the apparent motion of the same planet at perihelion.  But, again, these motions occur in an entire solar system, and since it is not a composite of one-on-one interactions, we must check whether the planets are in harmony with each other.  To do this Kepler compares the diverging and converging motions between the planets.

The results of this investigation are compiled in the following chart. 

chart

The corrections that are made in this image are of the translator's mistakes, not Kepler's

So, go back to the opening question.  What causes the change in Kepler’s case?  Is it ‘gravity’?  Well, what causes that?  How is it that there is a harmonic ordering to the solar system? Why?

For would that excellent Creator, who has introduced nothing into Nature without thoroughly foreseeing not only its necessity but its beauty and power to delight, have left only the mind of Man, the lord of all Nature, made in his own image, without any delight?

Mysterium Cosmographicum



Ellipse