Harmonies and Solids
Chapters 3 and 4
Chapter 5
Harmony of the World
Chapter 9
A Dissonant Harmony

©2006 LaRouche Youth L.L.P.
Intro to Book V

Introduction to Book V

In the midst of the present turmoil of global politics that we find ourselves facing today, Lyndon LaRouche issued the challenge of animating the discoveries of Johannes Kepler, in order that we may come to understand the fundamentals of physical economics, an understanding necessary to facing this crisis with the sense of optimism and authority that the crisis requires.  This particular site is dedicated to fostering a comprehensive work-through of Kepler’s Harmonices Mundi, published in 1619.  However, to achieve the most of such a work it is important to know the process that led to that work.  The discovery is not to be found on the pages of that, or any book, but rather, takes a great deal of concentration and can only be relived by retracing Kepler’s footsteps for yourself.  For as Kepler himself wrote, “indeed, the occasions by which people come to understand celestial things seem to me not much less marvelous as the celestial things itself.” 


Mysterium Cosmographicum

Our journey seems to begin on July 9/19 1595, when the 23 year old Kepler, while teaching, made the first of a profound series of discoveries.  Kepler fully elaborates this discovery in his Mysterium Cosmographicum, published less than a year later and dedicated to the 18 year old Archduke Ferdinand.  25 years later, in the footnotes to the 2nd publication, Kepler emphasizes the point that all his later achievements were already scattered in seed form among those pages.

But this work also is the product of an epistemological tradition that Kepler inherited from, most notably, the Pythagoreans, Plato, and Nicolas of Cusa.  This tradition informed all of Kepler’s life’s works.  And he indeed had no misgivings about passing that tradition on to the reader in that first work. 

As you plunge into the pages of the Mysterium, one of the first things you encounter is a greeting.  But in this greeting, Kepler tells you that God’s intention in creating the universe is revealed to you by Pythagoras with the five regular solids.  A bold statement?  The question that might arise in your mind: What exactly does this bold statement imply?

To begin to unfold the answer, Kepler tells you the basis of his method of investigation. The appearance of the universe is the effect of universal principles.  Meanwhile, the appearances are not the principles themselves.  How then does one contemplate the universe in all its splendor?  As Kepler suggests, referencing Paul, “like the Sun in water or in a mirror.”  This method is the same as described by Plato in the Socratic dialogues and it is further discussed by Cusa, in his De Docta Ignorantia.

The starting point for Kepler then, as for any scientist, is the idea that the Universe is constructed from principles and that mankind, made in the image of the Creator, could rediscover those principles and put them to use in perfecting that Universe. 

'For it neither is nor was right’ (as Cicero in his book on the universe quotes from Plato’s Timaeus) ‘that he who is the best should make anything except the most beautiful.’ Since, then, the Creator conceived the Idea of the Universe in his mind (we speak in human fashion, so that being men we may understand), and it is the Idea of that which is prior, indeed, as has just been said, of that which is best, so that the Form of the future creation may itself be the best: it is evident that by those laws which God himself in his goodness prescribes for himself, the only thing of which he could adopt the Idea for establishing the universe is his own essence…..

This pattern, this Idea, he wished to imprint on the universe, so that it should become as good and as fine as possible; and so that it might become capable of accepting this Idea, he created quantity; and the wisest of all Creators devised quantity so that their whole essence, so to speak, depended on these two characteristics, straightness and curvedness, of which curvedness was to represent God….For it must not be supposed that these characteristics which are so appropriate for the portrayal of God came into existence randomly, or that God did not have precisely that in mind but created quantity in matter for different reasons and with different intention, and that the contrast between straight and curved, and the resemblance to God, came into existence subsequently of their own accord, as if by accident.

It is more probable that at the beginning of all things it was with a definite intention that the straight and the curved were chosen by God to delineate the divinity of the Creator in the universe; and that it was in order that those should come into being that quantities existed, and that it was in order that quantity should have its place that first of all matter was created.

Kepler, Mysterium cosmographicum

Beginning with this conception of the "best of all possible worlds" is essential to any rigorous scientific work.  That idea is not only epistemological but, it is the basis for any sound ontology.  For, if the universe were not created according to the greatest wisdom, how could it have been created at all?

Plato’s Timaeus dialogue is one of the oldest, if not the oldest, documents that presents a comprehensive image of a self-sufficient Universe with a unique Creator.  From that starting point, the dialogue elaborates how the nature of the Universe is derived from this source, detailing the cosmos, musical harmony, geometric figures, human nature, and the elements.  So, it should come as no surprise that the discovery Kepler is sharing in the Mysterium is a re-affirmation of the method of the ancients.  Later in his life, writing in the Harmonices Mundi, Kepler hypothesizes that his discovery, that the distances between the planets were determined by the five Platonic Solids, may have already been known to the Pythagoreans, although they may have tried to obscure that fact.

As was just stated, on July 9/19 1595 Kepler realized that according to the Copernican data, which had to be reworked, the distances that determined the planetary circuits, were themselves determined by something else, which could not be directly seen in the heavens but could be known to the mind of man.  These are the solids called the cube, tetrahedron, dodecahedron, icosahedron, and octahedron.  All of this is unfolded to the reader of the Mysterium Cosmographicum, if they take the time to work it through.

However, ambiguity still manages to creep in, that is ambiguities other than the questionability of the Copernican data.  The distances of the planets from the sun are not uniform around the orbit, nor are the distances between the planetary orbits.  So, maybe Kepler was able to grab a hold of one aspect of what was determining these distances, but what about this eccentricity?  Besides that question there is another. An observer at any location (whether on the earth or on the sun or on Mars) can only measure changes in movements in terms of angles.  But if these occur at different distances, how can we know the true movements of the planets?  What is the relationship of the motions to the distances?  And why are the planets moving at all?

These are questions that Kepler begins to wrestle with in the concluding chapters of the Mysterium Cosmographicum.  On the question of the motion of the planets he conclusively refutes Aristotle, who thought that the orbital periods are in direct proportion to the distances of the planets assuming the speed of each planet to be the same.  But, why should the speeds of the planets be the same?  What cause would produce that effect? Does the planet even keep the same motion within its own orbit?

Finally, in the 22nd Chapter, at the end of the Mysterium, Kepler takes up the question of the motions, raising the issue of the equant, and of the physical causes for these motions being derived from the sun.  He refers back to this chapter in Book 3 of the New Astronomy and, in fact, one could argue that all of Book 3 of that work came out of this chapter. 


The Next Revolution

Once the Mysterium was completed, Kepler, devoted to seeking out the truth, attempted to contact Tycho Brahe, whose observations were unparalleled.  They communicated through a series of letters from 1597 to 1600, when they finally met in person. In October of that year Kepler came to live with Brahe, whose observations were being put together for the Rudolphine Tables, a task that Kepler took the next 20 years to complete, after Brahe’s death in 1601.

Kepler, seeing that Brahe had made use of the mean motion of the sun, as opposed to the truer apparent motion, which would have been more appropriate for Keplers investigations, asked to make use of the observations.  Brahe and his aide were busy then studying the oppositions of Mars to the Sun, and Kepler eagerly joined them.  This, he describes, turned out to be a ‘divine arrangement’ because the motions of Mars “provide the only possible access to the secrets of astronomy, without which we would remain forever ignorant of those secrets.”

The motions of Mars presented a paradox, which Kepler seized upon.  The results of these investigations led to the development of a whole New Astronomy, which he expounds in a book with the same name, taking up the particular case of the motions of Mars.  These motions produce an effect that cannot be explained by the long held dogma that the planet is moving around an equant along a perfectly circular pathThere arises a slight deviation that Kepler knew cannot be simply ascribed to observational error. 

In the spirit of Cusa, Kepler applies the method of Learned Ignorance. He first attempts to find the eccentricity of the circle which would give the correct longitudinal positions of the planet at the corresponding times, as dictated by the equant. In fact, he is so rigorous with this approach, that he is able to determine an orbit, for this purpose, more precise than anyone to date. But, if he were to leave it at that, he would be no true scientist. And so, in the persuit of truth, he makes another attempt at the eccentricity. This time, he turns to the story told by the latitudes, which takes into acount another factor, untouched by the longitudes, distance. Here, the hypothesis breaks down. The eccentricity arrived at in this manner hints at a hidden secret. If one were to combine the results of these experiments, they may see, as Kepler did, that the error increases from 2' of arc to 8', an error that Kepler says, led the way to a revolution.

Therefore, since his prevoius attempt failed, he knew that the truth must lay elsewhere. Being a man of reason, Kepler seeks out the physical causes that produce the orbit as an effect. In other words, the reason why the planet acts as it does.  

As he had already stated in the Mysterium Cosmographicum, the source of the planets' motions, he believed, must be the sun. As it spins it emits what he refers to as an immaterial species throughout the Solar System. This immaterial species moves the planets, according to its strength, around the sun as the sun rotates.  Borrowing the image of a lever, Kepler argues that the strength of the immaterial specie weakens proportionally to its distance. This also supposes that there is something substantial to the planets themselves.  They are not immaterial, as Aristotle had assumed.  The planet’s power and the strength of the sun’s motor virtue conspire to create the speed of the planet.  To get a sense of how this works, experiment. What is the difference between moving a weight with a short sick rather than a long one? Place a fulcrum at the center of a lever.  In order to keep the lever parallel with the floor or table, the weight at each end of the lever has to be equal.  If the weight on one side of the lever is different than the other side, the lever rotates. This may remind some of you of the experience at the playground when you were younger.  were you ever stuck up in the air on the see-saw?  Or, have you ever done that to someone else?  Where would we have to move the fulcrum to get it parallel once more?  Similarly, in the balance (if you don’t have one, see if you can come up with something creative to test this) if the weight on either side is equal, the balance will be even with the floor, but if not, it will rotate.  What principle governs this change? See what other experiments you can come up with.  If you apply this to the planet, you can see how the planet sweeps the same area in equal times, at any part of its orbit.

In this animation, we have the same body rotating around the sun in perfect circles, but at different distances.  The speed is inversely proportional to these distances.

Click here for animation on equal area in equal time

But, this still does not tell us what path is being traced out by the planet’s orbit. Nor, does it tells us what impels the planet to change its distance from the sun.  Why not travel in a perfect circle about the sun?  Perhaps there is something else about the sun-planet relationship that is determining this.  This is a question that had to be answered.  Without a hypothesis of why the planetary orbit takes the path that it does, there is no way of knowing why the planet appears to be where it is at any given moment.  There would be no way solving the mystery of the 8' of arc that Kepler keeps bumping into (For more on the mysterious 8' of arc, see the New Astronomy website and work through the book itself). 

By determining a principle moving the planets, we find a new method of measurement. The equant must yield this this principle if truth is to prevail. Utilizing the principle that the speed of the planet is inversely proportional to the distance, Kepler finds a new method for determining the position of the planet at a particular time. From the discrepancies that arise between the observations and the measurements derived from each particular hypotheses, Kepler can move toward the knowledge of what the true path must be.  Using a circle to calculate the position of a planet over a given period of time, yields results that suffice for the motions at the extremes, but makes the planet too slow in the middle longitudes.  It must be too far away by this model.  On the other hand, the oval that he tried next had the opposite effect.  By this, the planet was too fast in the middle longitudes.  It must have been too close by this hypothesis.  The truth must lie somewhere in the middle, but the only thing between a circle and an oval is another oval!  So, again he comes back to the physical causes.  What causes the planet to librate, or move away from the circle?

Kepler introduces the principle of magnetism. The arrow in this animation is meant to depict the magnetic threads of the planet.  Here, the threads stay in the same position as the planet moves around the sun. At different places in the orbit, the threads are exposed to the sun in different ways.

Here Kepler makes use of the recent discovery of magnetism and compares this to the planetary movements.  Although he expresses doubts about this, it seems reasonable given the proof by William Gilbert that the Earth is a giant magnet.The sort of phenomenon described by the speed/distance relationship is also exhibited by magnets, i.e. they approach one another with increased speed as their distance diminishes.  Therefore, since the planet’s body is magnetic, why not apply this to the relationship between the sun and the planet?  If the planet contains magnetic threads where one side seeks the sun and the other side flees it, as Kepler hypothesizes, the following effects coincide with observations. Another animation on the planet's magnetism is found here.

At aphelion and perihelion, both sides of the threads are equally exposed to the suns rays, so, neither side wins over the other and the planet is unaffected.  But, as the solar species sweeps the planet around the sun, the sun hits the magnetic threads at an oblique angle.  As it moves from aphelion to perihelion the friendly side of the magnetic threads is facing the sun and so, the planet is pulled toward it.  Once the planet passes perihelion the side that flees the sun causes the planet to be repulsed until it returns to its aphelial position.  Here Kepler, applies the principle found in the balance to measure out the strength that this magnetic power has over the planet as the planet revolves around the sun.  He measures this effect by taking the sine of the equated anomaly (angle at the sun), which is equal to the sine of the complement of the angle made by the threads and the suns rays. 

But, astronomy tells us that the maximum deviation of the planet from the circle occurs at quadrature, when the eccentric anomaly (angle at the center) is 90 degrees. And, the greatest power of the sun on the planet must occur when the threads are directly pointed at the sun.  But, according to our model, that occurs at 90 degrees of equated anomaly, rather than eccentric.  This signifies that something else is occurring.  Perhaps the magnetic threads are themselves moving as they approach and recede from the sun.

  Here the magnetic threads are being influenced by the sun in such a way that the resulting libration of the planet can be measured by the eccentric anomaly.  At quadrature, where the greatest libration occurs, the threads are directly facing the sun.

Kepler concludes that this can be corrected by taking the sine of the eccentric anomaly instead.  From this the actual magnitude of the libration can be found, and therefore also the distance of the planet produced thus. From these results, the planetary path, thus traced out, is elliptical. And applying the distances of the ellipse to determine the planet’s position at a given time causes the discrepancies between the measurements and the calculations to vanish. But, not by magic.  Rather it is Kepler’s looking into physical causes that does that.

But, upon making the revelation that the planetary orbits are elliptical, an entire new array of questions is posed.  Not all ellipses are similar.  In fact, the specific features of ellipses can be very different based on their eccentricities.  The combined effect of the relatively large eccentricity of Mars and its proximity to the Earth made the discovery of the elliptical orbit possible.  But, all the planetary paths are elliptical.  Why are they not the same ellipse?  Why must they have different eccentricities? What does this mean about the relationship of orbital periods to the distances? How did they get there in the first place?

Why are not the librations of the planets in the same ratio to their mean distance, that is, why is the eccentricity of Mercury greatest, next, that of Mars, and then those of Saturn, Jupiter, and the Earth, while that of Venus last?

The instrumental cause is the different strength of the threads, whether that is produced by nature or by posture.  But the final cause is the same as the eccentricities themselves, namely, in order that by reason of these eccentricities the movements of the planets should become very fast and very slow in such measure as would suffice for the harmonies to be exhibited through them.  Book V of my Harmonies has to do with this.

Epitome of Copernican Astronomy

Harmonies and Solids