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

©2006 LaRouche Youth L.L.P.
Proposition 13
"The universe of Riemann and Einstein, for example, is a dynamic system, of a type best described… as finite and self-bounded. That means, for example, that gravity, as discovered uniquely by Johannes Kepler (but not the modern sophists Galileo and Newton) is an efficiently universal physical principle. This means, in other words, a principle of action as extensive as the universe, in a universe which extends no further than is reached by the universal principle of gravitation. Our universe is therefore self-bounded, and finite in that sense. Its bounds are expressed in mankind's expanding accumulation of discoveries and applications of universal physical principles."

Lyndon LaRouche, Jr. EIR volume 33, number 29, July 21, 2006


The Personality of the Sun

Picture taken by Chris Landry

The sun, a choirmaster, conducts the system of the harmonies; he generates the motions of the planets, and receives the harmonic effect of these motions as the apparent angles the various planets make from him as they orbit him. He conducts and is the beneficiary of his conducting:

“Not only does light go out to the whole world from the Sun, as from the focus or eye of the world, as all life and heat does from the heart, all motion from the ruler and mover; but in return there are collected at the Sun from the whole cosmic province, by royal right, these, so to speak, repayments of the most desirable harmony, or rather images of the pairs of motions flowing to it are linked together into a single harmony by the working of some mind, and so to speak stamped into coin out of rough silver and gold."

Harmony of the World

What would this mind be that is repaid by the pleasure of the harmonies that it has created? Without delving into sublunary nature, let us say that the Sun is a creative personality, as Kepler’s entire conception of the orbits of the Earth and Mars in the New Astronomy rely on his perception that the Sun is the cause of their motion. So it is in how these motions appear from the sun that Kepler’s harmonies are created.

This idea, that the Sun should be more than a fiery ball sending its heat and light over millions of miles, that it should be more than its matter, that, in fact, it should be like a perceptive entity, sounds terribly strange, even mystical. The idea is even more troubling as Kepler clearly conceives of the “heavenly harmonies” from this standpoint.


Pictures from NASA website

Why should this be troubling? Because the very notion is entirely unscientific and verges on animism? But what do we mean by science that we might call this idea “unscientific?” Is science all that which is free of human emotion, of joy, of sorrow, of happiness? Is it the slow death of a human calculator, endlessly tabulating statistics in a dark hole somewhere? If science were this, what would the universe that it claimed to analyze be?

Is the Sun creative? If so, what is his relationship to the solar system?


A Three Body Problem

If the Sun sends out a species whose affect lessens as this species moves further from the Sun, if its affect on the planets varies as the inverse of their distances from the sun, or if its apparent affect, as viewed from the Sun, varies as the inverse of the square of these distances:

“Why should the relationship between speed and distance be different within a planetary orbit than it is between orbits? Why should this be so? If the proportion of two apparent daily arcs, taken as an infinitesimal measure of angular speed from the sun at two “moments” of the planet’s orbit have a proportion which is close to the inverse proportion of their corresponding distances squared, then why does this relationship not hold across two orbits? Why is the proportion of two apparent daily arcs of two different planets, like Mars and Jupiter, not close to the inverse of the ratio of their two corresponding distances squared?”

Kepler’s work to develop the musical harmonic system of the planets from the Platonic solids required that he work out the possible speed ratios that two planets could have, if their converging distances were fixed by a solid between them. That is, Mars and Jupiter are separated by the tetrahedron, which means the aphelial (farthest) distance of Mars is one-third the perihelial (closest) distance of Jupiter. So, what is the ratio of the speeds of the two planets at those positions? Well, it turns out that it depends on the rest of their orbits!

While the farthest distance of Mars is fixed by the tetrahedron, we could change the eccentricity and pull the perihelial distance closer to the sun. Also, we could increase Jupiter's eccentricity and push its aphelion farther away. We could do both while still maintaining the 1/3 ratio of their approaching distances.

For fun, imagine that we were to place a planet between Mars and Jupiter, whose orbit came down as low as Mars, and went out as far as Jupiter, with a ratio of one-to-three for its closest and farthest distances. Now, when this planet is at the distance of Mars, does it move faster or slower? How about when it is out as far as Jupiter &emdash; will it go faster or slower than Jupiter? Think about it, and come up with a hypothesis before you continue.

A Three Body Solution

Here you have an animation with two fixed planets, and a third that you are able to adjust using the keyboard. The numbers in the upper right adjust when you change the orbit, and indicate the distances of the planets as well as their apparent speeds. Play around and see what you figure out.

You can also get a full-screen version by clicking here

Surprisingly (to some people), the fake middle planet goes faster than the inner planet when it is at its distance, and slower than the outer planet. That is, even though the middle planet has an average distance that is greater than the inner planet, and, on average, goes slower as an entire orbit, the speed relationships inside one orbit result in its going faster at the closest spot. Between these two relationships, 1. that of one orbit to another, which slows the middle planet overall, and 2. that of the various positions in one planet’s orbit, which speeds it when closer, relationship #2 is stronger than #1, with the result that the fake planet moves more quickly than the inner planet.

This is what we find in planning planetary missions. We must first speed the spacecraft to escape the orbit of the earth and move further out to a destination planet, and then it must be sped up again to catch up with it and stick in its orbit. The later ideas of Leibniz and Gauss, creating potential, make this more intuitive &emdash; the spacecraft’s speed must be increased through an application of power to give it a great enough potential to reach the outer planet, and it must be increased again at that point, to prevent it from "falling" in an ellipse back toward the sun.

Absolute Space?

There is no absolute relationship between planetary speed and distance. In fact, there is no direct way of comparing these two seemingly equal quantities among the orbits. We can only directly compare orbits as wholes. It is tempting to want to say that at a given distance from the sun, any object would travel at a certain speed, but it is not so. Some might say that if mass were taken into account, then the problem could be solved, but remember, there was no consideration of mass in creating the middle planet here, only the harmonic considerations Kepler lists in chapter 3 of book 5 of the Harmony of the World. At that time, Kepler was only able to have the vaguest idea about the real size of the planets. He conceived of the planets' masses and volumes as reflective of the higher harmonics of the universe, but he did not conceive of these as causal. Mass is not the reason the planets orbit as they do.

But it is important to reiterate that Kepler did conceive of the planets as bodies in a direct break with Aristotle:

“the planetary bodies moving or revolving around the sun must be considered, not as mathematical points, but obviously as material bodies having, as it were, a certain weight… that is, insofar as they are endowed, in proportion to the bulk of their bodies and the density of their substance, with the capacity to resist motion imparted to them from without.”

From Gesam. Werke, VIII, 94:9-14, as quoted in Kepler’s Somnium

So, in the universe, there is no absolute time or space. There are regions of action defined by the relationship of the sun and the individual planets.

“The virtue (gravity) which lays hold (of the planet) does not overcome every whit: for the resistance of matter in the planetary body stands up against it and restricts it: hence the planet does not follow exactly the forward movement of the prehensive force, but is left behind and abandoned by it and in that mutual struggle there is place for time.”

J. Kepler, Epitome of Copernican Astronomy pg 62

Perhaps, in a way, time is the result of this struggle.


In this section, Kepler calculates the exact relationship between the competing powers of the sesquialterate (3/2-power) relationship between a planet's mean distance and its orbital time, and the inverse-squared distance between the distance and apparent speed within one orbit. His language is very difficult to follow, and modern readers, more familiar with algebra than with Kepler's notation, may find it very time-consuming to come to understand his work. The following PDF works through the application of the two competing powers using more modern notation, while reaching exactly the same conclusion that Kepler arrived at.

The problem is posed in calculating exactly which harmonies two planets can sing with each other, when their converging distances are fixed by the Platonic solid lying between them. By adjusting the eccentricities of the planets, there is a whole range of possible planetary systems that can be created, all with the same solids between the spheres. Basically, Kepler consideres how changing the thickness of the spheres containing the planetary orbits in his model from the Mysterium Cosmographicum will affect the speeds of the planets.

Click here for the pdf.

This relationship between the relative distances of the orbits implies what their potential converging and diverging harmonies could be. Within this region of possibility, the best possible harmonies with respect to the entire system of planets then dictate the planets’ eccentricities, which in turn, can violate the dictate of the distances to a certain degree. From the proportional relation of the planets’ mean distances to their converging extreme distances, Kepler can quickly check what harmonic relationships were possible for these planets. For instance, if the “proportion of the spheres” of two planets were 1000 to 795, how large would the proportion of their eccentricities need to be for these planets to make a major or minor third?


The nested solids define the large question of how the planets are ordered, “the number of the six primary spheres has properly been taken from the five solids alone, their proportion principally from the five geometrical solids; but it has conceded very small amounts all around to the motions, as it was the final cause which was accepted for the idea of the operation right from the start. And this is to be understood of the motions of each planet, its slowest on the one hand and its fastest on the other, that is of the motions considered as the cause of its particular properties. Indeed the periodic motions, that is to say, the number of days assigned to the revolutions of each individual planet, have both on account of the proportion of the orbits and on account of the eccentricities (which have been established from the harmonies) regressed further from the five solids.” (Mysterium Cosmographicum). Both the motions and the relative distances are coherent in one universe. “The bodies which make up the universe are not parts of one continuous quantity… the bodies of the universe are allotted spaces which are solid, or of three dimensions, to traverse.” (Mysterium Cosmographicum) The “space” between the planets is a function of distinct regions delimited by the solids, whose “nobility depends on their simplicity… for just as God is the model and rule for living creatures, so the sphere is for the solids… (and the sphere) is extremely simple, because it is enclosed by a single boundary, namely itself.” But the motions of the individual planets, their voices, were “the final cause which was accepted for the idea of the operation right from the start.” The harmony of the individual voices is the primary cause, yet the range of action which is possible for these harmonies is delimited by the boundaries of the solids, which in turn must give way for the harmonies. The relative distances and relative motions of the planets are subject to different domains, and are yet unified in a single idea.

Thus, the personality of the Sun acts through the entire solar system, like a conductor conveying an idea in music through the medium of the activity of the individual members of the orchestra. And each of those members is himself a sovereign in his own right, not a featureless automaton, but a unique individual with his own voice. The conductor then unifies these multifarious voices into a one.

This thirteenth proposition of chapter 3 is therefore concerned with potential. What were possible in this self-bounded universe of dynamic play?

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