Therefore, as matter strives for form, as a rough stone, of the correct size indeed strives for the Idea of the human form, so the geometrical proportions in the (solid) figures strive for harmonies; not so as to build and shape them, but because this matter fits more neatly to this form, this size of rock to this effigy, and also this proportion in a figure to this harmony, and therefore so that they may be built and shaped further, the matter in fact by its own form, the rock by the chisel into the appearance of an animate being, but the proportion of the spheres of the figures...by close and fitting harmony.
Book V, Chapter 9
Twenty-two years prior to the publication of the Harmony of the World, the 25 year old Kepler published his first major work, exhibiting his first major discovery, the Mysterium Cosmigraphicum. In an anniversary re-publication 25 years later, with added footnotes by the author, Kepler stated that he didn't want to change a thing about the original work, though many errors and false hypotheses had since been brought to light, because it contained in seed form, everything that was to be unfolded in his work throughout the rest of his life—especially raising questions and offering hypotheses about the relationship of the planetary distance to the motion, and even harmonies existing among the spheres. In that early work, Kepler unveils his discovery of the cause of not only the number of the planets (at least those known at the time), but also the reason why they lay at the particular distances that they did, with certain proportions between the sizes of the spheres. He boldly projects at the end that obtaining more accurate data will only confirm his hypothesis, eliminating the small discrepancies between the perfect proportions of the hypothesis, and the currently available observations.
However, in seeking out the all-star astronomer of the day, Tycho Brahe, to gain access to his massive tables of data, Kepler finds himself involved (as if “by divine arrangement”) in helping Tycho fetter the unruly Mars, and placing him into the bounds of a known orbit. This delay in his true purpose becomes a great boon to mankind, as he not only “captures” the planet, but does so by discovering the universal principles of gravitation and the elliptical motions of the planets, breaking the stranglehold of Aristotelean astronomy across Europe. Despite this “distraction”, Kepler gains access to Tycho's tables of observations, in fact being left them by the astronomer upon his unfortunate (and uncomfortable) death. In Book V of the Harmony, Kepler reveals that Tycho's accurate and trustworthy data has done just the opposite of what he had at first hoped. The data for the distances of the planets showed that the margin of error was too large to ignore, but that the error wasn't arbitrary—it was precise, and organized.
While the faces of the cube that would lie between Saturn and Jupiter should have touched the outer sphere (or farthest distance) of Jupiter (see diagram to the left), when tested against the more accurate data, they overstepped their bounds, reaching almost to the mean distances. The same was true of the cube's “mate”, the octahedron, which lies between Venus and Mercury. The dodecahedron and icosahedron (another “married couple”) fell far short of the outer distances of Earth and Venus. The tetrahedron, which is “celibate”, and couples with itself, is the only practically perfect fit (see diagrams below). Reflecting on this, Kepler is able to make one small correction, replacing the dodecahedron with the “hedgehog”, or star dodecahedron, a solid which he discovered. This solid fits nearly perfectly between the spheres of Mars and Venus.
Upon a brief re-examination of the “solids hypothesis", we can get a sense of why the “error” exists. While each orbit is not a perfect circle (as proved in the New Astronomy), but is an ellipse with an eccentricity, this means that at all but two particular moments in the annual orbit, the planet pulls away from and incurs upon its mean distance from the sun, going beyond and inside of the planet's mean sphere. In the animation, the dotted blue circles are circles with a radius of the planet's mean distance. You can see that as the red planet moves on its orbit, the distance of the planet from the sun is only equal to the mean distance at two points during the entire orbit.
The distances which are closer to and farther away from the sun than the mean distance give a certain “thickness” to the sphere. This was a fact which Kepler knew about 22 years earlier, since it was then believed by the followers of Copernicus, among whose number Kepler counted himself, that the planets' orbits were circles with an off-center sun, but he couldn't account for the cause of this thickness.
Look at the next animation. Here the eccentricity of the inner orbit (Mars) is being “stretched”, meaning that the proportion between it's aphelial and perihelial distances is increasing, going from a perfect circle (of zero eccentricity) to a very eccentric ellipse. Between the two, an infinite number of ellipses are possible. However, as is shown by the small dotted circle (whose radius is the planet's mean distance from the sun), the mean distance remains the same for every possible eccentricity.
Though the extreme distances are changing, the mean distances of Jupiter and Mars still maintain the 3:1 proportion of their intermediate tetrahedron. If one wishes to place the solid between the mean distances, then all of these very different ellipses would permit the tetrahedron to lie between Mars and Jupiter, and the only thing changing would be the thickness of the sphere. What, then, would tell us which eccentricity is most appropriate for the planet? However, one who insists that the solid is placed between the extreme distances of the two planets, must answer this question: what in the hypothesis requires there to be any eccentricity at all?
Confirming this, Kepler raises the fact that while there are only 5 intervals between the 6 planets, the five Platonic solids are able to give reason to their proportion. However, when one considers these greatest and least distances, we have not one, but four proportions between any pair of neighboring planets: greatest to greatest, least to least, greatest to least, and least to greatest. All in all, that's twenty proportions of the distances, which there aren't enough platonic solids to give cause for! Therefore, while the Creator would not have violated the archetype, or organizing principles which have given shape to the solids and is reflected in the sizes of the planetary spheres, we must seek another, higher causality which shaped and ordered both the spheres, and the eccentricities of the planets.
Look back at the animation. You can see that as the inner planet's eccentricity changes, so do the proportions of its extreme distances both with one another, and in relationship to those of the outer orbit. Kepler seeks some harmonic proportion in the four proportions of extreme distances between two planets, which would account for the eccentricity, but as can be seen in the diagram (from page 420 of the English translation), there is none to be found.
However, the distances are not the only thing changing here. According to the principles demonstrated in the New Astronomy (and reviewed here), as the planet moves closer and further from the sun, the planet's speed also changes (according to the inverse proportion of its distance from the sun), so we have a changing relationship of the motions as well as the distances. Kepler advises us in Chapter IV of Book V: “These distances, insofar as they are lengths without motion, are not appropriate to be examined for harmonies, because the harmonies are more intimately connected with motion, on account of its swiftness and slowness.”
Looking at the next animation, you can see that as the eccentricity of the Mars orbit is stretched, we're changing the proportion of the extreme distances of Mars and Jupiter. However, what you hear as the eccentricity changes is the change of the interval in the proportions of their motions. At certain moments in the continuous process, the slowest motion of Mars forms a harmonic proportion with the fastest motion of Jupiter, as you can hear, with the interval diminishing as the eccentricity grows. (The distance/speed relationships will be elaborated in the following pedagogicals.)
Kepler says in Book V, “For...the proportion of the spheres of each figure ought not to have been expressed immediately by itself, but through it there had first to be sought, and fitted to the extreme motions, harmonies which were very closely related to the actual proportions of the spheres.” Taking his cue from the Greeks (whose word “harmonia” means “to fit together”), Kepler points out that harmonies between the extreme motions of the planets serve us well, being the bounding parameters of the orbits individually (for the planet's own extreme motions are in harmony), but simultaneously bringing the planet into a harmonic relationship with the extreme motions of the neighboring planetary orbit.
So if harmonies potentially exist between motions of the planets as individual pairs, the question is then raised of there being one harmonic system, with a generating principle which accounts for the organization of all of the planetary motions as a unity, a single composition, and thus giving rise to both the distances and the eccentricities at once.
Now our question can no longer be: “Why are the distances what they are?”, but instead: “What is the ordering of the harmonies among the planets, in order to forge a single solar system from them?” In our specific example, we must be able to determine which harmonies are required of the motions between Mars and Jupiter, because it is the fulfillment of this requirement which calls upon the planets to assume the particular distances that they do, and to trace out a particular eccentric.
What the relationships are between the distances and the motions, as well as the ordering of the planetary harmonies will be elaborated in the following pedagogicals.