Johannes Kepler was incredibly prolific, he published over eighty books and booklets over a very wide range of scientific and mathematical topics during his life. As far as he was concerned his magnum opus was his *Ioannis Keppleri Harmonices mundi libri V* (*The Five Books of Johannes Kepler’s The Harmony of the World*) published in 1619 some twenty years after he first conceived it. Today in popular #histsci it is almost always only mentioned for the fact that it contains the third of his laws of planetary motion, the harmonic law. However it contains much, much more of interest and in what follows I will attempt to give a brief sketch of what is in fact an extraordinary book.

A brief glace at the description of the ‘five books’ thoughtfully provided by the author on the title page (1) would seem to present a mixed bag of topics apparently in some way connected by the word or concept harmonic. In order to understand what we are being presented with we have to go back to 1596 and Kepler’s first book *Mysterium Cosmographicum** (The Cosmographic Mystery)*. In this slim volume Kepler presents his answer to the question, why are there only six planets? His, to our eyes, surprising answer is that the spaces between the planets are defined by the regular so-called Platonic solids and as the are, and can only be, five of these there can only be six planets.

Using the data from the greatest and least distances between the planets in the Copernican system, Kepler’s theory produces an unexpectedly accurate fit. However the fit is not actually accurate enough and in 1598 Kepler began working on a subsidiary hypothesis to explain the inaccuracies. Unfortunately, the book that he had planned to bring out in 1599 got somewhat delayed by his other activities and obligations and didn’t appear until 1619 in the form of the *Harmonice mundi*.

The hypothesis that Kepler presents us with is a complex mix of ideas taken from Pythagoras, Plato, Euclid, Proclus and Ptolemaeus centred round the Pythagorean concept of the harmony of the spheres. Put very simply the theory developed by the Pythagoreans was that the seven planets (we are talking geocentric cosmology here) in their orbits form a musical scale than can, in some versions of the theory, only be heard by the enlightened members of the Pythagorean cult. This theory was developed out of the discovery that consonances (harmonious sounds) in music can be expressed in the ratio of simple whole numbers to each other (the octave for example is 1:2) and the Pythagorean belief that the integers are the building block of the cosmos.

This Pythagorean concept winds its way through European intellectual history, Ptolemaeus wrote a book on the subject, his *Harmonice* and it is the reason why music was one of the four disciplines of the mathematical quadrivium along with arithmetic, geometry and astronomy. Tycho Brahe designed his Uraniburg so that all the architectonic dimensions from the main walls to the window frames were in Pythagorean harmonic proportion to one another.

It is also the reason why Isaac Newton decided that there should be seven colours in the rainbow, to match the seven notes of the musical scale. David Gregory tells us that Newton thought that gravity was the strings upon which the harmony of the spheres was played.

In his *Harmony* Kepler develops a whole new theory of harmony in order to rescue his geometrical vision of the cosmos. Unlike the Pythagoreans and Ptolemaeus who saw consonance as expressed by arithmetical ratios Kepler opted for a geometrical theory of consonance. He argued that consonances could only be constructed by ratios between the number of sides of regular polygons that can be constructed with a ruler and compass. The explication of this takes up the whole of the first book. I’m not going to go into details but interestingly, as part of his rejection of the number seven in his harmonic scheme Kepler goes to great lengths to show that the heptagon construction given by Dürer in his *Underweysung der Messung mit dem Zirckel und Richtscheyt* is only an approximation and not an exact construction. This shows that Dürer’s book was still being read nearly a hundred years after it was originally published.

In book two Kepler takes up Proclus’ theory that Euclid’s *Elements* builds systematically towards the construction of the five regular or Platonic solids, which are, in Plato’s philosophy, the elemental building blocks of the cosmos. Along the way in his investigation of the regular and semi-regular polyhedra Kepler delivers the first systematic study of the thirteen semi-regular Archimedean solids as well as discovering the first two star polyhedra. These important mathematical advances don’t seem to have interested Kepler, who is too involved in his revolutionary harmonic theory to notice. In the first two books Kepler displays an encyclopaedic knowledge of the mathematical literature.

The third book is devoted to music theory proper and is Kepler’s contribution to a debate that was raging under music theorist, including Galileo’s father Vincenzo Galilei, about the intervals on the musical scale at the beginning of the seventeenth century. Galilei supported the so-called traditional Pythagorean intonation, whereas Kepler sided with Gioseffo Zarlino who favoured the ‘modern’ just intonation. Although of course Kepler justification for his stance where based on his geometrical arguments. Another later participant in this debate was Marin Mersenne.

In the fourth book Kepler extends his new theory of harmony to, amongst other things, his astrology and his theory of the astrological aspects. Astrological aspects are when two or more planets are positioned on the zodiac or ecliptic at a significant angle to each other, for example 60° or 90°. In his *Harmonice*, Ptolemaeus, who in the Renaissance was regarded as the prime astrological authority, had already drawn a connection between musical theory and the astrological aspects; here Kepler replaces Ptolemaeus’ theory with his own, which sees the aspects are being derived directly from geometrical constructions. Interestingly Kepler, who had written and published quite extensively on astrology, rejected nearly the whole of traditional Greek astrology as humbug keeping only his theory of the astrological aspects as the only valid form of astrology. Kepler’s theory extended the number of influential aspects from the traditional five to twelve.

The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. He finally has success with the angular velocities of the planets in their orbits at perihelion and aphelion. He then examines the relationships between the tones thus generated by the different planets, constructing musical scales in the process. What he in missing in all of this is a grand unifying concept and this lacuna if filled by his harmonic law, his third law of planetary motion, P_{1}^{2}/P_{2}^{2}=R_{1}^{3}/R_{2}^{3}.

There is an appendix, which contains Kepler’s criticisms of part of Ptolemaeus’ *Harmonice* and Robert Fludd’s harmony theories. I blogged about the latter and the dispute that it triggered in an earlier post

With his book Kepler, who was a devoted Christian, was convinced that he had revealed the construction plan of his geometrical God’s cosmos. His grandiose theory became obsolete within less than fifty years of its publication, ironically pushed into obscurity by intellectual forces largely set into motion by Kepler in his *Astronomia nova*, his *Epitome astronomiae Copernicanae* and the *Rudolphine Tables*. All that has survived of his great project are his mathematical innovations in the first two books and the famous harmonic law. However if readers are prepared to put aside their modern perceptions and prejudices they can follow one of the great Renaissance minds on a fascinating intellectual journey into his vision of the cosmos.

(1) All of the illustration from the *Harmonice mundi* in this post are taken from the English translation *The Harmy of the World by Johannes Kepler*, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997

Clearly Plato’s Timeaus retained a strong influence on Kepler’s work. Curious is the geometric theory of the elements became associated with the planets or planetary distance in his thought. Or did he accept the aether theory?

Kepler originally tried to fill the gaps between the orbits with two dimensional polygons, inspired by a diagram of the cyclical occurrences of the conjunctions between Jupiter and Saturn. (see picture in the middle of linked page). Not able to find any passing fit with polygons he then tried out polyhedra and hit pay dirt. The extension to Platonic philosophy via Proclus was after the fact, so to speak.

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Missing in this is any mention of what is central to Kepler’s method of hypothesis. And that is his polemic against algebraic method and insistence that any hypothesis governing the laws of planetary motion or any physical phenomena must be geometrically grounded.

It is germane that the correct heptagon construction developed by Archimedes consists of a projective geometry as likewise do the conic sections of elliptical planetary orbits. Looked at from this “perspective” the later revolutionary developments in complex non-Euclidean geometry are further developments of Kepler’s search for such physical geometric bases for astronomy. Riemann’s seminal Hypotheses which lie st the bases of geometry was the marvelous culmination of this Platonic/Kelerian method.

I recommend David L. Goodstein and Diana R. Goodstein’s “Feynman’s Lost Lecture – The motion of the planets round the Sun”. There were actually five Feynman lectures that did not make it into the trilogy, three on how to solve problems, one on inertial guidance and this one, which the Goodsteins recovered from Feynman’s lecture notes (it was not a trivial undertaking as the images of the notes in the Epilogue show). This slim volume illustrates why Newton used geometry, not the calculus that he had independently invented, to explain Kepler’s laws of planetary motion.

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About Kepler’s

Mysterium Cosmographicum: someone (I thought it was Weinberg, but I can’t find the quote) gave a quasi-defense of the polyhedral nesting theory. Modern particle physicists look for explanations of order in various symmetry groups. It was only natural for Kepler to do the same in terms of the symmetrical mathematical objects known to him, namely the Platonic solids.I did find this quote:

Perhaps it’s only presentism that makes Kepler’s theory seem bizarre.

OTOH, it doesn’t seem that any of Kepler’s contemporaries or successors found much value in his harmonic speculations, except for the 3rd law.

They completely ignored the

Harmonice Mundiand took the three laws of planetary motion from theEpitome astronomiae CopernicanAs I recall, the

Mysterium Cosmographicumreceived a bit more attention, but no takers.Small typo – it should be ‘planets’ not plants in the paragraph beginning

“The fifth book …”

“He then examines the relationships between the tones thus generated by the different PLANeTS, …..

Thx.