Concerning the Harmonies of the World

Overview

The Harmony of the World is five books and about two hundred pages in the modern translation, published at Linz in 1619. Kepler had been working toward it since his first book in 1596, and he regarded it, not the astronomy, as his real achievement.

That is the thing to understand before opening it. Kepler was not a modern scientist who occasionally lapsed into mysticism. He was a Pythagorean who believed God had constructed the world on geometrical and musical proportions, and he spent a career looking for them. His first book had proposed that the spacing of the planets was determined by the five regular solids nested inside one another, which is wrong and which he never entirely gave up. The three laws that made him famous were by-products of that search.

His circumstances while writing were appalling. He was an imperial mathematician who was frequently not paid, a Lutheran repeatedly displaced by the wars that became the Thirty Years War, and during the composition of this book his mother was arrested as a witch and held for fourteen months, chained, facing torture. Kepler conducted her legal defence himself and got her acquitted.

Books I to IV are geometry, music theory, metaphysics and astrology. Book V is the astronomy, and it is the one that gets assigned.

The argument

Books I and II: the geometry. Kepler classifies regular polygons by whether they can be constructed, and then by how they tile a plane and combine into solids. This is serious mathematics: he gives the first systematic account of the star polygons and discovers two of the regular star polyhedra now called Kepler solids. Nothing here is astronomy. It is the foundation, because Kepler holds that the proportions God used must be geometrically constructible ones, which is why the classification matters.

Book III: harmony. Music theory derived from geometry. Kepler’s claim is that the consonances, the intervals the ear finds agreeable, are exactly the ratios produced by dividing a string according to the constructible polygons. Where the Pythagoreans had derived consonance from simple number ratios, Kepler derives it from what can be constructed with compass and straightedge, and gets a set of intervals closer to the ones actually used in practice.

Book IV: the soul and astrology. Metaphysics and a reformed astrology. Kepler cast horoscopes for money and despised most of what passed for astrology, and here he tries to put it on a footing he can accept: the soul perceives the angles between heavenly bodies as it perceives musical intervals, and responds to the harmonious ones. He rejects the zodiac signs and the houses and keeps the aspects.

Book V: the harmony of the spheres. The part to read.

Kepler begins by asking where the harmonies are in the solar system, and works through candidates. Not the distances, not the periods, not the linear speeds. He eventually finds them in the angular velocities of each planet at its nearest and furthest points from the sun, as they would be seen from the sun. Each planet therefore has a range between two notes, and Kepler writes them out in musical notation. Saturn and Jupiter span roughly a third, Mars a fifth, Earth and Venus barely a semitone, and Mercury more than an octave. The earth’s two notes, mi and fa, he takes to be a pun: misery and famine.

The combined harmonies, he argues, are audible only to the intellect, and the full polyphony was sung once, at the creation.

Then, in chapter 3, among a list of thirteen propositions about the planets, comes the sentence the book is remembered for. The proportion between the periodic times of any two planets is precisely the ratio of the three-halves power of their mean distances. He adds the details of when he found it: he had tried a version in 1596 and rejected it, returned to it on 8 March 1618, made an arithmetic error, and got it right on 15 May.

The proem to Book V is where he lets himself go, and it is one of the great passages of scientific prose: he says he has stolen the golden vessels of the Egyptians to build a tabernacle to his God, that he is casting the die and writing the book, that it may wait a hundred years for a reader since God has waited six thousand for an observer, and that nothing holds him.

How to read it

Read the proem to Book V and then Book V itself, particularly chapters 3 to 7. That is the astronomy and the rapture, and it is perhaps fifty pages.

Do not read Books I to IV unless the mathematics interests you, but do look at them long enough to see what kind of book this is. The third law appears in a work whose main business is assigning tunes to planets, and that fact is more instructive about how science actually happened than any tidy summary.

Get the Aiton, Duncan and Field translation, published by the American Philosophical Society, which is the only complete English version and has the apparatus to make the geometry followable.

Notice the discipline underneath the mysticism. Kepler had a theory he loved and he tested it against Tycho Brahe’s observations and threw out every version the numbers refused. The harmonies he finally reports are the ones that survived that, which is why a correct law came out of a wrong programme.

Read it between the Epitome and Newton’s Principia. The first two laws come from the Astronomia Nova of 1609, the third from here, and the Principia derives all three from a single force. Seeing where each piece came from is the clearest available picture of how a scientific revolution is actually assembled.

Questions it raises

• The third law is correct and the framework that produced it is not. What does that say about the relation between a research programme and its results?
• Kepler tested his harmonies against data and discarded what failed. Does that make him a modern scientist, or does the motive matter as much as the method?
• He assigns notes to planets and says the music is perceptible only to the intellect. Is that a claim about the world or a way of describing a mathematical relation?
• God waited six thousand years for an observer. What does it mean to do science as an act of worship, and does it change the science?
• Kepler defended his mother against a witchcraft charge while writing this. How should the conditions of a work’s making bear on how we read it?
• Newton took the third law and derived it from a force. Did that explain Kepler’s harmony or dissolve it?