Classics of Astronomy by Johannes Kepler

Kepler’s Mysterium Cosmographicum (1596) is an extremely odd astronomical book by modern standards. Still, it is significant for a number of reasons. To begin with, it is virtually the first book to argue that heliocentrism is a physically true description of the world since Copernicus’s publication of his heliocentric system over fifty years before. As Kepler’s first book, it also represents his debut as a young astronomer. But he himself realized that it was far more than just that. It was the only one of his books that was reissued during his lifetime, a critical edition he published in 1621 which reproduced the original text verbatim along with copious footnotes explaining how his ideas had developed in the intervening years. In the dedication of the 1621 edition, he explained the significance of the Mysterium by noting that “almost every book on astronomy which I have published since that time could be referred to one or another of the important chapters set out in this little book.”

Ironically, it was disappointment in the career of mathematics teacher in which Kepler found himself that led him to write the Mysterium Cosmographicum. He had studied theology at the University of Tübingen, and had aspired to be a theologian. He had also been a promising student of astronomy and had argued for the truth of heliocentrism in public university disputations. His arguments were unusual, for whereas astronomers had generally been drawn to Copernicus’s theories for their mathematical elegance while dismissing the motion of the earth, Kepler’s approach was the opposite, dismissing the mathematical details of Copernicus’s work and arguing for heliocentrism on the basis of “physical, or if you prefer, metaphysical” arguments. When he found himself assigned to teach astronomy at the Protestant school in Graz, Styria, he cast his mind back to these arguments. It was important for Kepler that the heliocentric system be recognized as physically true, for the arrangement was to him a manifest symbol of God’s providential design. It was proper, he had argued, that the sun, the most resplendent of the planets, should be stationary in the center, distributing light, heat, and motion to the planets. And he saw a most significant parallel between the center, surface, and volume of a sphere; the sun, the sphere of stars, and the space of the solar system; and the three elements of the Holy Trinity, God the Father, God the Son, and God the Holy Spirit.

While Kepler’s “physical” arguments in the Mysterium Cosmographicum do not at all match our notion of what constitutes physics, they fare much better in the contemporary context of Aristotelian physics and causal explanation. There were, as Kepler saw it, several features of the Copernican system that needed explanation, primarily: Why were the planets at their specific distances from the sun? Why were there six and only six planets? and Why do the planets have their specific periods?

The answer to the first two questions he found in the five perfect Platonic solids. It had been known since antiquity that there were five and only five perfect solids, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Kepler argued that God had used these solids as archetypes when constructing the solar system. The five solids interposed between the planetary spheres determined their relative spacing, and the fact that there were only five solids explained why there were only six planets.

The problem of the planets’ periods did not yield such a satisfactory answer. Kepler had argued as a student that there should be a mathematical ratio between the periods and the distances. Now, he attempted to derive it from more recognizable physical principles, reasoning that the diminution of speed with distance was due both to the increasing size of the planets’ orbits and to a weakening of the planet-moving force as one got further from the sun. Even though he added rather than multiplied the effects, he produced a physical explanation that provided a rough correlation of distances and periods among the planets.

More surprising still was the penultimate chapter of the Mysterium, chapter 22, where Kepler extended his planet-moving force analysis to consider the variation in a planet’s speed around its own eccentric orbit. Here he argued that the equant in Ptolemaic theory or the largely equivalent eccentric epicyclet in Copernican theory were both merely slightly different mathematical models of the motion his physical theory described.

Kepler published the Mysterium Cosmographicum with the hope that astronomers would respond to his novel arguments for heliocentrism. The great Danish astronomer Tycho Brahe responded eventually that Kepler’s ideas were interesting, but that the problem of comparing the distances predicted by the polyhedra with their actual distances could be undertaken only with a body of very accurate observations, such has those he had been collecting for the past thirty years. It was the promise of using these observations to test and refine the cosmological argument of the Mysterium that was the primary motive for Kepler seeking out Tycho in Prague at the beginning of 1600.

Tycho did not trust Kepler with his life’s work of observations. But wanting to put him to work, he supplied him only with the observations of Mars, the planet whose data were then being reduced into theory. It is this restriction that accounts in the first place for the great contrast between the Mysterium Cosmographicum and the Astronomia Nova. However, a strong unity can still be seen in the research programs in the two books.

Not having the data to pursue any of the schemes in the Mysterium that compared the distances or periods of the planets among themselves, Kepler turned to the one question that concerned the detailed motion of a planet around its own orbit, that in chapter 22. There were a few qualifications he had had to make about his account of a planet’s motion around its own orbit, the most obvious of which was that the earth had never been assigned the same kind of orbit as the other planets by Ptolemy or Copernicus. It had always been simply a uniformly rotating circle slightly eccentric from the sun. (The inverse is the case in Ptolemy’s geocentric system, but the form of theory is identical.)

Kepler’s five years of research into Mars’ orbit therefore began with using the observations of Mars Tycho supplied to try to investigate the orbit of the earth. Kepler almost immediately began to find confirmation that the earth’s orbit, in principle, has the same form as the theories for the other planets. This turns out to be a very important finding, for even if one retains the basic form of Ptolemy’s theory of Mars, if one gives the earth the same form of theory, the errors in this pseudo-Ptolemaic theory will be slashed from a maximum of 3 or 4 degrees (or more) to more like 30 minutes. All of this reduction in error can be accomplished without Kepler’s well-known ellipses and area law, which are also presented in the Astronomia Nova. The painstaking working and reworking of Mars’s theory required to tease out, first, that the area swept out by a planet is equal in equal times (his so-called Second Law), and second, that the orbit of Mars is an ellipse with the sun at one focus (his so-called First Law), took about another four years.

The research on the orbit of Mars was not meant to be a book in its own right. It was, instead, intended to produce a theory of Mars for a great work of planetary tables that Tycho as imperial mathematician had promised the Emperor, which duty Kepler inherited after Tycho’s death in 1601. The greed of Tycho’s heirs, however, saw to it that this lucrative responsibility was temporarily taken away from Kepler. And the necessity of publishing something to justify his employment led to the decision to publish the Mars work by itself in the Astronomia Nova, whose title in English reads A New Astronomy, or Celestial Physics, Treated by Means of Commentaries on the Motion of the Star Mars. After having essentially finished the book in 1605, another four years of wrangling over rights and printing costs delayed its publication until 1609.

One of the byproducts of the book’s publication history was a quite unusual presentation of Kepler’s discoveries. Rather than the straightforward deductive manner of astronomical treatises, Kepler’s presentation is a narrative which details not only his successes but also the many failures and missteps along his path. But there is method to Kepler’s apparent madness. He knew that his idiosyncratic physical approach to astronomical theory would be met with disapproval by his fellow astronomers and viewed as an affront by Tycho’s heirs. So he carefully set up his story to give the appearance that he had been forced to pursue his physical analysis, and that this was the only approach that would yield agreement with Tycho’s very accurate observations. It is astonishing how effective this ruse was. Until only very recently Kepler continued to be seen as a brilliant but bumbling “sleepwalker” (to use Arthur Koestler’s characterization) who merely stumbled into the discovery of his laws.

Circumstances had thus conspired to force Kepler to publish two quite different books, one an exuberant laying open of God’s cosmological plan, the other a formidable, focused treatise on technical planetary theory. It is important to recognize, however, that there is a continuous motivation behind them, the physical proof of the truth of Copernicus’s heliocentric system, and that that proof was inseparable from Kepler’s religious calling to make plain God’s providential construction of the world.

When a time came in Kepler’s tumultuous life a few years later, when his spirit was too broken to carry on with his everyday scientific activities, he took solace in this original, overriding motivation. As a young man, shortly after the publication of the Mysterium, he had outlined to a correspondent the plan for an ambitious sequence of books of which the Mysterium was only a forerunner. One of these included an examination of “Pythagorean music,” i.e., the music of the celestial spheres, and another the astrological aspects, the geometrical alignments of the planets that were thought to be especially influential. Both of these shared the characteristic that they represented phenomena of nature that followed from the geometrical principles built in to the world at the creation by God. God was for Kepler the supremely rational being Who constructed the world based on rigorous geometrical principles. As creatures made in God’s image, men could perceive those mathematical underpinnings. And it was this perception that was the basis for our appreciation of these built-in mathematical relationships, such as musical harmony.

The Harmonices Mundi, or The Harmony of the World, published in 1619, was the outcome of Kepler’s examination into the mathematical basis of musical harmony, the spacing of the planets, and the efficacy of the aspects. It addresses these fundamental questions: Why are certain ratios harmonious and not others? Why are the planets at their particular distances from the sun and why do they have the eccentricities they have? Why are certain aspects important and not others?

The complexity of Kepler’s line of reasoning is not easy to summarize. In the case of the spacing of the planets, he began by examining the ratios of the planets’ aphelial and perihelial distances individually and among themselves to see whether the distances were in ratios that would correspond to musical harmonies, but found no satisfactory match. Eventually he hit upon the idea that the system might appear harmonious only to an observer at the center of the universe, the ever important sun, and that harmony had to involve not only the planets’ static distances but also their motions. His final conclusion is a tortuously reasoned tour de force. It is the extreme values of the angular motions of the planets as seen from the sun that embody all musical harmonies.

In the midst of this nearly intractable mathematical problem, Kepler did stumble upon a most remarkable mathematical relation between the planets’ periods and distances, that the ratio of the period of a planet squared to its distance cubed was a constant for all of the planets. This was exactly the mathematical relation he had sought since writing the Mysterium Cosmographicum over twenty years before. Kepler was ecstatic. He recorded the exact date of the discovery – May 15, 1618 – and wrote lyrically in the introduction to book V:

Thus was this eleventh-hour discovery, now known as the Third or “harmonic” Law, included in the Harmonices Mundi, where it fit only imperfectly with Kepler’s other elaborate lines of reasoning, and whence it took astronomers some time to extract it.

The work from which Kepler had turned to take the time to write the Harmonices Mundi was his monumental Tabulae Rudolphinae or Rudolphine Tables. Originally conceived by Tycho Brahe in 1601, these were to be only the third truly new set of planetary tables in western history. Just as the 13th-century Alfonsine Tables (based on Ptolemaic theory) had immortalized Alfonso X of Castile, and the 16th-century Prutenic Tables (based on Copernican theory) the Prussian patrons of Erasmus Reinhold, the Rudolphine Tables were sold to Emperor Rudolf II to immortalize him. At the time of their conception, Tycho had only theories of the sun and moon and the sure conviction that his unprecedented observations would produce the best planetary theories yet. He had no way of knowing that the Rudolphine Tables would become the embodiment of Keplerian planetary theory.

The publication history of the Rudolphine Tables was tortured. Kepler was originally hired chiefly to complete them. That responsibility was wrested away by Tycho’s heirs, but eventually returned to him. In the meantime, he had published the Astronomia Nova, which contained the theoretical basis on which all of the theories in the Rudolphine Tables would be based. Emperor Rudolf II died in 1611, and was succeeded by Emperor Matthias, and then by Emperor Ferdinand II in 1619. Both of these successors continued to fund Kepler’s monument to their Hapsburg forbear. Some years were needed for the reduction of observational data and the construction of the new tables. Kepler’s unsettled life (including time off to defend his mother against charges of witchcraft), other projects, and the outbreak of the Thirty Years’ War all contributed to delay the publication of the Rudolphine Tables until 1627.

The Rudolphine Tables were unquestionably the most accurate planetary tables produced to that time. Ptolemaic and Copernican theory had about the same accuracy, with errors for Mars ranging up to 4.5 degrees; Keplerian theory slashed that error by a factor of 50 to some minutes of arc. Surprisingly, the most significant element in that improvement was neither of Kepler’s first two laws (the third law was not used at all in the compilation of the Rudolphine Tables). It was instead Kepler treating the earth the same as all of the other planets, and having all of their apsidal lines intersect in the body of the sun. Those innovations alone could reduce the error to 30 minutes of arc. From there, the ellipse and the area law were needed to drive the error down to, for Kepler, undetectable levels.

Kepler’s innovations did not come without a cost in complexity. Unable to solve “Kepler’s Problem,” he was forced to tabulate the equation of center by whole degrees of eccentric anomaly, forcing (undoubted disgruntled) users to use a tricky reverse interpolation to proceed from mean to true anomaly. Worse still, his tables required time-consuming multiplication where previous Ptolemaic and Copernican tables had required only addition and subtraction. But this was required for calculating geocentric positions for an earth that could be at a range of longitudes and distances. Luckily, a brand new mathematical tool was at hand. Kepler’s tables became the first planetary tables to incorporate the use of logarithms. And he was also able to conceal within the logarithm table a table of sines for solving the earth-sun-planet triangles required to calculate geocentric positions.

Needless to say, the Rudolphine Tables were challenging. They retained the name of their original first author, Tycho Brahe, on the title-page, signaling to contemporary users that they were based on observations that were already legendary. But Kepler’s fussy implementation was a drawback. Vindication came four years later, already after his death, when his superior prediction allowed astronomers to view Mercury in transit – the first planetary transit ever witnessed.

The four books we have just examined trace Kepler’s career, from his debut as an astronomer through his revolutionary work in planetary theory to his triumph in producing tables of unparalleled accuracy. More importantly, they reflect the unity in his accomplishments, exposing different facets of his interests that still all belong to a single body of thought. Kepler’s work as an astronomer cannot be divorced from his work as a cosmologist (or “cosmographer,” as he would have said). And his work as a cosmologist cannot be divorced from his fundamental faith in God and in man’s ability to comprehend God’s mathematical construction of the universe. This faith found its first expression in the Mysterium Cosmographicum. The Mysterium also contained the seeds of Kepler’s physical analysis of the movement of the planets. That line of thought was continued and brought to a brilliant conclusion in the Astronomia Nova, but only in the form of a “commentary” on the motion of a single planet. Extended to all the planets, it became the basis for the breathtaking accuracy of the Rudolphine Tables. Throughout his career, however, Kepler never lost sight of his fundamental faith in God’s design, which was displayed in its most highly developed form in the Harmonices Mundi. Kepler’s creativity and genius were not confined to these four books, but they are surely the crown jewels of an illustrious and inspired career.

James R. Voelkel, Williams Class of 1984, is Curator of Rare Books at the Chemical Heritage Foundation, Philadelphia. He wrote this essay in April 2003.