In popular presentations of the so-called scientific or astronomical revolutions Galileo Galilei is almost always presented as the great champion of heliocentricity in the first third of the seventeenth century. In fact, as we shall see, his contribution was considerably smaller than is usually claimed and mostly had a negative rather than a positive influence. The real champion of heliocentricity in this period was Johannes Kepler, who in the decade between 1617 and 1627 published four major works that laid the foundations for the eventual triumph of heliocentricity over its rivals. I have already dealt with one of these in the previous post in this series, the *De cometis libelli tres I. astronomicus, theoremata continens de motu cometarum … II. physicus, continens physiologiam cometarum novam … III. astrologicus, de significationibus cometarum annorum 1607 et 1618 / autore Iohanne Keplero …*, which was published in 1619 and as I’ve already said became the most important reference text on comets in the 1680’s during a period of high comet activity that we will deal with in a later post.

Chronologically the first of Kepler’s influential books from this decade was Volume I (books I–III) of his *Epitome Astronomiae Copernicanae* published in 1617, Volume II (book IV) followed in 1620 and Volume III (books V–VII) in 1621. This was a text book on heliocentric astronomy written in question and answer dialogue form between a teacher and a student spelling out the whole of heliocentric astronomy and cosmology in comparatively straight forward and simple terms, the first such textbook. There was a second edition containing all three volumes in 1635.

This book was highly influential in the decades following its publication and although it claims to be a digest of Copernican astronomy, it in fact presents Kepler’s own elliptical astronomy. For the first time his, now legendary, three laws of planetary motion are presented as such together. As we saw earlier the first two laws–I. The orbit of a planet is an ellipse and the Sun is at one of the focal points of that ellipse II: A line connecting the Sun and the planet sweeps out equal areas in equal times–were published in his *Astronomia Nova* in 1609. The third law was new first appearing in, what he considered to be his opus magnum, *Ioannis Keppleri Harmonices mundi libri V* (*The Five Books of Johannes Kepler’s The Harmony of the World*) published in 1619 and to which we now turn our attention.

Kepler’s first book was his *Mysterium Cosmographicum* published in 1597 with its, to our way of thinking, somewhat bizarre hypothesis that there are only six planets because the spaces between their orbits are defined by the five regular Platonic solids.

Although his calculation in 1597 showed a fairly good geometrical fit for his theory, it was to Kepler’s mind not good enough and this was his motivation for acquiring Tycho Brahe’s newly won more accurate data for the planetary orbits. He believed he could quite literally fine tune his model using the Pythagorean theory of the harmony of the spheres, that is that the ratio of the planetary orbits build a musical scale that is only discernable to the enlightened Pythagorean astronomer. The *Harmonices Mundi* was that fine tuning.

The first two books of the *Harmonices Mundi* layout Kepler’s geometrical theory of music, which geometrical constructions produced harmonious musical intervals and which disharmonious ones, based on which are constructible with straight edge and compass, harmonious, and which are not, disharmonious. The third book is Kepler’s contribution to the contemporary debate on the correct division of the intervals of the musical scale, in which Vincenzo Galilei (1520–1591), Galileo’s father, had played a leading role. The fourth book is the application of the whole to astrology and the fifth its application to astronomy and it is here that we find the third law.

In the fifth Kepler compare all possible ratios of planetary speeds and distances constructing musical scales for planets and musical intervals for the relationship between planets. It is here that he, one could say, stumbles upon his third law, which is known as the harmony law. Kepler was very much aware of the importance of his discovery as he tells us in his own words:

*“After I had discovered true intervals of the orbits by ceaseless labour over a very long time and with the help of Brahe’s observations, finally the true proportion of the orbits showed itself to me. On the 8 ^{th} of March of this year 1618, if exact information about the time is desired, it appeared in my head. But I was unlucky when I inserted it into the calculation, and rejected it as false. Finally, on May 15, it came again and with a new onset conquered the darkness of my mind, whereat there followed such an excellent agreement between my seventeen years of work at the Tychonic observations and my present deliberation that I at first believed that I had dreamed and assumed the sought for in the supporting proofs. But it is entirely certain and exact that the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.”*

Translated into modern notation the third law is P_{1}^{2}/P_{2}^{2}=R_{1}^{3}/R_{2}^{3}, where P is the period of a planet and R is the mean radius of its orbit. It can be argues that this was Kepler’s greatest contribution to the history of the emergence of heliocentricity but rather strangely nobody really noticed its true significance until Newton came along at the end of the seventeenth century.

However they should have done because the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. There is nothing comparable for either a full geocentric system or for a geo-heliocentric Tychonic or semi-Tychonic system. It should have hit the early seventeenth-century astronomical community like a bomb but it didn’t, which raises the question why it didn’t. The answer is because it is buried in an enormous pile of irrelevance in the *Harmonices Mundi* and when Kepler repeated it in the *Epitome* he gave it no real emphasis, so it remained relatively ignored.

On a side note, it is often thought that Kepler had abandoned his comparatively baroque Platonic solids concepts from the *Mysterium Cosmographicum* but now that he had, in his opinion, ratified it in the *Harmonices Mundi* he published a second edition of the book in 1621.

Ironically the book of Kepler’s that really carried the day for heliocentricity against the geocentric and geo-heliocentric systems was his book of planetary tables based on Tycho Brahe’s data the *Tabulae Rudolphinae *(*Rudolphine Tables*) published in 1627, twenty-eight years after he first began working on them. Kepler had in fact been appointed directly by Rudolph II in Prague to produce these tables at the suggestion of Tycho in 1601. Turning Tycho’s vast collection of data into accurately calculated tables was a horrendous and tedious task and over the years Kepler complained often and bitterly about this burden.

However, he persevered and towards the end of the 1620s he was so far. Because he was the Imperial Mathematicus and had prepared the tables under the orders of the Emperor he tried to get the funds to cover the printing costs from the imperial treasury. This proved to be very difficult and after major struggles he managed to acquire 2000 florins of the more than 6000 that the Emperor owed him, enough to pay for the paper. He began printing in Linz but in the turmoil of the Thirty Years War the printing workshop got burnt down and he lost the already printed pages. Kepler decamped to Ulm, where with more difficulties he succeeded in finishing the first edition of 1000 copies. Although these were theoretically the property of the Emperor, Kepler took them to the Frankfurt book fair where he sold the entire edition to recoup his costs.

The *Tabulae Rudolphinae* were pretty much an instant hit. The principle function of astronomy since its beginnings in Babylon had always been to produce accurate tables and ephemerides for use initially by astrologers and then with time also cartographers, navigators etc. Astronomical systems and the astronomers, who created them, were judged on the quality and accuracy of their tables. Kepler’s *Tabulae Rudolphinae* based on Tycho’s data were of a level of accuracy previous unknown and thus immediately won many supporters. Those who used the tables assumed that their accuracies was due to Kepler’s elliptical planetary models leading to a gradually increasing acceptance of heliocentricity but this was Kepler’s system and not Copernicus’. Supported by the *Epitome* with the three laws of planetary motion Kepler’s version of heliocentricity became the dominant astronomical/cosmological system over the next decades but it would be another thirty to forty years, long after Kepler’s death, before it became the fully accepted system amongst astronomers.

Great post! However, Kepler’s second law is that the line from the Sun to a planet sweeps out equal areas in equal times (what we would now call conservation of angular momentum). I don’t think Kepler ever identified the location of the Sun as the focus of Mars’ elliptical orbit in Astronomia Nova. As far as I know, the Epitome is the first place where he states that clearly.

Also, you seem to imply that the Rudolphine Tables were built solely on Tycho’s data and not on Kepler’s elliptical theory. Is that correct? My understanding was that Kepler’s elliptical theory played a critical role in improving the accuracy of the Tables, particularly for Mars and Mercury (the planets with the most elliptical orbits). There’s a paper by Owen Gingerich where he analyzes the errors in the positions of Mercury using the Alphonsine Tables and the Rudolphine Tables and shows the tremendous improvement achieved by Kepler and I thought he claimed that the improvement was due to Kepler’s new theory (although, as I recall, a large part of the improvement could have been achieved solely by giving the Earth an equant).

Thanks for the heads up on the second law, what I wrote above was a brain fart. You are indeed correct on the fact that he only introduces the Sun at one of the foci in the Epitome; I just checked, so thanks for that too. It’s interesting, as well, because he coined the term focus of an ellipse in 1604 in his Optics. As your final line shows although Kepler’s models did play a role in the calculation of the Rudolphine Tables, the same accuracy could have been obtained from Tycho’s data by other means, so the accuracy was not dependent on his models as the people believed at the time. That slightly mistaken belief played a very major role in the acceptance of a heliocentric model for the solar system.

He did. See chapter 40, where he states the 2nd law, and the end of chapter 44, which discusses the connection between the elliptical orbit and the speed law. The assumption is also central to Kepler’s physics, and so pervades the later chapters.

You may be thinking of this fact: in the

Astronomia Nova, Kepler did not clearly distinguish between two speed laws: the so-called distance law (the speed of Mars is inversely proportional to its distance from the sun) and the area law we know and love. See the extensive discussion in Bruce Stephenson’s bookKepler’s Physical Astronomy.So far as accuracy goes, Kepler’s most important innovation was neither the first nor the second law, but giving the sun a non-uniform speed. Donahue remarks (in his translation of the

Astronomia Nova):Curtis Wilson makes a similar point in his article “Predictive astronomy in the century after Kepler”.

This innovation remained controversial for some time, until finally being decisively confirmed by the first Cassini. Heilbron devotes a whole chapter to this in his book

The Sun in the Church.Neither in Chapter 40 nor in Chapter 44 does Kepler identify the position of the Sun as at the focus of the ellipse. To quote Donahue:

In Chapter 40 Kepler is using an eccentric circular orbit for the Earth’s motion around the Sun. He notes at the beginning of the chapter that he will later need to modify this hypothesis for the other planets, but he says it is good enough for the Earth. He goes on to introduce the area law (which he still thought of as an approximation to the distance law), but the areas are measured using a sector defined by the physical location of the Sun, the location of the eccentric point (center of Earth’s assumed circular orbit), and the location of Earth. There is no way he could possibly identify the location of the Sun as the focus of an ellipse at this point, because he isn’t using an ellipse at all (although he does foreshadow a couple of times that he will later show that the orbits are not circular).

Likewise, in Chapter 44 Kepler gives his first clear statement that the orbits are not circular – but he doesn’t say they are elliptical. He says they are oval, basically a squished circle (using his analogy of squishing a sausage). So again, it is not possible that he says the Sun is at the focus of an elliptical orbit in this Chapter because there are no elliptical orbits in the chapter.

I’ve read Donahue’s recent translation pretty carefully and I don’t think Kepler ever says the Sun is at the focus of the elliptical orbit. It’s possible I missed it (or that is it in the original Latin but not Donahue’s translation), but it’s definitely not in Chapters 40 or 44.

I agree that the big improvement in accuracy was from giving the Sun a non-uniform speed. That’s exactly what I meant by giving the Earth an equant.

@Todd: Thanks. I was misled by Thony’s ascribing his “brain fart” to the second law; actually, it was about the first law. As you said, Kepler clearly states that the line sweeping out equal areas goes from the sun to the planet. Or in the distance law version, that the speed is inversely proportional to the distance from the sun. That’s central to his physics.

Then I completed the mix-up by claiming that he identified the sun as the focus of the ellipse; what I meant was that he identified the sun as the origin of the line sweeping out equal areas. I also didn’t pay enough attention to your parenthetical comment.

Almost simultaneously with your blog posting on Kepler is this article in

Physics Today, the American Institute of Physics house journal.https://physicstoday.scitation.org/doi/full/10.1063/PT.3.4388