Kepler’s divine geometry.

Egil, the host for this month’s edition of The Giants’ Shoulders, wants esoteric episodes from the history of science for his Giants’ Shoulders, nobody in the history of science is more esoteric than Johannes Kepler.  Almost single handedly, Kepler dismantled the whole clockwork apparatus of the encrusted Ptolemaic and Copernican astronomies at the same time, as the first to do so, combining the physical natural philosophy with the mathematical astronomy against the dictates of scholastic academia to create the modern physical astronomy. However the heuristic that he used to achieve this masterwork was anything but modern and even less ‘scientific’. Here I shall just outline his first steps along his path of discovery.

Kepler was born into a family that had known better times, his mother was an innkeeper and his father was a mercenary. Under normal circumstances he probably would not have expected to receive much in the way of education but the local feudal ruler was quite advanced in his way and believed in providing financial support for deserving scholars. Kepler whose intelligence was obvious from an early age won scholarships to school and to the University of Tübingen where he had the luck to study under Michael Mästlin one of the very few convinced Copernican in the later part of the 16^th century. Having completed his BA Kepler went on to do a master degree in theology as he was a very devote believer and wished to become a theologian. Recognising his mathematical talents and realising that his religious views were dangerously heterodox, they would cause him much trouble later in life, his teacher, Mästlin, decided it would be wiser to send him off to work as a school maths teacher in the Austrian province.

Although obeying his superiors and heading off to Graz to teach protestant school boys the joys of Euclid, Kepler was far from happy as he saw his purpose in life in serving his God and not Urania (the Greek muse of astronomy). After having made the discovery that I will shortly describe Kepler found a compromise between his desire to serve God and his activities in astronomy. In a letter to Mästlin in 1595 he wrote:

I am in a hurry to publish, dearest teacher, but not for my benefit… I am devoting my effort so that these things can be published as quickly as possible for the glory of God, who wants to be recognised from the Book of Nature… Just as I pledged myself to God, so my intention remains. I wanted to be a theologian, and for a while I was anguished. But, now see how God is also glorified in astronomy, through my efforts.

So what was the process of thought that led to this conversion from a God glorifying theologian to a God glorifying astronomer and what was the discovery that he was so eager to publish? Kepler’s God was a geometer who had created a rational, mathematical universe who wanted his believers to discover the geometrical rules of construction of that universe and reveal them to his glory. Nothing is the universe was pure chance or without meaning everything that God had created had a purpose and a reason and the function of the scientist was to uncover those reasons. In another letter to Mästlin Kepler asked whether:

you have ever heard or read there to be anything, which devised an explanation for the arrangement of the planets? The Creator undertook nothing without reason. Therefore, there will be reason why Saturn should be nearly twice as high as Jupiter, Mars a little more than the Earth, [the Earth a little more] than Venus and Jupiter, moreover, more than three times as high as Mars.

The discovery that Kepler made and which started him on his road to the complete reform of astronomy was the answer to both the question as to the distance between the planets and also why there were exactly six of them, as stated above everything created by God was done for a purpose.

On the 19^th July 1595 Kepler was explaining to his students the regular cycle of the conjunctions of Saturn and Jupiter, planetary conjunctions played a central role in astrology. These conjunctions rotating around the ecliptic, the apparent path of the sun around the Earth, created a series of rotating equilateral triangles. Suddenly Kepler realised that the inscribed and circumscribed circles generated by his triangles were in approximately the same ratio as Saturn’s orbit to Jupiter’s. Thinking that he had found a solution to the problem of the distances between the planets he tried out various two-dimensional models without success. On the next day a flash of intuition provided him with the required three-dimensional solution, as he wrote to Mästlin:

I give you the proposition in words just as it came to me and at that very moment: “The Earth is the circle which is the measure of all. Construct a dodecahedron round it. The circle surrounding that will be Mars. Round Mars construct a tetrahedron. The circle surrounding that will be Jupiter. Round Jupiter construct a cube. The circle surrounding it will be Saturn. Now construct an icosahedron inside the Earth. The circle inscribed within that will be Venus. Inside Venus inscribe an octahedron. The circle inscribed inside that will be Mercury.”

The diagram shown above and drawn to illustrate this extraordinary hypothesis is one of the most often reproduced in the history of astronomy and was originally published in Kepler’s first publication from 1596, the splendidly titled:

The Forerunner of Cosmographical Essays, Containing the Cosmographical Secret: On the Marvellous Proportion of the Celestial Spheres, and on the Time and Particular Causes of the Number, Size and Periodic Motions of the Heavens, Demonstrated by Means of the Five Regular Geometric Bodies.

What to us seems more than somewhat bizarre was accepted by Kepler’s contemporaries with admiring interest and although many rejected his hypothesis it established him as a serious Renaissance astronomer. Unfortunately for Kepler but fortunately for us and the development of astronomy his geometric model, whilst impressive, didn’t quite fit. However he didn’t abandon his flash of inspiration or sit back on his laurels, saying it’s near enough, but spent the next thirty odd years searching for ways to improve, refine and perfect his cosmological model in the process laying the foundations of modern astronomy. Kepler never abandoned his belief in divine geometry or in the truth of his model and over the years he added other another layer consisting of the harmony theories of the Pythagoreans that I will deal with in a separate post.