Johannes Kepler – A Mysterious Cosmographist

Kepler broke radically from authority and tradition by utilizing the ellipse (as opposed to a composition of circular motions) and non-uniform velocities. He hewed firmly to the position that scientific investigations are independent of all philosophical and theological doctrines, that mathematical considerations alone should determine the wisdom of any hypothesis, and that the hypotheses and deductions from these must stand the test of empirical confirmation. (Morris Kline in “Mathematical Thought from Ancient to Modern Times, p.245)

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first turns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.. (Galileo in “The Assayer.”)

Kepler’s contributions to cosmology in particular and science, in general, are fundamental. Although Copernicus had already suggested that heliocentric theory (Sun is stationary and the planets including Earth revolve around it) was better and more consistent than Ptolemy’s geocentric theory (Earth is stationary and the planets including Sun revolve around it) a couple of centuries earlier, it was not accepted by a majority of the astronomers. Even Kepler, who was highly religious, and knew of Copernicus’ thesis, believed, in the early part of his life, that Earth was stationary and Sun revolved around it. This belief followed from Aristotle’s postulation which suggested that not only that Earth was stationary and the center of the universe but also that the planets moved in circular orbits with uniform speeds. This was because circle was believed to be the best form and the one that was preferred by God. Since God was unchangeable, likewise the speeds of His created planets were also constant. According to Kline (p.246), “Copernicus and Kepler were highly religious; yet both denied one of the central doctrines of Christianity, that man, the chief concern of God, was at the center of the universe, and everything in the universe revolved around him.” Many renowned astronomers after Copernicus, including Tycho Brahe, continued believing in Ptolemy’s geocentric theory. Brahe however believed that all the planets revolved around Sun and the Sun revolved around Earth. This he called the “Tychonic System.”

Although Copernicus had placed Sun at the center of the solar system and made Earth like any of the other five planets (there were only six planets known at that time) revolve around it, he stuck to the concept of circular orbits and constant speeds. He adhered to the epicycles and epicycles on epicycles which Ptolemy and the earlier Arab astronomers had developed and used in order to secure agreement between observations and predictions. His book, On the Revolution of Heavenly Spheres, included one diagram exactly like the one Ibn-al-Shatir had used earlier and which was regarded a great improvement on Ptolemy’s method, although Copernicus did not specifically quote his name. He quoted al-Battani’s work 23 times in his book.

Kepler was born in December 1571 in Weil-der-Stadt in Germany. His parents were Heinrich Kepler and Katharina Guldemann. He started his schooling at Schreibgschule in Leonberg when his parents relocated there. He entered the Adelburg Monastery School and Maulbronn in preparation for his entrance to the University of Tubingen in 1586.

At Tubingen, he was greatly influenced by his astronomy professor, Michael Mastlin, with whom he remained in touch all his life. Mastlin privately believed in Copernican theory but it was not taught in the university. He introduced Kepler to Coperniocan theory. Kepler read Astronomy as a matter of course because it was prescribed in the curriculum but he was interested in theology and wanted to become a clergy. He stated, “..these were prescribed studies and nothing indicated to me a particular bent for astronomy,” (Jenny Hwang, Kepler and the First Law of Planetary Motion, http://math.berkeley.edu/~robin/kepler/). He continued studying theology after receiving his M.Sc. from Tubingen in 1591.

In his last year at Tubingen, a vacancy for a professor of astronomy occurred at the Lutheran School in Graz due to death of its mathematics professor. The University of Tubingen recommended Kepler for this position which he accepted rather reluctantly. It was at Graz that his interest in astronomy was to spark and shape his later life. In addition to his teaching duties, he was also appointed the district mathematician which was considered quite a distinguished appointment. This required him to prepare astrological horoscopes. “For 1595, he had predicted an exceptionally cold winter, an attack by the Turks from the south, and a peasant uprising. All of these prophecies came true,” (Kitty Ferguson, Tycho & Kepler, p.183). His interest and faith in astrology would wane in his later life but it did provide an incentive for him to continue his work in astronomy more diligently.

The path of scientific research is seldom straight. The scientists try various methods of developing scientific theories taking guidance from empirical data. Some of these may be quite weird but unless they are proven wrong, the scientists try them out anyway. Kepler worked with one of such methods and thought he had stumbled on to something great. But he was wrong.

He took his cue from Archimedes who had inscribed polygons in a circle to determine its area. The more numerous sides such a polygon had, the more accurate his approximation to the area of circle would be. Kepler used the five perfect polyhedrons circumscribed (inscribed) them around the spheres which, he believed, gave the orbits of the six planets. According to Jenny Hwang, “..Kepler concludes that the existence of six planets is due to the existence of five perfect polyhedrons. Of course this was based on the ‘fact’ that there were six planets in the universe and only five perfect polyhedrons. Around Earth’s orbit, Kepler circumscribes a perfect dodecahedron (a solid having 12 faces) and the sphere containing this is Mars’ orbit. Similarly, around Mars’ sphere, a tetrahedron (a polyhedron that has four faces) is circumscribed and the sphere containing this is Jupiter’s orbit. Inscribing an icosahedron (a polyhedron having 20 faces) in Earth’s orbital sphere, the resulting inscribed sphere is Venus. This is done with the remainder of the perfect polyhedrons. Astonishingly, the ratios of adjacent planetary orbits represented in Kepler’s nested spheres model coincides with Copernicus’ calculations. Of course, Kepler relied mostly on divine inspiration for this theory.” This was all fortuitous but it impressed many of his prominent contemporaries.

He published this theory in his book “Mysterium Cosmographicum.” This book put Kepler on the forefront of theoretical astronomy (Cosmology).

A copy of this book landed into Tycho Brahe’s hands who was greatly impressed by Kepler’s work. Brahe had designed and constructed a number of very accurate astronomical instruments and had accumulated numerous observations which were far more accurate than any available in his time. Brahe was a noble and was in the employ of kings who had bestowed on him not only honorific titles but also generous financial grants. He had a number of prominent scientists working for him whom he treated like traditional employees which they indeed were, albeit, he had great respect for some of them. He guarded his data jealously because he feared that others would steal it and rob him of credit that was rightly his.

He also wanted somebody well-versed in mathematics to use his data and develop a theory which would verify his Tychonic hypothesis. When he read Kepler’s book he thought Kepler was the right man for his coveted goal. Fate would land Kepler into Brahe’s lap shortly.

Kepler was Protestant and Graz was ruled by a Roman Catholic king. Religious prejudice was rampant there and non-Catholics were not tolerated. For this reason, all the Protestants were exiled from Graz including Kepler but he was allowed to return by virtue of his position of district mathematician and his newly acquired fame, But later he was compelled to convert to Catholicism which he refused. So he was in a bind and was looking for a job elsewhere. He wrote to his mentor, Mastlin, at Tubingen for help but there was no job available there. Then he wrote to Brahe who invited him to join him at his Benatky observatory outside of Prague in 1600. Brahe at that time was Imperial Mathematician for the Emperor Rudolph II.

Kepler’s relations with Brahe were quite tumultuous, to say the least. Kepler wanted to develop a physical theory of planetary movements and wanted to use Brahe’s data for this purpose. Brahe was testing Kepler and was hesitant to release his data readily. Moreover, there were problems at a more personal level as well. An instance of bitter feelings between the two is illustrated by Barbara’s (Kepler’s wife) letter which she sent to Kepler in Graz where he had gone to settle some property issues. According to Kitty Ferguson (Tycho & Kepler, p.280), “Barbara wrote to Kepler that she was not getting as much money from Tycho as he had promised. She could not buy wood for the fire. An angry exchange of letters ensued. Tycho told Kepler to calculate what was owed and he would be paid, but to behave in future more considerately toward his ‘benefactor’ and ‘have more confidence in him.’ Kepler bristled at the insinuation that Tycho was giving him charity instead of fair recompense for his work. The contretemps finally ended agreeably, but it was symptomatic of the dissatisfaction Kepler still felt with his working arrangement.”

The duration of their association was rather brief; Kepler had joined Brahe in 1600 and Brahe died in 1601.Brahe gave his data to Kepler shortly before his death and Kepler submerged himself in the work of developing a theory of the planetary orbits. He started with Mars and believed his theory for Mars should apply to all the other planets as well.

Kepler, working with Copernican’s heliocentric system, tried various techniques to develop a method to predict the planetary orbits but didn’t succeed. Gradually, he had to part ways with the complete Copernican formulation and realized that the circular orbits would have to go. He tried an elliptical orbit for Mars with the Sun at one of the two foci and it worked admirably. He tried his elliptical orbit with the other planets and they also fitted accurately. Regarding his endless efforts, Kepler wrote, “I was almost driven to madness considering and calculating this matter. I could not find out why the planet would rather go on an elliptical orbit. (And then), as if I were roused from a dream and saw a new light,” (Kitty Ferguson, Tycho & Brahe, p.318). This gave Kepler his first law which was published in “Astronomia Nova (New Astronomy)”. His three laws of planetary movements are as follows:

1. The orbits of the planets around the Sun are elliptical in shape with the center of the Sun located at one of the two foci.
2. An imaginary line drawn from the center of the planet to the center of the Sun sweeps equal areas in equal intervals of time.
3. The ratio of the squares of the time periods of any two planets is equal to the ratio of the cubes of their average distances from the Sun.

Later, Newton would derive all these laws elegantly from his theory of gravitation. Kepler had hypothesized that there was a force due to Sun that kept the planets in their orbits. He thought this was a force due to magnetism. Later, he conjectured about gravitational force which he didn’t have time to fully develop and formulate. According to Kitty Ferguson (p.310), Kepler conjectured, “if one would place a stone behind the Earth and would assume that both are free from any other motion, then not only would the stone hurry to the Earth but the Earth would hurry to the stone; they would divide the space lying between in inverse proportion to their weights.”

Kepler was harassed all his life by one worry or the other. His mother was accused of witchcraft and imprisoned. Kepler spent several years in bringing her case to the court with mixed results. Eventually she was released.

Kepler died on November 15, 1630. He lived his life in poverty and misery but he immortalized himself by formulating his three laws of cosmology. He was a humble person who said, “there was nothing I could state that I could not also contradict.” His epitaph read as follows:

I used to measure the heavens.
Now earth’s shadow I measure,
Skybound, my mind. Earthbound, my body rests.