Johannes Kepler, his quest, his tortures, and his legacy.

Johannes Kepler (1571-1630) discovered the true mathematical shapes (the ellipse) and mathematically described the motions of the planets (faster when nearer to the sun). He wrote about the joy of discovery as he realized that no one else in the world knew what he knew, “I feel carried away and possessed by an unutterable rapture over the divine spectacle of the heavenly harmony!” Indeed, “the music of the spheres” as an aesthetic ideal drove Kepler to his discovery.

The idea that the heavens were singing in some sense goes back more than twenty-five centuries. The Greek philosopher Pythagoras had thought that there was music in the spheres, meaning that the planets, stars, sun, and moon, must harmonize. Not in a way that the ears could hear, but music of the mind; harmony of the imagination.


This house stands in the town of Weil der Stadt, in the Baden-Wurttemburg state of Germany. It stands on the site of the supposed birthplace of Johannes Kepler, born there in the year 1571. The town of Weil der Stadt was completely destroyed in the 30-years war in 1648, some 18 years after the death of Kepler in 1630. It was rebuilt, and is now known quasi-officially as The Kepler Town. The political boundaries of the region today don’t resemble those of 1600. In those days, it was part of the Holy Roman Empire, whose seat of government was Prague in present day Czechoslovakia.

Those times were much harsher than those we enjoy these days. First, Feudal politics was rife, and Kepler and his family were several times forced to flee by the smoke and battle of war. Second, Plagues periodically swept across the lands. Kepler, his first and second wives, and all of his children for which the information is known all eventually died of sicknesses involving fevers. Third, religion in the region was a sharp source of disruption for Kepler and his family. The Roman Catholic church was being challenged by the Reformation of Martin Luther, but the Reformation itself was split between Martin Luther loyalists and those that followed the strict dogma preached by Calvin. These three camps each excluded the others, and tensions could often boil over. Finally, ignorance was rife. Few were educated, and superstition ran unchecked. For example, Katharina Kepler, the mother of the astronomer, was accused of being a witch, and would almost certainly have been found guilty and executed had it not been for the intervention of her son Johannes.

Despite these obstacles, the intellectual life of the times was lively and ripe for an individual like Kepler to contribute. The printing press had been invented, and ideas could travel far and wide in the form of books.

Young Kepler.

Let us review the cosmology of the day. First, let us dispose of the flat earth. A flat earth has never in all recorded history been a popular model of the earth. In 1600, all educated folk knew the earth was a sphere. This was due to a fascination with ancient Greek philosophy as written down by Aristotle. Among other things, he noted that when the moon passes through the earth’s shadow-cone during a lunar eclipse, the shadow the eclipse_shadow_apodearth presents on to the sky is a circle, and a circle that is about four times the diameter of the moon. That simple observation leads to the correct conclusion that the earth is about four times the diameter of the moon. The size of the earth and the moon were therefore known to within a few percent of the present day value, and the shapes of both were spheres. Spheres were considered the perfect shape due to their simplicity. The stars were located on a vast, distant sphere. The sun, moon, and planets likewise were thought to adorn different heavenly spheres, though the spheres themselves were invisible.

The motion of the planets was understood in several ways, but the two most common might be called the old way, and the new way. The old way was a system of moving spheres that came from the Greek tradition. In this picture, the Earth was the center of motion. The moon orbited the earth on a perfect circle, or sphere, and the sun orbited on a bigger circle. The outer planets Mars, Jupiter, and Saturn orbited on bigger and bigger circles until the most distant sphere of all was the home of the stars. The retrograde, backwards epicyclemotions of the planets were accomplished by having a little circle attached to the bigger circle. The smaller circle was called the epicycle, and the main circle the deferent. The center of the epicycle moved along the deferent at a steady pace, but the planet itself was attached to the epicycle. The planets moved backwards in this model because the epicycles were large enough and the cycling speed large enough, to simply make them move backwards.

This old picture is, of course, quite wrong, but it serves to show that the Greeks were attempting to think of geometric solutions to how the universe really behaves. It also, as a model, functions adequately to predict the future positions of the planets.

The new picture, published 28 years before the birth of Kepler, was a model by the Polish cleric and amateur astronomer Nicolas Copernicus. Copernicus put the sun at the center of the universe, and the earth was merely one of six planets. The order was what we learned in school: Mercury, Venus, Earth, Mars, Jupiter, and Saturn: all the planets known to the ancients, with the moon orbiting the earth as in the old model.copernican-model

Kepler learned of Copernicus’s model at the University in Tubingen, from an experienced teacher of mathematics and astronomy, Magister Michael Maestlin, and immediately favored it over the older model on grounds of being more elegant. Kepler had been a sickly child, but he did well at Latin School, and his parents decided to send him to seminary, so that he might escape labor in the fields, and perhaps bring some prestige to the family. Kepler absorbed theological thought eagerly and well, and was always devout. He ascended in due course through the levels of seminary until he arrived at the University in Tubingen. Although he learned of Copernicus and his model of the planets and their motions, he did not see a copy of the publication itself.

As his theological and philosophical training was proceeding, however, he got a summons from a Protestant seminary in distant Graz, in Styria, present-day Austria. It was to teach mathematics to boys, and taking the teaching position meant that Kepler himself would be giving up the chance to attain priesthood. But Kepler felt that God was calling him to this position, and went to Graz, thus leaving behind a rare, peaceful chapter in his life.europe00

Kepler’s seminary training was firmly Protestant, but in Graz, Protestantism was a distinct minority, and only the school Kepler was summoned to was a Protestant school. The rest were Catholic. This situation simmered for a while, making it difficult for Kepler to fit in. He was not popular as a teacher of his favorite subjects of mathematics and astronomy, but he found greater acceptance as a teacher of history and philosophy. He was tasked with the chore of making yearly calendars, which are more like the Farmer’s Almanac, full of predictions of the weather both meteorological and political. Amid these publications, and amid some travel, teaching ups and down, religious tension, and even a budding romance, Kepler also conceived and wrote his first book, called Mysterium Cosmographicum, “The Mystery of the Universe.”

The idea came to Kepler, in his words, “as though an oracle had spoken to him from heaven,” on July 19, 1595, and it concerned the five so-called Platonic solids. Each three dimensional shape has sides made of perfect triangles, squares, or pentagons, and there are exactly five such shapes. Kepler suddenly realized anew that there were six planets, and therefore there were five gaps between their orbits. He leapt to the conclusion that the sizes of the five gaps were determined by the five geometrical shapes.platonic_solids

“The earth is the measure for all other orbits,” he wrote, “Circumscribe a twelve-sided regular solid about it; the sphere stretched around this will be that of Mars. Let the orbit of Mars be circumscribed by a four-sided solid, the tetrahedron. The sphere which is described about this will be that of Jupiter. Let Jupiter’s orbit be circumscribed by a cube. The sphere described about this will be that of Saturn. Now, place a twenty-sided figure in the orbit of earth. The sphere inscribed in this will be that of Venus. In Venus’s orbit place an octohedron. The sphere inscribed in this will be that of Mercury. There you have the basis for the number of planets.”

Graz to Prague.

Kepler’s courtship and his book proceeded at approximately the same rate, both managing to resolve in the early parts of the year 1597. He married Barbara Mȕller in February and received his first printed copy of Mysterium Cosmographicum a couple of months later. The religious situation in Graz was deteriorating for Protestants, but Kepler always himself stayed above the fray as much as humanly possible. This period also saw his first visit with the formidable Tycho Brahe.

Tycho was a Danish astronomer, the most famous observer of the age, and a stately 53 years of age. He had been the favorite of the Danish king for years, and had amassed a catalog of star and planet positions from his private island observatory. Kepler began to realize that in order to test his idea about the spacings of the planets, he would need to use Tycho’s precise observations. A model of the planets, reasoned Kepler, should fit the sky positions of the planets at all times: the past, the present, and the future. This check sounds reasonable to we modern folk, but such checks were considered secondary in those days. Kepler was being persnickety … and inventing the practice of modern science.

At any rate, Kepler wanted Tycho’s planet positions. And Tycho wanted him to have them: Tycho had been ejected from Denmark after the death of the king, and had taken up residence in Prague. Tycho was impressed by Kepler’s book. Tycho saw promise that Kepler’s brilliant mathematical mind could take his planet numbers and turn them into a predictive model. Their first attempt to arrive at some kind of master-assistant relationship ended without resolution, but then disaster struck Kepler’s Protestant school in Graz. By order of the archduke, all citizens that would not convert to Catholicism were to be expelled from the city. Since Kepler would not so confess, he and his small family were sent packing. Tycho it was who saved Kepler. Tycho provided all the introductions and even some money for Kepler to travel to Prague, and thus it was that Kepler moved again, to the cosmopolitan court of Emperor Rudolph II.

Of key importance for Kepler was the orbit of Mars, because Mars had the most eccentric orbit, except for Mercury, but the observations of Mercury were spotty and inaccurate. Tycho was to pass away within a year of Kepler’s move, but no great collaboration was fostered. For many months after arriving in Prague, Kepler was quite ill, and even after he recovered, the details of how they were to collaborate caused much friction between Kepler and Tycho and some of Tycho’s assistants. But Emperor Rudolph was much taken with Kepler, and upon Tycho’s death, the emperor appointed Kepler his successor. Kepler’s title was “Imperial Mathematician,” and his commanded task was to take Tycho’s observations and produce planet position predictions from them. This list was already named, by Tycho, in honor of the emperor. When completed, the list of future planet positions was to be called the Rudolphine Tables. As it happened, it took Kepler 27 years to complete the tables, but they were, in the end, everything the emperor could have wished: predictions more precise than the old geometric methods by factors of hundreds.

But back in 1601 Kepler had sudden full access to Tycho’s life’s work – and no adequate model of the planets, as Kepler himself quickly realized. Neither the old Greek model nor the more recent Copernican model, both of which were based on steady orbital motion on perfect circles, could quite match the Mars positions that Tycho had patiently collected. Because Kepler was Kepler, he did not give up. He tried different geometric shapes until he tried an egg-shaped orbit. This allowed him to discover what we now call Kepler’s second law for teaching purposes, although it was the first Kepler discovered. Each planet moves more quickly when it is nearer to the sun, and more slowly when it is more distant. Mars’s speed variations amount to about twenty percent. Kepler quantified this by imagining a skinny triangle connecting the sun with two Mars positions a week apart in time. The spatial area of this triangle, think of it as the amount of paint need to paint the triangle, Kepler discovered this area was swept out at a constant rate. The amount of area swept out per week was the same, no matter where the planet was on its orbit.

Soon after the discovery of the equal-area in equal-times law, Kepler modified his egg shaped orbits to ellipses. Ellipses are squashed circles. Today we call the ellipse requirement Kepler’s first law. He found sudden, amazingly precise agreement with Tycho’s planet positions. The excitement of discovery was upon him as he double checked his new method against the other planets. He found precise agreement again, and in due course wrote a book called Astronomia Nova, the New Astronomy. He was not shy about the title: He knew he had cracked the puzzle of the motions of the planets.

To Linz, and the witchcraft trial.

Kepler’s decade in Prague was fruitful for the future of science – he also published a book laying out the foundations of optics with lenses and mirrors – but it also brought frustration and tragedy in good measure. He had enormous financial difficulties despite his lofty title, for the emperor’s treasury was usually empty, and his wife was an aristocrat and demanded a high standard of living during this decade. Barbara and Johannes had five children together, but three died in infancy. Emperor Rudolph, patron of arts and sciences though he was, was inept with matters of security, and abdicated the throne to his brother Ferdinand. Ferdinand was a Catholic, and began the process of counter-reforming Prague, making the city more and more uncomfortable for Kepler. Kepler stayed in Prague for a while at the request of the deposed Rudolph, but when Rudolph died of plague, Kepler arranged to move to Linz, to provide a more congenial environment for his wife Barbara. Alas, as he returned from finding a house in Linz, he found that Barbara was feverish and raving. She shortly died. Her estate did not go to Kepler, but to her daughter from a previous marriage and her two surviving children with Kepler. The grieving widower and his children moved on to Linz after the funeral.

His title in Linz was District Mathematician, though he was still in the direct pay of the Emperor. He was not very welcome in the city. The Lutheran pastor excluded him from taking communion, which grieved Kepler the whole 14 years he stayed there. He published steadily, working toward the grand tables of planetary positions already pre-titled the Rudolphine Tables, but he kept getting distracted, publishing volumes on the history of the calendar, the physics of flotation, and a textbook on his New Astronomy for students and teachers. Too, toward the end of his second year, he remarried. His marriage to Susanna Kepler was much more harmonious than his first marriage, and she cared for his three children as if they were her own. Kepler and Susanna had at least five children together, but only the two girls Cordula and Anna Maria survived to adulthood, leaving  5 out of 14 children born from Barbara and Susanna that lived to adulthood. Even then the oldest stepdaughter Regina died at age 27, a year after a celebrated and joyful marriage.

Unexpectedly, a vexation that was not disease, war, or religion suddenly came to plague Kepler. It was sheer, ugly ignorance and superstition. His mother had been accused of witchcraft. He received the news, he writes, “With unutterable distress, nearly causing my heart to burst out of my body.” The letter came as the year 1615 was waning. Kepler witch2immediately wrote back to Leonburg, a town a few miles from Weil der Stadt where Kepler was born and where his mother now resided. Kepler’s books were making him famous, and his formidable title and reputation made the local tribunal think twice. But the witch trial was only delayed, not stopped. Soon, Kepler was forced to leave his family in Linz and travel to Leonburg to testify at his mother’s trial, where she faced possible torture and death.

The tale is like many from that dark age. An aggrieved former friend accused old Katharina Kepler, a small-framed woman with a sharp tongue, of poisoning her. An additional accusation came from a 12 year old girl who said that the touch of Katharina’s hand upon her upper arm cause it to break out in boils. Suddenly, the tailor was blaming Frau Kepler for the death of his two children, and the butcher said that he had felt pains in his his thighs once, when Frau Kepler had glanced his way. It was a sign of the times. Between 1615 and 1629 in Weil der Stadt, no few than 38 women were executed upon accusations of witchcraft.

The whole story of the trial would fill many hours. The legal wranglings carried on for five full years until they reached a crisis. Katharina Kepler was abducted by night, and stuffed into a chest to keep her quiet during transport to the jail in Leonburg for immediate trial. She stuck to her story of innocence, and the magistrate ordered her scheduled for torture. Kepler was informed of this by letter from his sister.

Kepler being Kepler, this letter arrived at a bad time, namely as his present home town of Linz was being invaded by soldiers of the counter-reformation. Giving up on his profession for the moment, Kepler hastily moved his family to temporary accommodation in Regensburg, then hurried on as fast as he could to intervene on behalf of his mother. He was able to get her better living quarters, and he was able, by virtue of his reputation, make sure that the legal process was followed. A year later, finally, a trial was underway. Kepler’s contributions included oral remarks, and a 128 page written document at the least. The scribe for the prosecution remarked, “The prisoner appears, alas, with the support of her gentleman son, Johannes Kepler, the mathematician.”

The conclusion of the trial was that Frau Kepler be shown the instrument of torture, but not tortured. If the terror of this experience did not cause her to confess, then she was to be declared innocent. And so it came to pass, that Katherina Kepler was brought to the torturers room amid the instruments of torture, but since she maintained her innocence, she was then released. She had spent 14 months in prison. She could not return to her home in Leonburg because of the superstition there. Max Kaspar, Kepler’s biographer, guesses that she went to a town called Heumaden, and about half a year later Katharina Kepler passed away.

The Music of the Spheres.

Kepler’s life was characterized by sorrow, suffering, and death. Yet, along with the string of tragedies, came his great intellectual triumphs. It was the death in 1618 of his little daughter, named Katharina after Kepler’s mother, that turned his mind from the completion of the Rudolphine Tables to a side project that he found suited his mental state much better. It concerned the music of the spheres.

Circa 550 BCE, Pythagoras had made clear that mathematics and music were closely related. The smaller the bell, for example, the higher the pitch. Moreover, the pitches one hears as harmonious are related mathematically, too. The easiest way to visualize this is the string. Pluck half of a string, and you get an octave higher in pitch compared to the full string. Pluck a 2/3 of a string and you get the interval known as a fifth by musicians.

Pythagoras extrapolated from the evident connection between mathematics and the physical universe. Mathematics and music are intimately connected, as was math and physical proportions. So Pythagoras convinced himself there was music in the spheres, meaning the planets, stars, sun, and moon in their slow, predictable motions, must in some sense harmonize with each other and with Nature on earth.

Kepler had gotten a copy of the book named Harmony by Claudius Ptolemy in the original Greek and applied his own imagination and mathematical expertise to his New Astronomy, his elliptical orbits and equal-area in equal-time laws. It wasn’t long before certain patterns emerged that resembled musical geometry. For example, Kepler looked at the ratio of the speeds of the planets at closest approach (perihelion) and farthest approach (aphelion) to the sun. He found something that resembled some of the intervals of the musical scale.


He also discovered what we now call Kepler’s third law.


This clean mathematical relationship between the distance that a planet is from the sun and how long it takes to orbit the sun is very impressive. Today, it is used to find the masses of distant planets, stars, black holes, and galaxies. Kepler’s book on the topic is called Harmonice Mundi, the Harmony of the Worlds.

Kepler’s journeys were not finished. War forced him from Linz, and his last years see him alight in Ulm, Sagan, and Regensburg. He did manage, at long last, to publish the Rudolphine Tables, a testament to Tycho’s observations and Emperor Rudolph’s vision as well as Kepler’s genius and doggedness. He was still busy traveling and publishing when he caught a fever and died in 1630. His wife Susanna, with ever-decreasing means, died in poverty in 1638. It is known that three of his children, at least, had children. Descendants of Kepler and Barbara and Susanna live among us today.

Kepler’s intellectual legacy lives on, too. It fueled Galileo’s zeal for the Copernican system, and it inspired Isaac Newton to solve the motion of planets using basic physical principles. Furthermore, it marks the first application of modern science, namely that one requires one’s model to fit the observations. And one throws away all models that do not fit. Few among us have been forced to toss away as many cherished notions as Kepler did. This continued winnowing of ideas leaves us with what works, and discards that which does not work. It is the engine of science.

-Based on a planetarium script written by Guy Worthey. The primary source is Max Kaspar’s biography about Kepler.

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