At Gratz, in the course of his lectures Kepler began to wonder why there were only six planets in the Copernican system: Mercury, Venus, the earth, Mars, Jupiter and Saturn. Then on July 9, 1595, as he drew a geometric figure on the blackboard, he remembered that Euclid proved that there exists only five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and the icosahedron. The Platonic solids are regular figures having identical faces. The tetrahedron has four faces which are identical equilateral triangles; the cube has six faces which are identical squares; the octahedron has eight faces of identical equilateral triangles; dodecahedron has twelve faces of identical equilateral pentagons; and finally the icosahedron has twenty faces of identical equilateral triangles. Euclid had proven that there are no other regular solid geometrical figures. Kepler noticed that the radii of the spheres, when they are successively inscribed and circumscribed about the five solids, are nearly proportional to the planetary distances in the Copernican system, if they are arranged concentrically. That is, being perfectly symmetrical, each of these regular solids can be inscribed into a sphere, so that all there corners (vertices) lie on the surface of the sphere; similarly, each can be circumscribed around a sphere, so that the sphere touches the center of each face. Now Kepler applied this geometrical arrangement to the planets. Into the orbit, or sphere, of Saturn, in his imagination, he inscribed a cube; and into that cube another sphere, which was that of Jupiter. Into the sphere of Jupiter he inscribed a tetrahedron, and in it he inscribed the sphere of Mars. Between the spheres of Mars and the earth he inscribed a dodecahedron; between the spheres of the earth and Venus the icosahedron; between Venus and Mercury the octahedron. And Behold! We have discovered the mystery of the universe. The radii of these spheres are proportional to the planetary distances from the sun at the center of spheres, almost. Kepler thought that he had found a bit of the Divine order of the world. Of course Kepler was mistaken; the orbit of the planets are not circles lying in the crystalline spheres, as he will later discover. But this idea became the subject of his first book, Mysterium Cosmographicum, or Cosmic Mystery, which he feverishly worked on for the next six months. In the first part of the book, he builds a philosophical case for his idea, and in the second he looks for scientific verification of it. He says at the start of part two:
"What we have so far said served merely to support our thesis by arguments of probability.Kepler begins his task of confirmation by checking the propositions of his model against the observed data. He found that the planets do not revolve about the sun in circles but in oval-shaped orbits (which years later he found to be ellipses), the planet's distance from the sun varies within limits. This variation (or eccentricity) he accounted for by allotting to each planet a spherical shell of sufficiently thickness to accommodate the oval orbit between its walls. The inner wall represents the planet's minimum distance from the sun and outer wall represents its maximum distance. Kepler does not think of these spherical shell as physically real, but as mathematical limits of the orbits. The thickness of each shell and intervals between them, were laid down in Copernicus' data. Using this data, he could not get the orbits to fit. There was fairly good agreement for the orbits of Mars, Earth, and Venus, but not for Jupiter and Mercury. In the following chapters Kepler tries various different methods to explain the discrepancies. And he had problems what to do with the orbit of moon. After twenty chapters, Kepler persuades himself that the model fits the data, and the existing discrepancies are due to Copernicus' faulty data.
Now we shall proceed to the astronomical determination of the orbits and to geometrical considerations. If these do not confirm the thesis, then all our previous efforts have doubtless been in vain."
By February, 1596, the rough draft was finished and Kepler ask his superiors for a leave of absence to visit his native Wuertemburg and make arrangements for it publication. He asked for two months and stayed away for seven. During this time Kepler married Barbara Mechleck, the daughter of a rich miller, on April 27, 1597. After many difficulties, he published his book at the age of twenty-six in the spring of 1597.
As wrong as this book was, its attempt to explain why there are only six planets eventually lead to his discovery of the three laws of planetary motion. In fact this book, which he distributed freely and widely, was sent to Tycho Brahe and lead to Kepler working for Tycho and having access to Tycho's data. In 1598 Tycho wrote to Kepler and invited him to join him. The last year of the century, his last year at Gratz, was a precarious one for Kepler. The young Roman Catholic Archduke Ferdinand of Hapsburg (later Holy Roman Emperor Ferdinand II) was determined to cleanse the Austrian provinces of the Lutheran heresy. In the summer of 1598 Kepler's school was closed down, and in September all Lutheran preachers and schoolmasters were ordered to leave the Province of Styria within eight days or forfeit their lives. Only one was allowed to return and that was Kepler. His exile lasted less than one month. Within a few years the people of the province were given the choice of either becoming Roman Catholics or leaving the province. Since his school was closed, Kepler was trying desperately to find a new position. He begged his old university of Tuebingen for a post, any position whatever. But nothing was offered. He wrote his acquaintances all over Germany, but nothing was found. It was as if he was being forced into the employ of Tycho Brahe. Tycho in the meantime was appointed Imperial Mathematicus by Rudolph II. But Kepler could not wait for word from Tycho. When a certain Baron Hoffman, Councilor to the Emperor, had to return from Gratz to Prague, Kepler asked him if he could travel with him, and he agreed and they left on January 1, 1600. On February 4, 1600, Tycho and Kepler met face to face at the castle Benatek, near Prague. He became Tycho's assistant. Tycho assigned him to work on the observations of Mars. A year and half later Tycho died on October 24, 1601, and a few days later on November 6, 1601, Kepler was appointed to Tycho's position of Imperial Mathematicus. Immediately Tycho's assistants and relatives began to squabble over the division of Tycho's wealth and the expensive equipment. In the confusion, Kepler quietly made off with the data.
Kepler settled down to work, free from the stress from arguments with Tycho, to fulfill Tycho's last wish: that Kepler use the data to prove Tychonic system of heavens. Kepler had boasted to Tycho that he could solve the orbit of Mars in eight days. It actually took him eight years; eight years of frustration because the orbit of Mars would not yield to the data. Kepler came to refer to his struggle with the data as his "war on Mars." Kepler finally won his war on Mars; and in 1609 he published a detailed chronicle of the war called Astronomia Nova -- A New Astronomy or the Physics of the Skies. It was difficult to read, because he recorded every blind lead, every false start, every speculation and all the computations which he made for eight years. It made his reputation as an astronomer but not his popularity with the non-professionals.
Kepler was an ardent supporter of the heliocentric theory, worshipped Copernicus, and credited him with a "stoical firmness" which he did not have. When he discovered that Osiander had written the Preface to The Revolutions, he rejected Osiander's hypothetical treatment of Copernican system. He says at the beginning of his book, Astronomia Nova:
"It is a most absurd fiction, I admit, that the phenomena of nature can be explained by false causes. But this fiction is not in Copernicus. He thought that his hypotheses were true, no less than did those ancient astronomers of whom you speak. And he did not merely think so, but he proves that they are true. As evidence, I offer this work. Do you wish to know the author of this fiction, which stirs you to such wrath? Andreas Osiander is named in my copy, in the handwriting of Jerome Schrether, of Nuremberg, Andreas, who supervised the printing of Copernicus' work, regarded the Preface, which you declare to be absurd, as most prudent (as can be inferred from his letter to Copernicus) and placed it on the title page of the book when Copernicus was either already dead or certainly unaware [of what Osiander was doing]."Kepler here may be referring to the change of the title of Copernicus' book reported by the Johannes Praetorius, Professor of Mathematics, who was an intimate friend of Rheticus, in a letter to a correspondent.
"The title also was changed from the original beyond the author's intentions, for it should have been: De revolutionibus orbium mundi, whereas Osiander made it: Orbium coelestium."As a preliminary, Kepler made three revolutionary innovations, simplifying the heliocentric system, by eliminating some confusing and complicating assumptions:
(1) The sun was the center of the circular planets' orbits, not only physically but geometrically,The first innovation was to shift the geometrical center of the circular orbits of the planet to the sun, making the distances and positions of the planets relative to the sun. With this innovation Kepler simplified the calculations by making all the planetary distances with reference to the sun, not some geometrical point as the center of the orbit. Kepler here departs from Copernicus, who placed the geometrical center of the planets' orbits not at the sun, the physical center of cosmos.
(2) the orbits of the planets lie very nearly, but not entirely, in the same plane,
(3) the speed of the planets traveling about their orbits is not uniform (not constant).
The second innovation embodied the recognition that all the planes of the orbits of the planets did not lie in the same plane, or in the plane of the earth called the plane of ecliptic, but that the plane of their orbits are inclined with respect to the plane of the earth's orbit by very small angles. Since Mars, for example, was observed sometimes above the circle of ecliptic, sometimes below it, Kepler concluded that the plane of the orbit Mars was, not in the earth's orbital plane (the plane of ecliptic) as Copernicus assumed, but had its own plane centered on the sun and not on the earth. This meant that the angle between the planes of the orbit of Mars and of the orbit of the earth was fixed, not oscillating, as Copernicus postulated. Using Tycho's data he set about to find this angle between the planes of Mars and of the earth. And he found the angle to be 1 degree and 50 minutes of arc. He was delighted.
The third innovation was the most radical. The Copernican view had made the traditional assumption that the motion of the planets was circular and uniform. Kepler discarded the uniform speed. (For the time being, he kept the circular motion; although by the end of his book that would be discarded also.) This meant that the epicycles, which Copernicus had introduced to save the uniform motion, could also be discarded. Instead eccentrics he reverted to the equants as a calculating device.
This simplified theory, containing these three innovations, had to be tested. We shall not go through his calculations of the orbit of Mars. After using specially rare observations from Tycho's data, Kepler found that the observed position of Mars and the calculated position according to the simplified theory differed by eight minutes of the arc. This was catastrophic. Kepler at this point modified his theory and recalculated the position but that just made matters worst; he finally threw out his simplified theory and started over. Now at this point Kepler suspected that the cause of the discrepancy of eight minutes of the arc was due to the circular orbits and the shape of the planetary orbits had to be some other geometrical closed curve, instead of the circle. Instead of choosing some geometrical curve and computing the planetary position according to this curve, Kepler makes an immense detour. From the Tychonic data he will determine the shape of the curve. Now for a circle that was simple; he only needed three points on its circumference to establish the circle. But for any other shaped curve more points would be needed. The task before Kepler then was to construct the orbit of Mars, without any preconceived ideas regarding its shape, from the raw observations. This was no simple task. In his book in Book III, Kepler goes through the laborious steps of this task, solving the preliminary problem of the motion of the earth. He found what he had expected; the earth, like the other planets, did not revolve about the sun at a uniform or constant speed. And more importantly, he found that at the extremes of its orbit, the aphelion and the perihelion, the earth's speed was proportional to its distance from the sun. He generalizes this observation that the speed of earth, and all the other planets also, in its orbit varies inversely as its distance from the sun. This theorem was not true; his extension of the relation of the speed of the planet to the distance from sun at the extremes of its orbit, to all points in its orbit was an incorrect generalization. Kepler knew this and admitted at the end of chapter 32. But this theorem leads him into a side track.
Before he gets to the solution of his primary problem of the shape of the orbit of Mars, at the beginning of Book III in chapter 33, Kepler goes off on a tangent, into what he calls Himmelsphysik, or celestial physics, and in the next six chapter he tries to answer the question: what force moves the sun? Since he had no concept of momentum to keep the body in motion and vague intuition what is gravity which bends the orbit into a closed path, he has to invent two new kinds of forces to account for the orbit motion. A force that causes the orbital motion, like a broom sweeps the planet around its orbit and an opposing force, a "laziness" that resists the sweeping force. The sweeping force he explains as like a vortex (like Descartes' explanation) or as a magnetic force (like Gilbert's suggestion). He assumes that this sweeping force diminishes by distance between the sun and the planet, instead of the square of distance. He sensed that there was something wrong here, because he knew that light diminished by the square of the distance. But he was stuck with it because his theorem on the inverse ratio of speed to distance was equally in error.
After this excursion into celestial physics, Kepler returns to his investigation of the motion of the planets. He now derives the law that has come to be known as Kepler's Second Law of planetary motion: an imaginary line from the planet to sun sweeps out equal areas in equal times. But Kepler believed that he had made a second error of assuming that the orbit of the earth was circular, and that it had canceled out his first error in his theorem on the inverse ratio of speed to distance. Even though he derived this law assuming that the orbit of the earth is a circle, the law does hold true for a circular orbit, and in fact for any closed orbit. Kepler was thus wrong about thinking that a second error had canceled his first error. The first error led him to consider the imaginary line from the planet to the sun sweeping out equal areas in equal times. Even though his derivation and proof of this second law was faulty, the law was true. It does correctly described the variations in the speed of the planets in their orbits.
But this law did not describe the shape of their orbits. In the Fourth Book he resumes the search for the shape of the orbit of Mars. This research took about two years and occupies chapters 41 to 60 of His New Astronomy. We will not recount his attempt to impose the egg shaped oval path on the planets. He had derived the formula of the curve but he did not recognize it. It was an ellipse. The orbits of the planets are ellipses and the sun is at one of the foci of the elliptical orbit of a planet. This is now known as Kepler's First Law of planetary motion. The circle is a special case of an ellipse, where the two foci coincide at the center of ellipse; the circle is a reduced ellipse. The orbits of some planets, namely the earth, are almost a circle. Now the planetary motion could be calculated exactly for any time in the past or the future. But more importantly, the discovery of these two laws of planetary motion by Kepler is the confirmation of the heliocentric view of the planetary motion. They at last eliminate the ancient assumption that planetary motion must be circular and uniform (constant speed). Kepler's system was heliocentric like Copernicus' system, but it was heliocentric in a new way: a new method of formulating its laws with new assumptions about planetary motions. Kepler's book The New Astronomy is the watershed in the history of astronomy; it is the end of the ancient Ptolemaic geocentric view of the universe and it is the beginning of the true heliocentric view of the planetary system.
It took Kepler six years to do the writing of the New Astronomy; the writing was finished in 1608 and the printing in 1609. Kepler's contemporaries did not appreciate what Kepler had accomplished; they were too controlled by the ancient assumptions of uniform and circular motion of the planets, even when they had accepted the heliocentric system of Copernicus. Galileo, far example, rejected Kepler's three laws as impossible and ignored Kepler and his work. Even Kepler's patrons and well-wishers could not comprehend what he was up to. The first to comprehend the significance and implications of what Kepler had done, were not the Germans, nor the Italians, but the English: the traveler Edmund Bruce, the mathematician Thomas Harriot who tutored Sir Walter Raleigh, the Reverend John Donne, and finally Isaac Newton.
Kepler came very close to discovering Newton's explanation of the motion of the planets described in Kepler's three laws, the Law of Universal Gravitation. In the introduction to New Astronomy in a passage where he sets about to demolish the Aristotelian doctrine that bodies are by nature "heavy" because they contain the element earth and are striving to move toward its place at the center of the world, whereas other bodies are "light" because they contain those elements whose place is not at the center of the world and thus are striving to move away from the center of the world, in conclusion he writes:
"It is therefore clear that the traditional doctrine about gravity is erroneous.... Gravity is the mutual bodily tendency between cognoate [i.e. material] bodies towards unity or contact (of which kind the magnetic force also is), so that the earth draws a stone much more that the stone draws the earth.... Supposing that the earth were in the centre of the world, heavy bodies would be attracted to it, not because it is in the centre, but because it is a cognate [material] body. It follows that regardless where we place the earth... heavy bodies will always seek it.... If two stones were placed anywhere in space near to each other, and outside the reach of force of a third body cognate body, then they would come together, after the manner of magnetic bodies, at an intermediate point, each approaching the other in proportion to the other's mass. If the earth and moon were not kept in their respective orbits by spiritual or some other equivalent force, the earth would ascend towards the moon fifty-fourth part of the distance, and the moon would descend the remaining fifty-three parts of the interval, and thus they would unite. But the calculation presupposes that both bodies are of the same density. If the earth ceased to attract the waters of the sea, the seas would rise and flow into the moon... If the attractive force of the moon reaches down in the earth, it follows that the attractive force of the earth, all the more, extends to the moon and even farther.... Nothing made of earthy substance is absolutely light; but matter which is less dense, either by nature or through heat, is relatively lighter..."In this remarkable passage, Kepler gives the first correct explanation of the tides as a motion of the waters "towards the region where the moon stands in the zenith". In a later work (the Somnium) he explained the tides, not by the attraction of the moon alone, but of the moon and of the sun combined; he thus realized that the attraction of sun reached as far as the earth.
Even though Kepler here recognizes correctly that the force of gravity is a mutual attraction between two bodies and that that mutual attraction is proportional to the attracting masses, he makes the wrong assumption, expressed in his later excursion into celestial physics, that the sun's force of gravity diminishes in ratio to the distance between the bodies. So that when he tried to work out the mechanics of the solar system, these insights get lost again in confusion. Even the concept of gravity as attracting force gets replaced, in the celestial physics section, with the concept of force of the sun like a broom, sweeping the planets around their paths. And since the sun causes all motion, the sun handles the broom. And this required that the sun rotates on its axis - a guess that was confirmed later by the discovery of the sunspots - and the force radiated out from the sun, like the spokes of a wheel, and swept the planets along. This means that all the planets would have the same angular velocity about the sun, and therefore would all have the same orbital period, which they do not have. To explain why they do not all have the same period, Kepler introduces the concept of laziness, or "inertia" of the planets, who desire to stay in the same place, resisting the sweeping force. The sweeping brooms of force are not rigid, but are flexible, allowing the planets to lag behind. This wrong concept of inertia, as tendency to stay in the same place, will be later corrected by Newton, as the tendency to stay in the same state of motion when the net force on body is zero, that is, if at rest, stay at rest, if in uniform motion, to continue in uniform motion.
When Galileo made his telescopic observations and published them in Venice during March, 1610, in a short booklet entitled, Sidereus Nuncius, The Messenger of the Stars, Kepler, when he received a copy from Galileo on April 8, although he did not have a telescope, accepted Galileo's claims on trust and wrote by April 19th a short pamphlet in the form of open letter entitled Conversation with the Star Messenger. It was published in Prague in May, 1610, and in Florence shortly afterward. And it was what Galileo needed and turned the tide of astronomical opinion in his favor. Kepler's authority as the Imperial Mathematicus and as the first astronomer of Europe was uncontested. Although Galileo boasted about Kepler's letter privately, he neither thanked Kepler or publicly acknowledged it. Using a telescope that was lent to him, from August 3 to September 9, 1610, Kepler was able to see with his own eyes Jupiter's moons and he published a short pamphlet, entitled Observation-Report on Jupiter's Four Wandering Satellites, in which Kepler confirms with first hand experience Galileo's discoveries. Also this is the first appearance in print of term "satellite", which Kepler had coined in the previous letter to Galileo.
In 1604 Kepler published a work entitled Optics in which he showed that the intensity of light diminish by the square of the distance from the source. During August and September, 1610, Kepler had use of a borrowed telescope. Within a few weeks he wrote a theoretical treatise in which he laid the foundation of a new science of the refraction of light by lenses and coined a name for it: dioptrics. Although he did not discover the law of refraction he was able to develop a system of geometrical and instrumental optics and deduce the principles of the Astronomical or Keplarian Telescope. In his book, Dioptrics, published in 1611, Kepler explained the operation of the telescope which Galileo seems never able to do.
Kepler formulated his Third Law of planetary motion in a book entitled, Harmony of the World, which was finished in 1618 and published in 1619, when he was forty-eight. The first two laws describes the properties of the orbits of each planet; the third law gives the relation between the orbits of the planets: it gives a relation between a planet's period and its mean distance from the sun. Kepler's Third Law of planetary motion says that the square of the time of revolution of planet about the sun is directly proportional to the cube of the mean distance of the planet from the sun. To use Kepler's example, let the distance of the earth from the sun be our unit of measure, now called the Astronomical Unit (AU), and the length of the earth's year, the time it takes the earth to travel once around the sun, as our unit of a planetary period. For the earth the cube of 1 is 1 and the square root of 1 is 1, therefore the ratio of square of period of the earth to the cube of its distance from the sun, that is, the constant of proportionality, is 1 year squared per AU cubed. Now the mean distance of the planet Saturn from the sun is a little over nine units, 9 AU. Since the cube of 9 is 729 and square root 729 is 27, the period of Saturn is 27 years; actually it is 29.46 years at a mean distance of 9.54 AU. This Third Law is what Kepler had been searching for twenty-two years. As he formulated the problem in his Cosmic Mystery, if the sun governs the planets' motion, then their motion must somehow depends on their distance from the sun; but how? Kepler was the first to see the problem and after twenty-two years of searching to find its solution. This is was what Kepler thought that he had found in the five Pythagorean solids and published in his Cosmic Mystery in 1597. Kepler's Harmony of the Worlds is the continuation of his Cosmic Mystery, and the climax of his lifelong obsession. Kepler attempts here to lay bare the ultimate secret of the universe, a Theory Of Everything.
This third law like the first two are based on the observations that was made by Tycho Brahe. But Kepler was not able to explain the their raison d'etre. He could not see how they fit together. He seems to be almost embarrassed with the elliptical orbits of the planets. Why did God choose that shape for the planet's orbit? The Second law he regarded as merely a calculating device, and constantly avoided using it, relying on faulty approximations. The Third law is merely a link to the harmonics, and nothing more. Without the Law of Universal Gravitation and the Calculus, how could he appreciate and explain his three laws? Later Isaac Newton with his Law of Universal Gravitation will provide that explanation.
In remaining eleven years of his life he continued to pour out a torrent of books, pamphlets, annual calendars and ephenerides, books on comets and on logarithms, and two more major works: the Epitome Astronomiae Copernicanae in 1621 and the Rudolphine Tables in December of 1623. The title of Epitome is misleading; it is not an epitome of Copernican astronomy, but of an epitome of Keplarian astronomy. It is based on his three empirical laws of planetary motion, not on a metaphysical principle of circular and uniform motion. The first two laws originally referred to Mars only, Kepler here extends them to all the planets, to the moon and to the satellites of Jupiter. Gone are the epicycles, deferents, eccentrics and equants of the Ptolemaic system. It was his most voluminous work, a systematic exposition and textbook of astronomy, since Ptolemy's Almagest. The Rudolphine Tables was his crowning achievement in practical astronomy, based on Tycho's lifelong labors. The work was long delayed for nearly thirty years by Tycho's death, quarrel with his heirs and the chaotic conditions caused by war and most of all by Kepler's own reluctance to undertake the huge task. Astronomers, calendar-makers and horoscope-casters, impatiently waiting for the long promised tables, were making angry demands. Finally in 1623 he started the task which he only dabbled at for 23 years. It took four years to get it printed owing to lack of money and the chaos of the Thirty Years War. At last in September of 1627 the work was finished. It remained the indispensable tool for the study of heavens, of the planets as well as of the fixed stars, for over hundred years.
He attempted to write a "science fiction" story called Somnium, a dream about a journey to the moon; he did not finish it and the fragment was published posthumously in 1634. It was the first science fiction story in the modern sense of the word. During the last three years Kepler wandered about southern Germany and died at Ratisbon on November 15, 1630.