
Before we're done with this program we will have seen how a combination of events, both human and astronomical, took Europe into the seventeenth century full of turmoil, recoiling from seventeen hundred years of oppression and forced thought. At the time of Bruno's death at the stake for heresy, for considering the implications of heliocentrism, a working arrangement between Tycho Brahe and Johannes Kepler was beginning which would last for only one year, but which would mark the beginning of the end of geocentrism and start the real Copernican revolution rolling.
Here are the objectives for today's lesson. These objectives are also in the study guide at the beginning of the lesson. Before you begin to study the lesson, take a few minutes to read the objectives and the study questions for this lesson. Look for key words and ideas as you read. Use the study guide and follow it as you watch the program. Some students find it helpful to make a note in the margin which pertains to a particular objective or a study question. Focusing on the learning objectives will help you to study and understand the important concepts. Compare the objectives with the study questions for the lesson to be sure that you have the concepts under control.
The sixteenth century is sometimes called the age of wonder, and for good reason. The rate of sociological change was beginning to increase to the extent that new ideas flourished as explorations and conquests of the new world brought back empire building loot. Magellan sailed around the world, the Protestant reformation cut further into the authority of the Church, and the leadership of Elizabeth the first elevated England to world power in the new Protestant world. The writings of Shakespeare in England and the work of Michelangelo in Italy carried the world of literature and art forward. The printing press, invented around 1450, had created an information explosion, not unlike the computer revolution of our own time. It is interesting to note that some of the critiques of the printing press included a number of ideas that would sound familiar to us in the era of television and computers.
"It will destroy the art of conversation because people will spend too much time reading."
"There will be so many things to read. How will we know what we should read?"
"There will be so many books, how can anyone read them all?"
Political, social, and economic ideas were changing, and astronomy became a precision science.
For the first time the positions of the planets could be measured to a high degree of accuracy, and had been tracked over long periods. The result was that it became quickly apparent that neither the Ptolemaic nor the Copernican system did a very good job of predicting the planetary locations.
What do we mean a by a very good job? The error in the location of Mars was very small by anyone's standards. About the width of a quarter two thousand feet away. This was well within the accuracy of Brahe's instruments.
1.1. Sixteenth century is also called "The Age of Wonder"
1.1.1. new ideas flourished
1.1.2. explorations and conquests of the new world brought loot to build and sustain empires
1.1.3. Magellan's crew sailed around the world for the first time
1.1.4. Protestant Reformation eroded Church authority
1.1.5. England became a world power
1.1.6. Art and literature flourished1.1.6.1. Shakespeard
1.1.6.2. Michaelangelo1.2. Information explosion as printing press came into common usage
1.2.1. Critics claimed it would destroy society
1.2.1.1. will destroy the art of conversation
1.2.1.2. will create too many books
1.2.1.3. will not know what to read1.3. Instruments allowed astronomy to be a precise science
1.3.1. neither Copernican nor Ptolemaic system worked very well
Tycho Brahe [1546  1601].
One of the more colorful characters in the history of science, Brahe is noteworthy for two main reasons. First, he designed and built an astronomical observatory and sighting instruments. Second, he gave his accurate data on the movements of the planets to Johannes Kepler who bent the circular paradigm and used them to determine the true orbits of the planets.
Brahe became the first professional astronomer when he persuaded Frederick II to fund the building and maintenance of an observatory on Ven, off Denmark. This island is so desolate that Denmark and Sweden each have claimed for centuries that the other owns it.
It was not a great place if you like the night life and excitement of the city, but if you like the night life of the sky, it's perfect.
Remote, dark and desolate, the perfect place for astronomy. Here, night after night for twenty years, Brahe and his assistants tracked the motion of the planets throughout their orbits. No one had ever spent this much time looking at the planets.
Brahe's goal was to come up with detailed tables of the locations of the planets which could then be used to bring the old Ptolemaic tables, as modified by Copernicus up to date.
2.1.1. to king of Denmark
2.1.2. built Royal Observatory in 1576
2.1.3. 20 years of detailed observations
Brahe's sarcastic manner and brash personality made him quite persuasive on one hand, but on the other he made some powerful enemies. His metal replacement nose was necessary to replace the original nose which he lost (or rather had accidentally removed) during a fight. One story says it was sliced off with a sword which narrowly missed sending him to an early grave. Another account claims it was bitten off in a fight over some inconsequential thing.
Whatever the nose story, Brahe finally left Denmark after the King's death. His rights on Ven and his pension were withdrawn by Frederick's son, Christian IV, and he left Ven in 1597.
2.2.1. made powerful enemies
2.2.2. metal replacement nose
2.2.3. exiled to Bohemia in 1597
Brahe was not much of a mathematician, but he was a great instrument designer. His observatory and his instruments made it possible to keep detailed records of the locations of the planets on a full time basis.
2.3.1. accurate to 4' (1/15 degree)
The instruments he designed were accurate to four minutes of arc. That's about onefifteenth of a degree, or about the size of a quarter dollar three quarters of a mile away. No one had ever made observations of the heavens with anything near that accuracy before.
2.3.2. tracked planets through orbit, not only at certain locations
Not only that, but no one had ever tracked the planets completely through their orbits on a nightly basis before.
All of the previous observations, including those of Hipparchus and Ptolemy, had been made sporadically.
So all of the work of the centuries developing tables of motion, theories of astrological mysticism, religious dogma, etc. had been based on naked eye observations taken sporadically.
It's a wonder that there were any theories at all about the heavenly motion, let alone that they worked as well as they did. Remember that the conjunction of Jupiter and Saturn in 1524 occurred a full month earlier than predicted by the Ptolemaic theory, but that's still not a bad prediction. Copernican calculations were only a few days different, and that was based on the old observations of unknown origin.
If you don't agree that this is pretty good, try to predict such an event for yourself.
2.3.3. roomsized sighting tubes and pointers
Kepler's instruments used the circumference of large circles to precisely pinpoint the location of a star or planet in both the horizontal and vertical planes.
The video program shows how twos protractor, one horizontal and one vertical, can be used to locate the coordinates of a planet or star.
Because all circles enclose the same measure of three hundred and sixty degrees, the size of a degree on the circumference gets larger as the circle gets larger. That's why you can cover the moon with your thumb, even though it's nearly three thousand miles across.
The size of the arc of a circle for any given degree measurement is larger when the circle is larger (this must be true because we know that the circumference of a circle, which inscribes 360 degrees depends on the diameter of the circle.)
2.3.3.1. chair quadrant
This instrument, one of the earliest in the observatory allowed Brahe to sit on a short stairway and observe a planet through a hole in the wall of the observatory. An assistant could then line up a sliding marker on the protractor scale and read the angle of altitude of the planet.
This quadrant did not have the accuracy of the later sighting tubes.
2.3.3.2. large quadrant
This sighting tube was six feet long. It sat on a pedestal which was recessed into the floor. Brahe stood or sat on the steps while sighting through the tube. Once a particular star or planet was centered in the tube, its altitude above the horizon was read off of the protractor scale. At the same time its orientation from north was noted on a horizontal scale attached to the pedestal. The date, time, and coordinates of the heavenly object were logged into a table for each observation. The speed and accuracy of this and other instruments allowed Brahe to take many measurements in a night, sometimes only a few minutes apart.
Don't forget the the earth's rotation carries a star or planet westward with the hours. Absolute locations in the sky must compensate for the diurnal movement.
In 1599 the emperor Rudolph II offered him the position of imperial mathematician of the Holy Roman Empire at his court in Prague. He gladly accepted, and in 1600, the same year that Bruno was burned at the stake for his heretical speculations, Brahe hired a young and promising mathematician, Johannes Kepler to be his assistant. He had read some of Kepler's publications and was impressed with his mathematical talents. They worked together for a year, and Brahe grew even more impressed with his young assistant's talents and work ethic. Upon Brahe's death on Oct. 24, 1601, Kepler became his successor and inherited his large collection of astronomical observations and data and equipment.
We'll learn more about Kepler in a few minutes.
2.4.1. Year of Bruno's death
2.4.2. Worked together for a year
2.4.3. Kepler impressed Brahe, got his data and his job
Another interesting and somewhat relevant contribution made by Tycho Brahe is his geoheliocentric model. That's right geoheliocentric, or I guess you could say heliogeocentric.
The absence of stellar parallax, that annual shifting of the star background as the earth moves in its orbit, prevented Brahe from accepting heliocentrism outright. However, the motions of the planets fit much more nicely into a heliocentric model.
Brahe's solution to the problem is a creative way of saving the appearances while still preserving circular motion.
2.5.1. recognition of failure of Ptolemaic model
In addition to the creativity employed by Brahe in visualizing this model is that it points to a general recognition by Brahe's time of the failure of the Ptolemaic model. No one, at least not yet, was ready to give up the stationary earth because that would contradict Aristotle's principles as they had been canonized in the Scholastic philosophy.
2.5.2. spherical, Earthcentered
Brahe's universe was spherical and earthcentered, thereby preserving the integrity and sanctity of the ageold circular paradigm and geocentrism.
2.5.3. sun & moon revolve around Earth
In the Tychonic universe, the earth is the center of all motion, and remains stationary as the sun and moon revolve around it. Meanwhile, the other five planets revolve around the sun as it goes around the earth.
Another way to say this is that the sun's orbit becomes the deferent, and the other planets do epicyclical loops around it. Do you see how this combines the best of both worlds. It is really quite clever, and works much better than Ptolemy's system.
Considering the seeming sensitivity of the Church on these matters, one would think that it would have caused quite a stir.
2.5.3.1. other planets revolve around sun
Not so. In fact astronomers of Tycho's time not only accepted the Tychonic system, many of them actually favored it because it made the calculations easy, it agreed with the observations quite closely, and it did not require a shift in paradigm quite as radical as a pure Copernican system would have.
2.5.4.1. several versions, some have Earth rotating
2.5.4.2. used routinely to produce tables
So here's a brief review of the contributions of Tycho Brahe to the growing and maturing river of our scientific heritage.
First professional astronomer
Built observatory
Designed and constructed precision astronomical instruments
Made regular and continued measurements of planetary motions
Created Tychonic system
Gave data to Kepler
Now let's focus for awhile on the times. No, not the New York Times!
Besides the rapid changes which had already taken place earlier in the century, several astronomical events occurred just at the time when Tycho Brahe was pointing his new instruments to the sky. As much as Brahe's instruments set the stage for Kepler's discovery of the harmony of the planets, his use of the instruments to challenge the sanctity of the heavens was just as important.
Since Plato's time Western philosophers had held that the heavens were perfect. From Aristotle came the idea that perfection was unchanging, except for the heavenly motions which were regular and perfect.
3.1.1. Novae in 1572, 1604, Comet in 1576
A nova in 1572 was recorded by Brahe at Uraniborg. His precise instruments allowed him to observe the nova over a period of time. From these measurements he was able to prove without a doubt that this "new star" was definitely cosmic and not sublunar. A comet in 1576 was likewise shown to be outside the moon's orbit and therefore behaving in a very non heavenly manner.
These were not the first novae and comets to be observed, but prior observations had been qualitative rather than quantitative. They were assumed to be sublunar, like clouds and meteors because everyone knows that anything that changes in the heavens must be sublunar.
3.1.1.1. Brahe's precision showed them to be Heavenly
3.1.1.2. such events reported earlier
3.1.1.3. observations were unreliable
3.1.1.4. thought to be sublunar3.1.2. Challenge to Paradigm
Brahe's proof, published in 1573 in a booklet called, appropriately De Nova Stella, or The New Star, presented a serious challenge to the concept of heavenly perfection.
Luckily for Brahe, the Protestant reformation in Central and Northern Europe had taken much of the wind out of the inquisition which was effective in preventing the spread of ideas in southern Europe.
More important than the challenge to the paradigm, was the possibility, that if the heavens were not perfect, then maybe they were also not circular.
3.1.2.1. change in Heavens ==> imperfection
3.1.2.2. imperfection ==> maybe not circular
It is not clear when the telescope was invented, or who invented it. It is said that the basic principle was fairly well known in Holland in the early 1600s.
3.2.1. 1608  earliest record of use
The earliest recorded use of the telescope was in 1608.
3.2.2. 1609  Harriot (English) looked at the moon
In 1609 an Englishman named Harriot looked at the moon, saw a few spots, and said, "that's cool", or some seventeenth century equivalent, made a notation in his journal, and that was that. He made no further observations, or at least no further records of them.
3.2.3. 1609  Galileo saw the heavens
In the same year Galileo learned of the new instrument from a trader and began experimenting to build his own. The earliest was about ten power. Within a few months he had increased the power to about thirty. Here is a wonderful example of the importance of the individual in the growth of knowledge. Harriot saw the moon up close and thought little of it.
3.2.3.1. saw many new things
Galileo was so fascinated by what he saw that he looked again and again, discovering more and more evidence that convinced him that the geocentric concept could not be correct. He saw that Aristotle and Ptolemy were both wrong, The scholastic philosophy was wrong.
Galileo saw the same moon that everyone else had always seen but instead of a heavenly object made of ethereal matter, he saw one that was earthlike, with craters and mountains.
3.2.3.1.1. craters and mountains on the moon
Here is one of Galileo's drawings of the moon with craters and mountains, like Earth, full of imperfection. Galileo saw the shadows cast by the mountains and craters change in the changing angle of sunlight and even went so far as to estimate their height and depth.
3.2.3.1.1.1. moon is imperfect and earthlike
3.2.3.1.2. sunspots
He turned the telescope to the sun, being careful not to look directly into it. He saw sunspots on an imperfect sun. Not only that but the sunspots moved across the face of the sun, changing from day to day.
3.2.3.1.2.1. sun is not perfect
3.2.3.1.4. new stars
He could also see many more stars than with the naked eye, especially the stars of the milky way, which to the naked eye appear as a creamy streak across the night sky. Of course, the existence of these extra stars could not be explained unless
3.2.3.1.4.1. universe is bigger than previously thought
the universe is bigger than previously thought and has a lot more stars which means:
3.2.3.1.4.2. heavens are not perfect
that the heavens are not perfect.
3.2.3.1.4.3. stars are smaller through the telescope
Galileo also noticed that while the planets get bigger when viewed through the telescope, the stars actually get smaller. From this, he concluded that they must be very far away compared to the planets.
This turned out to be true, but the explanation for the stars getting smaller would not come until the nineteenth century, some two hundred years later.
If you're curious about this, look up "diffraction". As it turns out, this is due to the optics of the eye and the telescope, a topic we will not have time to cove r in this course.
3.2.3.1.4.3.2. must be very far away
3.2.3.1.5. crescent of Venus
Probably the most convincing of all is that Galileo observed the planet Venus to go through a complete set of phases.
3.2.3.1.5.1. bigger when crescent
and the planet is biggest when it is crescent, and smallest when it is full.
3.2.3.1.5.2. cannot happen in Ptolemaic system
This cannot happen in the Ptolemaic system, no matter what kind of modifications.
3.2.3.2. changed him and the world forever
When Galileo looked through the telescope, it changed him and the world forever
3.2.3.3. convinced him that Ptolemaic theory could not be correct
because it convinced him, without a doubt, that the Ptolemaic theory could not be correct, because no geocentric theory could explain what he had observed.
We will study Galileo and his contributions in future lessons, but now let's turn our attention to Kepler.
Johannes Kepler not only worked for Brahe, but also exchanged correspondence with Galileo, who had become somewhat famous throughout Europe for his lectures at the University of Padua in Italy. He had read Galileo's Starry Messenger, published in 1610, and the two had exchanged correspondence. It is impossible to know precisely the influence the two of them had on each other. Whatever the influence, Galileo's contributions with the telescope notwithstanding, his greatest work was in the behavior of matter in the earthly realm, whereas Kepler's was in the heavenly realm.
Kepler was definitely a man of his times,
" . . . rooted in a time when animism, alchemy, astrology, numerology and witchcraft presented problems to be seriously argued."
Well schooled in the classics, including Euclidean geometry and infused with Pythagorean mysticism, he undertook to prove once and for all the harmony and perfection of the universe, using Brahe's highly accurate and precise planetary tables.
Kepler died feeling that he had failed in a major part of his mission which was
4.2.1. to reconcile Pythagorean mysticism and Ptolemaic system with precise measurements
to vindicate the Ptolemaic system by using Brahe's data to ascertain the circular planetary motions to a precision never before attempted and prove the truth of Ptolemy's geocentric model against the rising tide of heliocentrism..
4.2.2. to unify a mechanical universe
Kepler also hoped to reconcile Ptolemy's astronomy with Aristotle's cosmology by finding the links behind planetary motion which would once again allow it to function like a big machine, with the motion of each part somehow influencing and being influenced by each other part.
Not since Aristotle had anyone seriously attempted such a grand synthesis.
Although Kepler failed in his overall mission, he made some great discoveries which, with Newton's help, propelled Europe headlong into heliocentrism and more half a century later.
It is interesting to take a look at the personality behind Kepler's discoveries. There is the danger of losing the perspective that science is a human activity where personality as well as the social, religious, and political environment influence the course of history.
4.3.1. Right Man at the Right Time
Kepler was definitely the right man at the right time. Not everyone would have done what Kepler did, even if they could have.
4.3.1.1. new awareness of nature and the sky
For one thing there was a new awareness of nature and the sky, part of it a carryover from the awakening of the Renaissance, part of it from the excitement of the novae and comets. A second nova, this one in 1604, was also confirmed, by Brahe's method, to lie outside the sublunar realm in the Cosmos. This, the second in a little more than thirty years, is a somewhat rare occurrence Top that off with five comets in ten years, and you get the picture that coincidence played a major role in the timing of these events.
4.3.1.2. regarded 8' discrepancy as significant
Of all the planetary orbits, the one which was hardest to reconcile with Brahe's data was the planet Mars. The discrepancy was small, a little more than onetenth of one degree. With less accurate data, it would never have been noticed.
And such a small discrepancy might not have bothered most people. Most of us would have simply said, "It's close enough," and left it at that.
Not Kepler, considering the accuracy of the data and the precision of the instruments, he thought it to be highly significant.
4.3.1.3. had both the interest and the ability
Kepler had the interest to persevere, but he was also a good mathematician, much better than Brahe had been. Another person in the same position might have looked at the details of the data with a little less precision and let the discrepancy pass.
4.3.1.4. good Pythagorean mathematician
He was in fact, a good Pythagorean mathematician, but also well aware of Euclid's geometry, in which it is apparent that there is really not much difference between a circle and ellipse, and the ellipse is only slightly imperfect, like the heavens themselves.
4.3.1.4.1. easy to adopt ellipses as slight variation of circle
4.3.1.4.2. the ellipse is only slightly imperfect4.3.2. gifted flair for mathematics
Kepler was in fact, a gifted mathematician, inventing many new techniques of analysis and calculation as he worked his way through the planetary motions. Carefully he reconstructed the motions of the planets in their orbits from the twenty years of Brahe's data.
4.3.3. ill health, domestic
Kepler's health was never good, even as a child, so he learned to discipline himself indoors, learning music, mathematics, and other domestic arts rather than the outdoor activities of many of his contemporaries.
4.3.4. financial and personality problems
Throughout his life he had financial problems, continually falling in debt and having episodes of paranoia and odd behaviors.
4.3.5. voluminous and candid writings in theology, philosophy, astronomy
Although he wrote volumes and volumes of books on a variety of topics in theology, philosophy, and astronomy, his
4.3.6. Discoveries received little attention
discoveries received little attention. This was largely because he wrote in a very obtuse and formal style. OK, it was a boring style. And he rambled, burying the significant findings in with the trivial, as if he couldn't tell the difference. Unlike Newton, who would state his laws of motion as axioms, the statements that we now know as Kepler's Laws are deeply hidden in Pythagorean statements about planetary numerology. As such, his writings did not receive wide readership until
4.3.6.1. Newton read and popularized Kepler's works
Newton read and popularized them half a century later. Newton saw the significance of the laws and their relationship to the force of gravity which holds the planets in their orbits.
4.3.6.2. Newton deduced the laws mathematically from laws of motion & gravitation
In fact, one of the cornerstones of Newton's theory of universal gravitation was that Kepler's laws could be deduced from the mathematics of gravitation.
More on that later when we study gravity.
Kepler published several works, but the ones which contained his laws of planetary motion (the topic of the next program) are the ones which concern us. The first two laws are deeply imbedded in a book with a title longer than many essays. The full title of the book is
A New Astronomy Based on Causation or a Physics of the Sky Derived from Investigations of the Motions of the Star Mars, Founded on Observations of the Noble Tycho Brahe.
You can probably guess why we usually just call it The New Astronomy
It was published in 1609, the year that Galileo first turned the telescope to the heavens.
Kepler had spent about five year working on the discrepancy in the orbit of Mars before he cracked the planetary code and understood the true shapes of the orbits.
4.4.1. first two laws of planetary motion
4.4.2. A New Astronomy Based on Causation or a Physics of the Sky Derived from Investigations of the Motions of the Star Mars, Founded on Observations of the Noble Tycho Brahe
4.4.3. published in 1609, the same year Galileo first used the telescope
4.4.4. five years to understand planetary orbits
In 1619, ten years after the publication of The New Astronomy, Kepler published his Pythagorean masterpiece, entitled Harmony of the World.
4.5.1. case after case of Pythagorean and Platonic relationships
It contained case after case of Pythagorean relationships and analogies between planetary motion and music, planetary motion and geometry, planetary motion and numerology. In fact, there is a connection between planetary motion and just about everything.
4.5.2. stated the third law of planetary motion
In there is contained, as one of many such relationships, what we know today as Kepler's third law of planetary motion which relates the period of a planet's orbit to its distance from the sun.
We will study these three laws and their meaning in the next program.
Kepler published one other major work, called the Rudolfine Tables. These were tables of planetary motions calculated from Tycho's data.
4.6.1. while Royal astronomer for Rudolf of Prussia
Kepler worked on them as part of his job as successor to Brahe as Royal Astronomer to Rudolf of Prussia.
4.6.2. standard planetary tables for fifty years
The Rudolfines were highly accurate, the best ever, and remained the standard planetary tables until Newton's gravitational equations made them obsolete.
4.6.3. based on Tychonic system
The calculations of planetary locations was based on the Tychonic system. You remember, the heliogeocentric one where the sun and moon go around earth while the other planets circle the sun.
Interesting, that Kepler would use Tycho's method after stating his own laws of planetary motion which could have given much more accurate locations had he taken the time and effort.
Why do you suppose he did not, assuming that his own laws would have been more accurate, and they would have?
Part of Plato's own Pythagorean fascination with the geometry of the universe was in making three dimensional polyhedral shapes out of the flat Pythagorean polygons. (Polygon: many sides. Polyhedron: many faces)
Here
is an excellent page which will offer more insight to the Platonic solids.
Of course, there is the sphere, which was the most perfect three dimensional shape having an infinite number of faces. The other simple polyhedrons were constructed from equilateral polygons: triangles, squares, and pentagons. These are flat figures with three, four, and five sides of equal length.
For example the cube is made from six squares at right angles, thereby establishing a relationship between the numbers four (sides of the square) and six (faces of the cube). The other figures are constructed similarly. The tetrahedron (Greek: four sides) is four equilateral triangles, so a relationship exists between three (sides of triangle) and four (faces of tetrahedra). I think you can see that similar numerical relationships exist in the octahedron (3 sides, 8 faces), the dodecahedron (5 sides, 12 faces), and the isocahedron ( 3 sides, 16 faces)
If you have difficulty remembering these numbers that is good. It shows that your mind is occupied by something else. Hopefully, it is wondering what these shapes have to do with Kepler and the planetary motions.
4.7.1. why five planets?
Kepler thought it to be more than simple coincidence that there were exactly five planets, not counting the sun and moon, one for each of the Platonic solids.
4.7.2. why spaced?
Furthermore, what mystical relationship was behind the spacing of the planets. By Kepler's time the distance to the planets had been determined with a fairly high level of accuracy, although not nearly as good as today.
How do you suppose they did that, found the distance to the planets?
4.7.3. why specific motions?
Kepler believed that there must be some connection, mystical or otherwise, between the motions of the planets and their spacing.
In The Harmony of the World, Kepler relates amazing discoveries of harmonious relationships between the Platonic solids, the number of planets, their spacing, and musical notes, which are after all intervals of tone. Having discovered these relationships Kepler believed that the planets actually made music constantly. As to why we can't hear it, the only explanation was that since it is always present, and we have heard it since birth, our senses are not attuned to it, much in the same way that a fish is probably not aware of the water in which it swims.
Quite amazingly, Kepler discovered a very Pythagorean relationship, which to this day has no explanation. He discovered that if the Platonic solids are nested one inside the other in a certain way, their geometric properties naturally recreate the spacing of the planets.
Kepler proclaimed this to be his greatest triumph in searching for Pythagorean meaning in the universe.
Today, we guess that it is a coincidence, which like Bode's Law (a numerical relationship which also predicts the spacing of the planets, but not very well) works less well the more accurate are the planetary distances. Nonetheless, the fact that these relationships exist at all, and the fact that Kepler, or anyone else would uncover them, is incredible.
Two of Kepler's models (1596) of the plan by which the six planets were placed in the heliocentric system. In the first model the outermost sphere, corresponding to Saturn's path, is circumscribed around a cube. A sphere inscribed in the cube corresponds to Jupiter's orbit, and in turn encloses another of the five Platonic solids, namely a tetrahedron. A sphere erected inside the latter gives Mars' orbit, and so forth.
4.8.1. nested platonic solids to match spacing of planets
4.8.2. radii of planetary orbits fit perfectly in nested solids
Of all of the planets, none gave Kepler as much trouble as Mars. In trying to fit the orbit of Mars into Brahe's data, he failed miserably no matter which system he used. The other planets fit quite nicely, and, had it not been for the highly eccentric orbit of Mars compared to the other planets, it is not likely that even Kepler would have noticed it.
4.9.1. hardest to fit to Brahe's data
Mars was the hardest of all to fit to the data, and in fact, no matter what Kepler tried, he just couldn't get Mars to work with a circular orbit.
4.9.2. spent years to remove 8' discrepancy
He worked for years trying to fine tune this eight minute discrepancy, a little more than one tenth of a degree. Here's how small that is.
Suppose you started in Los Angeles and flew in a straight line at New York, 3000 miles away. If your course was off by 8 minutes you would arrive in New York only seven miles off course. If you could do that, you would probably say you were pretty close.
But Kepler was such a perfectionist that he could not let it go.
4.9.3. tried various combinations of eccentric, deferent, epicycle, equant
He tried various combinations of Ptolemaic devices, the eccentric, deferent, epicycle, equant. He tried using Tycho's model and Copernicus's methods and nothing worked.
He could not get Mars to fit into any existing system. Something was wrong.
Either Tycho's data was wrong, which wasn't likely, not consistently wrong for twenty plus years.
Or all of the models, Ptolemaic, Copernican and Tychonic, were wrong.
Wait! It had to be one of them didn't it? Doesn't it have to be either geocentrif or heliocentric or some combination? What are the other choices?
That's what Kepler must have asked.
4.9.4. set out to determine "true shape"
He set out to determine the "true shape" of the orbits.
Picture yourself in his place. You start out trying to prove that Brahe's observations supported the Ptolemaic system. You fail at that. Then you try the other systems which had been proposed, namely the Copernican and the Tychonic systems. Still you fail.
Five years has now passed and you still have no clue. Is it surprising that at some point you are willing to throw the whole thing out and start over?
This is basically what Kepler did. He didn't start over with the data. That was obtained with good instruments and had been checked and double checked, and in fact predicted the planetary motions better than anyone before in history.
But there was that nagging eight minutes of arc.
Before we get to Kepler's solution, we must take a little side trip to the land of shapes to study the properties of the Conic Sections.
What are conic sections? They are sections of a cone, what else.
Actually the conic sections are geometric shapes which are obtained by slicing a right circular cone. OK, I can see that your geometry is a little weak. That's all right. If you already knew this stuff, you wouldn't need to be here, right.
A right circular cone is a cone which is constructed like this.
Before I go on with this, I have to remind you that you are not expected to know the mathematical details of these conic sections. It is important to see the how these shapes are related as a family of curves, whose properties have so much in common with one another that Kepler was willing to consider them to replace the circular motion which had been part of the paradigm for two thousand years. It was not an easy thing for him to do.
So here is how we construct a right circular cone. First we take a circle. Then we take a line, called the axis, and place it perpendicular to the circle. So far we have the terms right (as in right angle, or perpendicular) and circular (as in circle). Now if we draw a line from the circle to a point on the axis, we have a triangle, formed by the radius of the circle, the axis and our line.
Now spin the whole thing and you have a right circular cone. Note that it is constructed from a triangle, a good Pythagorean figure.
]So what, we have a cone, now what? Now\ cut the cone at various angles to see what shapes are generated.
First cut it perpendicular to the axis, that is parallel to our original circle. You see, what we get is another, smaller right circular cone. The cross section of the cut is a circle.
OK, what did you expect, magic. It's a circle.
Now cut the cone at an angle to the axis, but at an angle smaller than the angle of the original triangle.
Before we do this, take a guess what the shape will be. You've done something similar before if you've sliced carrots on the bias. You don't get circles, you get . . .
What we get is an ellipse.
OK. What about the ellipse. It's almost a circle, isn't it. In fact if we cut the cone at any angle other that perfectly perpendicular to the axis we don't get a circle, we get an ellipse instead.
In fact, we might have a difficult time deciding whether a particular section was a circle or an ellipse unless we made very careful measurements of it. The sharper the angle we cut it, the less it looks like a circle, but the essential difference between the two shapes is a matter of degree and not a matter of kind. You will notice that the axis of the cone is not centered on the ellipse, but rather is offset to one side.
The circle and the ellipse are not the only shapes that we get when sectioning the cone. What happens if we cut the cone parallel to its sides? Now the angle of the cut is the same as the angle of the cone. See what happens. Now we get a section which does not close on itself. It is called a parabola. At first glance it does not look like the circle or the ellipse, but you can see that it just represents a threshold angle of cutting where the cut no longer forms a closed figure. In that respect it is very similar to the circle and the ellipse. You will also note that the axis still forms a trace which is not in the center of the parabola, but rather is closer to the closed end.
The fourth and final conic section is the hyperbola. I'll keep this brief, cause at this point it is the circle and the ellipse that we are concerned with. The parabola and the hyperbola will return later with Newton, so don't forget about them.
The hyperbola results from sectioning the cone parallel to the axis. The shape is similar to the parabola, but broader.
Without going into any more detail at this point, we notice that the circle and the hyperbola are rather special cases, requiring cuts at precisely the proper angle. If we make a random cut of the cone, the chances are very slim that we would get a circle or a hyperbola because there is only only plane which will generate either. The odds are overwhelming that we would get either an ellipse or a parabola depending on whether we cut the cone at a greater or lesser angle that its generating triangle.

You may want to study the figures in the text and study guide, or if you feel adventurous, make some cones out of paper and cut them in various ways and see what you get. To make a cone cut a circular piece of paper. Then cut out and discard a pieshaped wedge. Roll the rest into a cone shape then cut it as shown. 
Can you think of situations in which you might see these shapes in nature, related
to cones?
Watch the cone movie if your browser can handle quicklime movies. Here's a website with a very good treatment of the conic sections. Don't get baffled by the math, but look at the descriptions of the various sections.
Now we are ready to go back to Kepler to see the relevance of the conic sections.
Kepler felt justified in trying an elliptical orbit for Mars because the ellipse is really not so different from the circle. It is only slightly imperfect, and related to the circle. It is still possible to save the appearances. Besides it's not breaking the paradigm, it's only bending it a little.
"God can do whatever he wants" and if God wanted to give the planets elliptical orbits He could, regardless of what Plato, or Aristotle, or anyone else said.
The remarkable thing is that when he tried the elliptical shape for Mars, it fit perfectly. Going back to the other planets, including Earth, and trying ellipses for their orbits, Kepler found them to fit perfectly, without the need for any devices whatsoever
Here we see a perfect example of data challenging a paradigm. Brahe's observations were so good that it was no longer possible to design a geocentric system which fit them well. Without the accuracy of his sighting tubes the discrepancy in the orbit of Mars might have gone unnoticed for decades. Sure, the discrepancy was small, but it was significant enough in Kepler's eyes to be of such concern as to ultimately warrant a change in the paradigm.
Besides, he didn't really break the paradigm, he just sort of bent it. An ellipse is just a squashed circle, and the orbits of the planets, even Mars, were not very far from being circular.
In the next lesson we will study Kepler's laws of planetary motion. We will learn more about the ellipse and the elliptical orbits of planets around the sun.
In this lesson we have traced the demise of the circular paradigm, brought about by measurements of planetary location about as well as could be done by the naked eye. This was accomplished with the design and construction of extremely accurate sighting tubes by Tycho Brahe.
Tycho Brahe was the first professional astronomer, having convinced the King of Denmark to give him the money to build and fund a Royal Observatory.
From this observatory he would observe and track the planets night after night for twenty years, using instruments he invented and built which allowed observational precision never before possible.
This allowed him to determine that Ptolemy's calculations did not predict the paths of the planets nearly as well as had been thought previously. The observational precision also allowed him to determine that two comets and a supernova were heavenly events and did not occur in the sublunar realm where they were allowed by Aristotle's cosmology, and where it had been previously assumed that they did occur.
Brahe hired a young assistant named Johannes Kepler. Upon Brahe's death Kepler was given his data and his job.
Kepler, being a good Pythagorean mathematician, and a diligent worker, set out to prove once and for all the truth of the Pythagorean harmony and the Platonic perfection of the universe.
Although he failed to do so he did discover that the motion of the planets could best be described by a heliocentric model which had all of the planets, including earth, orbiting the sun in elliptical orbits rather than in circles. He did find that the planetary orbits contained geometric and harmonic relationships previously undiscovered, although he could not explain them.