Program 10 - "The New Astronomy: Brahe and Kepler"

Music Ohhh, this little thing won't allow be to find any new planets.Haahh...I wish I had a telescope.What am I thinking?I don't even know where the old planets are because they aren'twhere Ptolemy's book said they should be,and telescopes haven't even been invented yet.So.Oh, if only I had a telescope I could really find some neat stuff then.Oh, boy.

Music(Explosion.)We're back with the Nature of Physical Science,the telecourse that serves you snacks while it racks your brain.Siloco: "This is Program 10, Lesson 2.2.We call it the New Astronomy."That's it exactly.But to find out why we call it that, stay with us.Eh, nice voice, Silico.Before we're done with this program we will have seen howa combination of events, both human and astronomical,took Europe into the 17th century, full of turmoil, recoilingfrom 1700 years of oppression and forced thought.At the time of Bruno's death at the stake for heresy,for considering the implicationsof heliocentricism, a workingarrangement between Tycho Brahe and Johannes Kepler was beginningwhich would last for only one year, but which would mark thebeginning of the end of geocentricism and startthe real Copernican revolution rolling.Be sure to read these Objectivesin the Study Guide and refer to them as you study the lesson.

Focussing on the Learning Objectives will help you to study and understand the important concepts. Compare the Objectives with the study questions for this lessonto be sure that you have the concepts under control.You know, the 16th century is sometimes called the Age of Wonder, and for good reason.New ideas flourished. Explorations and conquests of the New World brought back empire building loot.Magellan sailed around the world, and the Protestant Revolutioncut further into the authority of the Church.And in England, the leadership of Elizabeth I elevated Englandinto a world power in the new Protestant world.

Meanwhile, the writings of Shakespeare in Englandand the work of Michelangelo in Italycarried the world of literature and art forward.And the printing press, invented around 1450 had created an information explosion not unlike the computer revolution of our own time.It's interesting to note that some of the critiques of the printingpress included a number of ideas that would sound quite familiar to us.Silico: "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?"I'm sure you've heard some of those before.

Anyway, political, social and economic ideas were changingduring this time, and astronomy had become a precision science.For the first time, the positions of the planets could bemeasured to a high degree of accuracy, and they'd been tracked over long periods.The result was that it became quickly apparent that neitherthe Ptolemaic nor the Copernican system did a very good jobof predicting the planetary locations.What do we mean by a good job?The error in the location of Mars was very small by anyone's standards.It amounted to about the width of a quarter, one inch, about 2000 feet away.This was well within the accuracy of Brahe's instruments.And all of this without a telescope.In fact, it was the accuracy of Brahe's sighting devices that...Silico: "Wait, you are getting ahead of the story again.Stay with the script, please."Tycho Brahe has got to be one of the more colorfulcharacters in the history of science. His great contribution is that he made the most accurate astronomical observations in the era before there were telescopes.Among other things, he's the first, if not the only astronomerof record, to wear a metal replacement nose.

He was born in Nustrup in Denmark on December 14, 1546,three years after the death of Copernicus.He was raised by his uncle who sent him to Copenhagen to study law. But in 1560, he observed the solar eclipse which turned hisinterest to astronomy, and against his uncle's objections,he studied the subject independently using Ptolemy's Almagest.Remember, the Almagest.In 1563, he observed the conjunction of Jupiterand Saturn which demonstrated to him about the inaccuracyof the existing records of the planetary positions.After his uncle's death in 1565, Brahe traveled through Europeand studied science at several universities.He returned to Denmark in 1571 and installed a chemicallaboratory in a castle of a relative at a nearby abbey.There he observed in 1572 a new star.Today we would call it a nova. In the constellation of Casiopia.

In his first published work, called "De Nova Stella," in 1573,he established that this nova was a star beyond the moon's orbit,in direct contrast to the cosmic perfection of the Scholasticparadigm which required that things in the heavens were perfect.In 1576, Frederick II, the King of Denmark, granted Brahean island all of his own, the island of Ven, to be seenbetween Sweden and Denmark along with a generous pension.There Brahe constructed his observatory, which he called Uraniborg, where during the next 21 years he built his own instruments and carried out an extensive program of observations.Now, where do you suppose that name, Uraniborg, came from?We heard that Uranus thing anywhere before?Even with these new precise measurements, Brahe could notdetect any stellar parallax which you may remember, should bevisible if the earth revolves around the sun.On this basis, just like Hipparchus, Brahe rejectedthe Copernican system and adopted the view that the earth is stationary. He discarded the Ptolemaic system, however, and because of his incompatibility with his observationsof planetary motions and favored instead his own new modelfor planetary motion, the Tychonic system, which was halfgeocentric and half heliocentric. And you might want to begin to think about how you can have amodel that's half geocentric and half heliocentric.

So, by proving that the nova in 1572 was in the Cosmos and notsublunar, Brahe demonstrated a basic flaw in the Erastotheleanworld view of perfection and unchangeability of the heavens.His observation of a comet in 1577 and five other subsequentcomets also convinced him that their orbits were farfrom the sublunar realm and way out in the Cosmos.This discovery, along with his accurate observations laida firm basis for the break through of the Copernican world view later in the 17th century.Unfortunately, Brahe's sarcastic manner and brash personalitymade him quite persuasive on one hand, but on the other hand, he made some very powerful enemies. His metal replacement nose was necessary to replace hisoriginal nose which he lost, or I should say hadaccidently removed, during a fight.

One story says it was sliced off with a sword, which narrowlymissed sending him to an early grave, and another account claims it was bitten off in a fight over some inconsequential thinglike what kind of beer to have with dinner or something like that.Well, whatever the nose story was, Brahe finallyleft Denmark after the King's death. His rights on Ven and his pension were withdrawn by Frederick'sson, Christian IV, and he left Ven in 1597.So, let's go back and look at this story a little bit closer. Because the story of Brahe is a very interesting one and he isa central character here in our unfolding drama.So, Brahe became the first professional astronomer whenhe persuaded Frederick II to fund the building and maintenanceof an observatory on Ven, off Denmark. The island's so desolate that both Denmark and Sweden claim that the other one owns it.It was not a great place if you like the night lifeand excitement of the city, but if you like the night lifeof the sky, it's a perfect place for astronomy.It's remote, it's dark and desolate.So, here, night after night for 20 years, Brahe and his assistantstracked the motions of the planets throughout their orbits.No one had ever spent this much time looking at the planets, no one.Brahe actually had a pretty good life here on the island.He built a castle, he built all kinds of instruments.He actually built a library. He built a printing press so he could publish his own documents.He had it really pretty good there.His goal was to come up with detailed tables of the locationsof the planets which he could then use to bring the old Ptolemaictables, having been modified by Copernicus, up to date.

Now, Brahe was called a mathematician.In fact, he worked as a mathematician,but he was not really much of a mathematician.What he was, was a great instrument designer with a lotof diligence and a lot of perseverance.The observatory and the instruments that he built madeit possible to keep detailed records of the locationsof the planets on a full time basis.Everybody else who looked at the stars and had done it sort as adilettante, you know looking here and there, but Brahe did it every night.The instruments he designed were accurate to four minutes of arc.Four minutes of arc.Now, think about that.That's 1/15 of a degree.That's about 1/15 of a degree, about the sizeof a quarter, three quarters of a mile away.

So, imagine a quarter, somebody holding it three quarters of a mile away.You get an idea how precise the instruments were.No one, no one had ever made observations of the heavenswith anything near that accuracy before.In fact, his accuracy was about ten times greater than theaccuracy of anybody else at the time.So, not only that, that he had the great accuracy, but no one hadever tracked the planets completely through theirorbits on a nightly basis, hour after hour.No one, no one had ever done this. All of the previous observations, including those of Hipparchusand Ptolemy had been made rather sporadically. After all, these guys were geographers and they were looking at other things and they did this whenever they had a chance. But Brahe's full time job was astronomer.

So, all of the work that people had done, all the centuries,developing tables of motion and theories of astrological mysticism, and religious dogma, and so forth, had been basedupon these very bad observations madewith the naked eye and taken only sporadically. It's a wonder there were any theories at all about the heavenly motion.Let alone that they worked as well as they did.Now, they did work pretty well, remember.That the conjunction of Jupiter and Saturn in 1524 occurreda full month earlier than predicted by the Ptolemaic theory.But, that's still not a bad prediction.The Copernican calculations were only a few days differentand that was based upon these old observationsof unknown origin and unknown reliability.If you don't agree that that's pretty good, only being offby a month, go out without any other basis and try to predictsuch an event yourself, and see how close you come.

Kepler's instruments used the circumference of large circlesto precisely pinpoint the location of a star or aplanet in both the horizontal and vertical planes.He basically used a large protractor.Something like this. And with a protractor, it's marked off in degrees, so you canmeasure the location of a planet from some fixed reference pointlike northeast, south or west and you can measure the anglein degrees around along the horizon and then you can usea vertical protractor and measure the angle vertically above the horizon.So what you wind up with is a very precise locationof the locations of the planet in termsof a horizontal arc and a vertical arc. Degrees horizontal and degrees vertical, if you like. OK.

Brahe's system, his instruments used a basic fact of geometry. You see, because all circles enclose the same measureof 360 degrees, the size of a degree on the circumferenceof the circle gets larger as each circle gets larger.That's why you can cover the moon with your thumb.Even though the moon is nearly 3000 miles across.So, let's go here to the ELMO for a minute and I'll showyou what I mean by the size of the degree.So here's the large protractor, and youcan see each one of these marks is one degree.So imagine trying to divide this into 15 segments.Now, if I put the small protractor on top of this, you see that notonly is the diameter of the protractor smaller, but so are the size of the degree.In fact, the size of each degree is proportionalto the diameter or to the radius of the protractor.Look at the size of the degrees on the small one.Can you imagine trying to subdivide each of those into 15?So, the bigger the protractor gets, the longer the arcon the outside of the protractor represents the size of each degree.

So in principle, if you could make a very, very large protractor,you could read the location of something very accuratelyby lining it up, like this, for example the sighting of a planetand mark off the fraction of that degree and read itdown to a very high level of accuracy.This instrument, one of the earliest in Brahe's observatoryallowed him to sit on a short stairway and observe a planetthrough a hole in the wall of the observatory.An assistant could then line up a sliding marker on the protractorscale and read the angle of altitude of the planet.This quadrant did not have the accuracy of the later sighting tubes.This sighting tube was 6 feet long.It set 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,it's altitude about the horizon was run off the protractor's scale.At the same time, it's orientation from north was notatedon a horizontal scale attached to the pedestal.The date, time and coordinates of the heavenly objects werelogged into a table for each observation.The speed and accuracy of this and other instruments allowedBrahe to take many measurements in a night, sometimesmeasurements only a few minutes apart.Now, thinking about this.Don't forget that the earth's rotation carries a staror planet westward with the hours, so absolute locationsin the sky must be compensated for the diurnal movement.

In 1599, the Emperor Rudolph II offered Brahe a positionof imperial mathematician of the Holy Roman Empire at his court in Prague.Brahe gladly accepted this and in 1600 the same year that Brunowas burned at the stake for his heretical speculations,Brahe hired a young and promising mathematiciannamed Johannes Kepler to be his assistant.Brahe had read some of Kepler's publications and was quiteimpressed with his mathematical talents.Kepler was quite a bit younger than, about 30 years younger,than Brahe so it was sort of like a mentor-protege relationship.They worked together for a year, and Brahe grew even moreimpressed with his young assistant's talents and work ethic.So upon Brahe's death in October 24, 1601, Kepler became hissuccessor and inherited his large collection of astronomicalobservations and his data and his equipment and also his job.Another interesting and somewhat relevant contribution madeby Tycho was his geo heliocentric model.That's right, geo heliocentric. Or, I guess you could say, helio geocentric.See, there's a whole lot of things going on here.

The absence of stellar parallax which Brahe thought he shouldbe able to detect with his sensitive instruments, that's,you know, the annual shifting of the starbackground as the earth moves in its orbit.It prevented him from accepting the heliocentric theory outright.However, the motions of the planets fit much more nicelyinto a heliocentric model than they did into a geocentric model. So, how do you solve this problem?Brahe's solution to the problem is a very creative way of savingthe appearances while still preserving circular motionand still corresponding to the data.The genius of Brahe in visualizing this model, in additionto the creativity, is that it points to a general recognitionin 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'sprinciples as they had been canonized in the Scholastic Philosophy.So what do you do with this?I mean on one hand you've got to have the earth stationaryand on the other hand the Ptolemaic doesn't work.Well, Brahe's universe was spherical and earth centeredthereby preserving the integrity and the sanctity of the age old paradigm.In the Tychonic universe it works like this:The earth is the center of all motion and it remainsstationary as the sun and the moon revolve around it, therefore, it's geocentric.Meanwhile the other five planets revolvearound the sun as it goes around the earth.It's geocentric because the earth is still stationary.It's heliocentric because all the planets actually goaround the sun while the sun goes around the earth.

Another way to say this is that the sun's orbit becomes thedeferent and the other planets do epicyclic loops around it.You see how this combines the best of both worlds?It's really quite clever, and, in fact, in works muchbetter than Ptolemy's system.And actually it even works better than the pureCopernican system because largely of Brahe's data.You know, considering the seeming sensitivity of the Churchon these matters, one would think that it would have had quite a stir.Not so.In fact, the astronomers of Tycho's time not only acceptedthe Tychonic system, many of them actually favored it,because it made the calculations easy.It agreed with the observations quite closely, and it did notreally require a shift in paradigm as radical asa pure Copernican system would have.It still, after all, left the earth as the center and left the earth unmoving.

So, here's a brief review of the contributions of Tycho Braheto the growing and maturing river of our scientific heritage.He was the first professional astronomer.He built the first full-time astronomical observatory.He designed and constructed precision astronomicalequipment that was probably as accurate as you couldever get without building a telescope.He made regular and continued measurements of the planetarymotions compared with other people who hadsimply looked at them from time to time.He created this Tychonic system which preserved both theheliocentric and the geocentric ideas.In fact, worked very accurately and became the standard modelfor his time, and probably as important as anything else,is that he wound up giving all of this 20 years worthof detailed data of observations to Kepler.OK.

Let's focus for a while on the times.Silico: "This is not a good time to read the newspaper.You have to finish the program."What?Oh, not the New York Times.The times.Besides, the rapid changes which had already taken place earlierin the century, several astronomical events occurred just at the time that Tycho Brahe was pointing his new instruments to the sky.Now was this a coincidence, that things happened justto be there when there was someone with accurateinstruments watching them?I don't know.But as much as Brahe's instruments set the stagefor Kepler's discovery of the harmony of the planets, his useof the instruments to challenge the sanctityof 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, of course, were regular and perfect.A nova in 1572 was recorded by Brahe at Uraniborg.His precise instruments allowed him to observethe nova not just once, but over a period of time.And from these measurements, he was able to provewithout a doubt that this "new star" was definitely cosmic and not sublunar.A comet in 1576 was likewise shown to beoutside the moon'sorbit and, therefore,also behaving in a verynon-heavenly manner.

Now, these two things were not the first novae or the firstcomets to be observed, but prior observations hadbeen qualitative rather than quantitative.This means that no one really bothered to take down numbersor were they able to take very accurate measurements.These things were assumed to be sublunar.Like clouds and meteors, because after all, everyone knows thatanything that changes in the heaven must be sublunar.Right?Brahe's proof, published in 1573 in a booklet which was called,appropriately, "De Nova Stella," or "The New Star," presenteda serious challenge to the concept of heavenly perfection.

Now, luckily for Brahe, the Protestant reformation in Central and Northern Europe had taken much of the wind outof the Inquisition which had been so effective in preventingthe spread of ideas in Southern Europe like the execution of Bruno. 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.It's not clear exactly when the telescope was invented, or who invented it.It's said that the basic principle of the telescope was fairly wellknown in Holland in the early 1600s and even in the late1500s, but the earliest recorded use of the telescope was in 1608.A year later in 1609, an Englishman named Harriotlooked at the moon, saw a few spots and said,"Hey, that's cool," or something like that.He made a notation in his journal and that was that.No further observations or no further records were keptby Harriot or at least if he made any, he never mentioned it.In that same year, 1609, Galileo in Italy learned of this new instrument from a trader and began experimenting to build his own.His earliest telescope was about 10 power.That's equivalent to a decent set of binoculars.

Within a few months he had increased the power of his telescope to about 30.Now, here's a wonderful example of the importanceof the individual in the growth of knowledge.Harriot saw the moon up close and thought little of it.Galileo was so fascinated by what he saw that he looked againand again and again and again, discovering more and moreevidence that convinced him that the geocentric concept could not be correct.He saw that Aristotle and Ptolemy were both wrong and he sawthat the Scholastic Philosophy was wrong.Think about this.What would you do in a situation like this?You just looked at this instrument, you've takena casual look at the heavens and suddenly you see that 2500years of thought by the best thinkers and greatestphilosophers in the world is completely wrong.What would you do?

Well, we'll return to Galileo in a later program and see what he did about it.But the point is here that he saw the same moon that everybodysees and the same moon that Harriot saw, but Galileo did notsee a heavenly object, he saw an earthlike moon with cratersand mountains like earth full of imperfection.Galileo saw the shadows that were cast by the mountainsand the craters and he saw them change in the changing angleof sunlight as the moon changes phases.And even went to so far as to estimate the height and the depth of those craters.Then he turned his 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 movedacross the face of the sun, changing from day to day.

So, the sun is not perfect either.The sun has zits,OK.It has imperfections.Then he turned the telescope to the planet, Jupiter, and therehe saw tiny dots of light moving back and forth around the planet.He assumed that these were moons and, in fact, these foursatellites or four moons of Jupiter are today called the Galilean satellites.Now, this, of course, showed him that earth was not the only center of motion.And, in fact, if there could be more than one center of motion,that means that earth lost its special placeas the center of all motion in the universe.Galileo also could see many more stars through the telescopethan he could see with the naked eye, especially the stars of the Milky Way.To the naked eye it appears simply as a band of creamy streak across the night sky. Of course, the existence of these extra stars that Galileo could see could not be explained under the current paradigm or else the universe is a lot bigger than previously thought.It has more stars.

So Galileo noticed that while the planets got bigger when viewedthrough the telescope, the stars actually got smaller. From this, he concluded they must be very, very far away compared to the planets.He had no other explanation for it.What it turns out that this is true, the stars are,indeed, much further away than the planets. But the explanation of why they got smaller would not be founduntil the 19th century, nearly 200 years later.By the way, if you're curious about this, look up the word, "diffraction."D I F F R A C T I O N.Diffraction.As it turns out this fact that the stars get smaller is dueto the optics of the eye and the telescope, and that's a topic we won't have time to cover in this course.Probably the most convincing of all the things that Galileo sawis that he observed the planet, Venus, to go through a complete set of phases.

Now, in the geocentric theory it's impossible for this to happen. And so seeing the full set of phases meant that thegeocentric theory simply could not be correct.Not only that, but he'd observed that when Venus isin a crescent phase, it's biggest and when it's in a full phase,it's smallest, indicating that it was further away when it wasfull, and closer when it was crescent. Exactly what you'd expect to happen in a heliocentric system. If you want to see this, look in the Study Guide, and there's a nice diagram there of the phases of Venus and how they look in the two systems.We'll return to this later on when we study Galileo, but for now,we just want to understand that this full set of phases of Venus cannot possibly happen in the Ptolemaic system no matter what kind of modifications you make.In fact, it cannot happen in any kind of geocentric system.

So, when Galileo looked through the telescope and saw all these different things, the imperfect moon, the moons of Jupiter,and sunspots on the sun, the crescent of Venus, the newstars, it changed him and the world forever.Because it convinced him without any doubt at all that thePtolemaic theory could not be correct and that no geocentrictheory could explain what he had just seen in the heavens.We'll study Galileo in several later programs, but for nowit's Kepler to whom we want to turn our attentions.Kepler had not only worked for Brahe, but also knew of Galileo,who had become somewhat famous throughout Europe for hislectures at the University of Padua in Italy.Kepler had read Galileo's "Starry Messenger," published in 1610where Galileo talked about his observations through thetelescope and the two had exchanged correspondence.It's impossible to know precisely the influence that the two hadon each other, but no matter what the influence was, Galileo'scontribution with the telescope, notwithstanding, his greatestwork 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.

As the quote on your screen says, ". . . he was a man rootedin a time when animism, alchemy, astrology, numerologyand witchcraft presented problems to be seriously argued."Kepler was very well schooled in the classics, including Euclidiangeometry and he was well infused with the Pythagorean mysticism.He undertook to prove once and for all that the harmonyand the perfection of the universe was as Plato had said,using Brahe's highly accurate and precise planetary tables.Kepler died feeling that he had failed in a major partof his mission, which was to reconcile the Pythagoreanmysticism and the Ptolemaic system with these precise measurements to vindicate the Ptolemaic system by usingBrahe's data to ascertain this circular planetary motionsto a precision never before attempted and at the same timeto prove Ptolemy's geocentric model against this rising tideof heliocentrism that was growing in his time.Kepler also hoped to reconcile Ptolemy's astronomywith Aristotle's cosmology by bringing back those linksbetween planetary motion which would once again allow thewhole universe to function like a big machine, with the motionsof each part somehow influencing and being influenced by each other.Not since Aristotle had anyone seriously attempted such a grand synthesis.

You know, although Kepler failed in his overall mission,he did make some great discoveries which,with Newton's help, propelled Europe headlonginto heliocentrism more than half a century later.It's very interesting to take a look at the personalitybehind Kepler's discoveries.There's a danger when we go through this material so fastof losing the perspective that science is a human activitywhere the personality of individuals as well as thesocial, religious and political environmentinfluence the course of history.Kepler was definitely the right man at the right time.Not every one would have done what Kepler did, even if theyhad the abilities and could have done it.For one thing, in Kepler's time, there was a new awarenessof nature and the sky, and part of it was a carry over from the awakening of the Renaissance, part of itfrom the excitement of the novae and comets.

A second nova in 1604, also confirmed by Brahe's method,to lie outside the sublunar realm in the Cosmos.This was the second in little more than 30 yearsand somewhat a rare occurrence.In a typical lifetime there might be only one nova,if even that, in a person's lifetime.Top that off with five comets in ten years and you get a picturethat coincidence played a major role in the timing of these events.Of all the planetary orbits, the one which was hardestto reconcile with Brahe's data was the planet, Mars.The discrepancy was small, a little more than one-tenth of a degree, in fact. And with less accurate data from Brahe, it would never have been noticed at all.And, such a small discrepancy might not have bothered mostpeople, even if they had noticed it.Most of us would simply have said, "Hey,it's close enough," and left it at that.But not Kepler.Considering the accuracy of the data and the precisionof the instruments, he thought it to be highly significant.So, Kepler had the interest to persevere in resolving this discrepancy, but he was also a really good mathematician.In fact, he was a much better mathematician than Brahe had been. He was, in fact, a good Pythagorean mathematician,but he was also well aware of Euclid's geometry, in which itappeared that there's really not much differencebetween a circle and an ellipse, and an ellipse is only slightly imperfect, like the heavens themselves.

So, Kepler was, in fact, a gifted mathematician,and he invented many new techniques of analysisand calculation as he worked his way through the planetary motions.Carefully he reconstructed the planets orbits from the 20 years of Brahe's data.Well, Kepler's health was never very good and this contributed to his personality as well.Even as a child, he couldn't go outside and play and do thethings that most kids do, so he learned to discipline himself indoors.He learned music, mathematics and other domestic arts ratherthan going outside and hunting and doing the thing that other guys did.He also, throughout his life, had financial and personality problems.He was continually falling in debt, couldn't pay his bills.

He had episodes of paranoia and rather odd behaviors.In other words, he was a pretty weird guy.What I think we'll see as we go through the history of sciencehere that many geniuses do have rather odd personalities,not only in the arts, but also in the sciences.Although Kepler wrote volumes and volumes of bookson a variety of topics in theology, philosophy, and astronomy,his discoveries received little attention in his lifetime.This was largely because he was a lousy writer.He wrote in a very obtuse and formal style.OK, it was boring.OK, it was downright boring.And he rambled burying his significant findingsin with the trivial, as if he couldn't tell the differencebetween which things were important and which weren't.And most scholars think that's the case, that he wasn't really surewhich things were significant, which things weren't.Unlike Newton, who we'll see, would state his laws of motionas axioms, very clearly, very precisely, right up front.The statements that we now know as Kepler's laws are hiddendeeply in this Pythagorean statements about planetarynumerology and the music of the spheres and harmonies and that sort of thing.As such, his writings did not receive wide readershipuntil Newton read and popularized it half a century later.

Newton, who was also a genius of the first caliber,saw the significance of these laws and their relationshipto the force of gravity which he later went on to showthat holds the planets in its orbits.In fact, one of the cornerstones of Newton's theory of universalgravitation was that Kepler's laws could bededuced from the mathematics of gravitation.But, more on that later.Well, Kepler published several works, but the ones that we'reconcerned with are the ones that contained his laws of planetarymotion, that's the topic of the next program.The first two laws are deeply imbedded in a bookwith a title longer than many essays.A full title of the book is, "A New Astronomy Based on Causationor a Physics of the Sky Derived from Investigationsof the Motions of the Star Mars, Foundedon Observations of the Noble Tycho Brahe."You can probably guess why we usually just call it, "The New Astronomy."The book was published in 1609, the same year thatGalileo turned the telescope to the heavens.

Now, Kepler had spent about five years workingon this discrepancy in the orbit of Mars before he trackedand cracked the planetary code and understood the true shapes of the orbits.Ten years after the publication of "The New Astronomy," in 1619,Kepler published his Pythagorean masterpieceentitled, "Harmony of the World.""Harmony of the World."What a nice title.What a nice Pythagorean title.It contained case after case of Pythagorean relationshipsand analogies between planetary motion and music, planetarymotion and geometry, planetary motion and numerology.In fact, there's a connection between planetary motion and just about everything.In there is contained as one of many such Pythagoreanrelationships, what we know today as Kepler's third lawof planetary motion, which relates the periodof a planet's orbit to its distance from the sun.But, again, I'm getting a little bit ahead, and we'll study thesethree laws and their meanings in the next program.Kepler also published one other major work, called the "Rudolphine Tables."There were tables of planetary motion calculated from Tycho's data.Kepler worked on them as part of his job as successor to Braheas the Royal astronomer to Rudolph of Prussia.The Rudolfines were highly accurate tables, the best ever,in fact, and remained the standard planetary tables until Newton'sgravitational equations made them obsolete 50 years later.The calculations of the Rudolfine system werebased upon the Tychonic system.

You remember, the helio geocentric one where the sunand moon go around earth while the circle the sun.Interesting, isn't it, that Kepler would use Tycho's method afterstating 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 didn't use his own laws to figure this out,assuming that his own laws would have been more accurate,and we know today that they would have?Part of Plato's own Pythagorean fascination with the geometryof the universe was in making three dimensional polyhedralshapes out of the flat Pythagorean polygons.The word, polygon, means many sides.The word, polyhedron means many faces. Of course, of all the geometric figures, there is the sphere,which was the most perfect three dimensional shapehaving an infinite number of faces.And, of course, it was made from a circle, having an infinite number of sides.The other simple polygons, or polyhedrons, I'm sorry, were constructed from equilateral polygons: triangles, squares and pentagons.These are all flat figures with three, four and five sides of equal length.For example, the cube is made from six squares at rightangles, thereby establishing a relationship between the numbers four and six.That's four sides of the square and six faces of the cube.The other figures are constructed similarly.

The tetrahedron, the word comes from the Greek meaning foursides, is four equilateral triangles. So now a relationship exists between the number three,the sides of the triangle, and the number four, the faces of the tetrahedron.I think you can see that similar numerical relationships existin the octahedron, three sides and eight faces, and dodecahedron,five sides and 12 faces, and the isocahedron, three sides and 16 faces.Remember the Pythagorean fascinationwith the relationships between numbers.So here we have more relationships between thesenumbers three, and eight, and five and 12 and so on.If you have difficulty remembering these numbers,that's good, because it shows that your mind is occupied by something else more important.Hopefully, it's wondering what those shapes have to dowith Kepler and the planetary motions.You don't have to remember the names of all these shapes. Here's why we care about these.

Kepler thought that it was much more than simple coincidencethat there were exactly five planets, not counting the sunand the moon, one for each of the Pythagorean, or the Platonic solids.Five planets, five solids.A coincidence?Kepler thought not.Furthermore, Kepler wondered what mystical relationship wasbehind the spacing of the planets.Why should there be five planets, but why shouldthere be space exactly as they are?Well, by Kepler's time, the distance to the planets had beendetermined with a fairly high level of accuracy, although notnearly as well as we know them today.How do you suppose they did that?How do you suppose you find the distance to the planets?Well, Kepler believed that there must be some connection,mystical or otherwise, between the motions of the planets and their spacing.In the book, "The Harmony of the World," Kepler relates someamazing discoveries of harmonious relationships whichhe discovered between the Platonic solids, the numberof planets, their spacing and musical notes, which,after all, are just intervals of tone or pitch.Having discovered these relationships, Kepler believedthat the planets actually made music constantly.As to why we can't hear it, the only explanation he had wasthat since the music is always present and we've heard it sincebirth, our senses are simply not attuned to it, much in the sameway that a fish is probably not aware of the water in which it swims.

Well, quite amazingly, Kepler discovered a very Pythagorean relationship, which to this day, no one has been able to explain.We still don't know whether this is a coincidenceor whether it means something deeper.He discovered that if the Platonic solids are nested one inside theother in a certain way, their geometric properties naturallyrecreate the spacing of the planets.That's amazing, if you think about it.Kepler proclaimed this to be his greatest triumph in searchingfor the Pythagorean meaning of the universe.He thought that this was the most important thing he had discovered.Today, we guess that it's just a coincidence, which like Bode'sLaw, which is a numerical relationship which alsopredicts the spacing of the planets, but not very well,works less well the more accurate the planets are,the distances are known, I should say.So, the better we know the distance, the less we see thatthese relationships really work, but nonetheless, the fact thatthese relationships exist at all, and the fact that Kepler,or anyone else would uncover them, is really incredible.

So, here's a picture of Kepler's model which he made in 1596of the plan by which the six planets were placed in the heliocentric system.The outermost sphere, corresponds to Saturn's path,and it's circumscribed around a cube.A sphere inscribed in this cube corresponds to Jupiter's orbit,and in turn encloses another of the five regular bodies, in this case a tetrahedron.A sphere erected inside the latter gives Mars' orbit and so on.This is amazing.You might want to look at the picturesin the textbook to get a clearer picture of this.I should point out that Kepler actually built physical copiesof many of these models, as well as doing the geometrybefore he found one that actually measured the spacing.Next I want to turn attention to the planet Mars because it wasthe planet Mars that gave Kepler the most trouble.In fact, of all the planets, none gave Kepler anywherenear as much trouble as Mars.In trying to fit the orbit of Mars into Brahe's data, he failedmiserably no matter which system he used.

Now the other planets fit quite nicely and, had it not beenfor the highly eccentric orbit of Mars compared to the otherplanets, it's not likely that even Kepler would havenoticed it, even with his diligence.Mars was the hardest of all to fit to the data,and in fact, no matter what he tried, Kepler simply could notget Mars to work with a circular orbit.He spent years, in fact, five years, trying to fine tune an 8 minute discrepancy, a little more than one-tenth of a degree. Here's how small that 8 minute discrepancy is.Suppose you started in Los Angeles and flew in a straightline aimed at New York, 3000 miles away.If your course was off by only 8 minutes of arc, you wouldarrive in New York only 7 miles off course.OK, so a discrepancy that small, 7 miles out of 3000miles, represents 8 minutes of arc.So, if you did that and flew to New York that way, you'dprobably say when you got there that you were pretty close.But, Kepler was such a perfectionist that he simplycouldn't let go of this 8 minute discrepancy.He tried various combinations of Ptolemaic devices.He tried the eccentric, he tried the deferent,he tried the an epicycle, he tried an equant.He tried using Tycho's model and he tried Copernicus' method, nothing worked.He simply could not get Mars to fit into any existing system.Something was really wrong.Either Tycho's data was wrong, which wasn't likely,considering that he spent 20 years taking these very accuratemeasurements and everything else worked. So, either that, or all of the models, not just oneof the models, but all the models, the Ptolemaic model,the Copernican model and the Tychonic model were wrong.Silico: "Wait, it has to be one of them, didn't it?What are the other choices?"That's exactly what Kepler must have asked.In fact, he set out then to determine the true shape of the orbits.

Now picture yourself in Kepler's place.OK.You start out trying to prove that Brahe's observationssupported the Ptolemaic system.You want to prove the ultimate PlatonicPythagorean harmony of the universe.You have this fantastic set of data.You fail at that.You keep trying, you keep trying, you keep trying.You simply cannot make it work.So, then you finally give in and you say, "OK, maybe it's notPtolemaic, let's try the other systems."So you try the other systems.The Copernican system, it doesn't work either, and then you trythe Tychonic system, and still you fail.So, now five years has passed, and you still have no clue.Five years, folks, that's a long time to work on one problem.After that, is it surprising that at some point you're willingto throw the whole thing out and simply start over?This is basically what Kepler did.He didn't start over with the data.The data was good.That was Brahe's 20 years worth.That was obtained with good instruments and had beenchecked and double checked, and in fact, it predicted the planetarymotions better than anything else before in history.There was that nagging eight minutes of arc.So, before we get to Kepler's solution, we need to takea little side trip to the land of shapes to discoverthe properties of the Conic Sections.Silico: "What kind of music do they play?"What kind of music does who play?Silico: "The Conic Sections."The Conic Sections?What are the Conic Sections?The sections of a cone, what else?No, actually the conic sections are geometric shapes which areobtained by slicing a right circular cone.OK, OK, I can see your geometry's a little weak.But 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.

Oh, wait. Before I go on with this, I have to remind you that you're notexpected to know the mathematical details of these conic sections. It's important to see how the shapes are related as a familyof curves, whose properties have much in common with one another.In fact, so much in common that Kepler was willing to considerthem to replace the circular motion which had beenpart of the paradigm for 2000 years.It was not an easy thing for him to do.So the cone is constructed like this.A right circular cone.Well you start with a circle and then you draw a lineperpendicular to it, sort of a circle like this, and you drawa line perpendicular to it like this, and then you make an anglelike that and you rotate the whole thing around so you wind up with a cone shape.Now everybody knows what a cone shape is, everybody'seaten an ice cream cone before.So, a right circular cone is simply a cone which has a circular baseand which the axis of the cone is a line that passesthrough the center of the circle.OK.So we have a cone.So now what?So now I'm going to cut the cone at various anglesto see how these shapes are generated.So, let's go to the ELMO.

OK, so here I have a cone and this is actually a cone of Mozzarellacheese and it's not a perfect cone, but it's not a bad sculpturing job.So what I want to is first of all to point out to you the geometry of the cone.The axis of the cone is a line that runs vertically up and down.So, if I turn the cone like this, I've now got the axis lineparallel to the table and if I make a cut of the cone along here,I'm cutting the cone perpendicular to the axis.OK.So, I'm cutting it parallel to the original circle.And you see that what falls off of here is circular.Well, it's not perfectly circular because I didn't cut the cheeseperfectly, but you can see it has a basically circular outline.OK, so what did you expect, magic?It's a circle.No problem, it has to be a circle because we cut italong the right part of the cone.So, this time I want to cut the cone now at an angle to the axis.Oops, who did that?Get back up there.Silico, did you, quit playing around with stuff.So, if I now cut it not parallel to that original circle, but cut itat an angle, like this, what do you suspect's going to happen?What do you think the shape's going to be like?Well, we've all done something similar like this before, I'm sure.I'm sure you've sliced a carrot or a bean along the bias.You don't get a circle, what do you get?Anybody?You get an ellipse.It's an elongated circle.Doesn't really show up much as an ellipse, does it?But, well you get an elongated circle.That's an ellipse.You can try this at home with a piece of clay or something.Or, if you want to have a party, you can up pieces of cheese.

OK, now what about the ellipse?The ellipse is almost a circle, isn't it?In fact, if we cut the cone at any angle along the bias, we'll get ashape that looks more and more elliptical.OK, the more I cut it the longer the shape would be.The longer the ellipse would be.In fact, we might have a problem, like you did with this piece,deciding whether a particular cut wasa circle or an ellipse, because how do you tell the difference?Well, an ellipse is longer and sort of like a squashed circle, right.In fact, sometimes you can't tell the difference between anellipse and a circle, unless you make very careful measurements of it.So, the sharper the angle, the less it looks like acircle and the more it begins to look like an ellipse.

OK.So, you notice that by the way that on this cross section thatthe cone is not really centered on the ellipse.In fact, the center of the cone, let me turn it this way, the side.You can see that the axis of the cone which runs through here isnot in the center of the ellipse, but rather it's offset to one edge.In fact, it intersects the ellipse somewhere around here.OK?So, unlike the circle which has the axis right through the center,the ellipse has the axis sort of offset from the center this way.Silico: "Offset to one side.Doesn't that sound like a Ptolemaic device, like an eccentric or an equant?"Well, what do you think?Is it like an eccentric or an equant?So, the circle and the ellipse are not the only shapes weget when we're cutting the cone.What happens when we cut the cone parallel to one of the sides?In other words, suppose I now take this and cut the coneparallel like this to one side, what kind of a shape are we going to get?Notice that the angle of the cut here is the same as the angle of the cone, and what sort of shape do we get?Well, look at this. It's a different sort of shape. In fact, it's no longer an open shape, but rather it's a,no longer a closed shape, but it's an open shape.You can see that it doesn't really look like the circleor the ellipse, but it has something in common with them.It's elongated. The difference is it doesn't close on itself on one end.

You also notice that the axis still forms a trace which is notin the center, if we can go back to the original cone here. The axis still forms a trace here which is not in the centerbut offset to one side of the parabola that results. The final section that I want to cut here is a hyperbola.We're not going to spend a lot of time on this,and I'll go through this pretty quickly. But, the if I cut the hyperbola, if I cut thisexactly parallel to the axis. Let me do it this way. If I cut this exactly parallel to the axis, like this,I also wind up with a curved shape.Right, there's the section.You notice that it has a lot in common with the parabola. This is a hyperbola or hyperbolic shape.It's a lot wider at the base and not nearly as pointed on the end.

OK, I'll keep this brief and to the point because the figuresthat we're really interested in as far as our effortsat this point go are the circle and the ellipse.Because, after all, it's the circle that was consideredto be the perfect Pythagorean model.OK.Well, let's, let's...done with the ELMO, let's go back over here.So, you might notice at this point that a couple of things,the circle and the hyperbola, as conic sections go representvery special cases, because there's only way you can cut the cone to get a circle.That is, perpendicular to the original axis.There's only one way you can cut the cone to get a hyperbola.That is, parallel to the original axis.So, what would happen if you randomly cut the circle,or the cone, what kind of shape would you get?Well, the chances are it would either an ellipse or a hyperbola. So the odds are overwhelming, in fact, that you would get eitheran ellipse or a hyperbola or a parabola, depending on whetheryou cut the cone at a greater or lesser angle. So, this is not the best way to see this.It's hard to see the three dimensions here on the television screen.So you may want to study these figures in the textand Study Guide and if you feel adventurous, make some conesand cut them in various ways and see what you get.By the way, can you think of other situations in naturewhere you might see these shapes?In terms of being related to cones?This would make a great topic for a writing exercise for today.

Well, now I think we're ready to get back to Kepler and to tryto see the relevance and significance of these cones and conic sections.So, now we're ready to go back and look at Kepler and the circle to see why it was that Kepler felt justified in abandoningthe circle after all those thousands of years of sphericaland circular perfection based upon the Pythagorean and Platonic ideals.Kepler felt justified in trying an elliptical orbit for Mars for the reason that I've already mentioned, the fact that nothingelse seemed to work and in desperation he knew that therehad to be some explanation for it.But, it turns out from what we saw with the conic sections that the ellipse is really not so much different from the circle.In fact, it's only slightly different and it's onlyslightly imperfect, and it's related to the circle. It's in the same family as the circle.It's a Euclidian thing, right?In fact, Euclid was the first one to deal with the conicsections and talk about their properties.

So, it's still possible to save the appearances by making the slightly imperfect circle correspond with what appearsto be a slightly imperfect universe. Besides, Kepler was a religious guy, right?Kepler thought, "If God wants to make elliptic orbits instead of circular orbits, he can do it because God can do whatever he wants," regardless of what Plato or Aristotle or anyone else had said. This is kind of an interesting approach here because God hasnever come into the picture in this particular way before.Remember that Kepler was a religious man as well asbeing a Pythagorean mystic.

So, the remarkable thing is that when he tried theelliptical shape for Mars, it fit perfectly.Can't you imagine Kepler sitting there sort of scratching hishead saying, "Ah, geez, circles are... what shape can I try.Let's see what's close to a circle.Well, it's not a cube, it's not a square, let's see, ah, the ellipse."So, you know, the fact that he was a good Pythagoreanmathematician, a good Euclidian mathematician enabled him to do this.Most of us probably would not know the propertiesof these figures well enough to have been able to do this in the first place.So, going back to the other planets, including earthand trying ellipses for their orbits, Kepler also found them to fit perfectly without the need for any devices whatsoever. No devices, whatsoever.So, what we see here is that Kepler discovered that you could fit these planets into a very few, in fact, five, well six,if you count the earth, elliptical shapes, rather than the 30 or socircles and epicycles and so forth that Copernicus had come up with.

So, what have you got here?I mean, on one hand you've got keeping the perfectionof the circle, but adding complexity.On the other hand, you make the circle slightly imperfectand then you come up with a real simplicity down to one sphereor one, I should say, one orbit per planet.So, here we see a perfect example of data challenging a paradigm.Brahe's observations were so good that it was no longer possibleto design a geocentric system which fit them at all.Sure, the discrepancy was small, but it was significant enoughin Kepler's eyes to warrant a change in the paradigm.We'll see this happening all the way through the course thatwhen it comes down to deciding between one theory,one hypothesis, one possibility or another, that it's very oftenthat numerical or the quantitative numbers that make thedifference in deciding between one theory and the other.And it's the precision of the theory. The more accurate the numbers become, the more able you areto decide between the truth or the reality of one theory versus another. So, Kepler changed the paradigm, or did he?Is this really a change in the paradigm?The ellipse is very close to the circle, they're related families.I would suggest that he didn't really break the paradigm at all. He just sort of "bent" it. Or, if you prefer, he sort of squashed it, squashedthe circle down into an ellipse.

OK, so a brief summary of the program. We covered a lot of ground in here.Let's go all the way back and sort of give a little summary.Tycho Brahe was the first professional astronomer.He convinced the King of Denmark to give him money to buildand fund a Royal Observatory where he would observeand track the planets night after night for 20 years using veryprecise instruments that he invented.With a staff full of assistants, he invented and built theseinstruments which allowed an observational precision never before possible.He also had a full life support system basically on the Islandof Ven including a printing press and a castle and full servant support and so on.So he was very well set there for 20 years. His data allowed him to determine that Ptolemy's calculations and Ptolemy's methods did not predict the paths of the planetsnearly as well as had been thought previously.

The observational precision of his instruments also allowed him to determine that two comets and a supernova were heavenlyevents and did not occur in the sublunar realm where they wereallowed by Aristotle's cosmology, and where it had been previously assumed that those kind of events occurred. Brahe hired a young assistant named Johannes Kepler.Upon Brahe's death Kepler was given his data and his job,largely because Kepler proved his genius in working along with Brahe.Kepler, being a good Pythagorean mathematician and a diligentworker, set out to prove one and for all that the truthof the Pythagorean harmony and the Platonic perfectionof the universe did really describe the reality of things.Although he failed to do so, and considered that he wasa failure, he did discover that the motions of the planets couldbest be described by a geocentric model which had all the planetsincluding earth orbiting the sun in elliptical orbits rather than in circles.He did find that the planetary orbits contained geometricand harmonic relationships previously undiscovered,and although he could not explain them, he thought that he tapped in here to this sort of mysterious Pythagoreanmysticism that he was looking at.Kepler's laws which is the topic of the next programwere really significant in several respects.

So in the next program we will examine the laws of planetarymotion and discuss their significance in this river of scientific heritage. Well, here we are, I guess, out of time again, so don't forgetto write a response of some kind for this program and send it inbefore the deadline and check the schedule in your Syllabus for the deadline.So, don't forget, when it comes to science, get physical.OK, guys, it's snack time, come let's have some cheese, come on Russell.I cut all this cheese up, let's have some cheese.There you go, take this good stuff, good stuff.Come and get them.You can have some too Alan.Well, we're gone.Bye.Silico: "I liked that program. Finally, we get down to some real science."Music