The Recovery of the Ancient Learning

Astrology  & Astronomy in the Middle Ages

By Michael Johnathan McDonald.

• Tycho Brahe

• Johannes Kepler

• Why are they Important?

Tycho Brahe

Significance: He is credited with the most accurate astronomical observations of his time, and the data was used by his assistant J. Kepler to derive the laws of planetary motion.

Key Phrase: so many redundant observations.

Key system: modified Geocentric.

Contemporary cultural relevance: Tychonic system became the major competitor with Copernicanism, and was adopted by the Catholic Church for many years as

 its official astronomical conception of the universe.

Tycho Brahe , born Tyge Ottesen Brahe (December 14, 1546 – October 24, 1601), was a Danish nobleman astronomer as well as an astrologer and alchemist. He was granted an estate on the

 island of Hven and the funding to build the Uraniborg, an early research institute, where he built large astronomical instruments and took many careful measurements. As an astronomer, Tycho

 worked to combine what he saw as the geometrical benefits of the Copernican system with the philosophical benefits of the Ptolemaic system into his own model of the universe, the Tychonic

 system. From 1600 until his death in 1601, he was assisted by Johannes Kepler, who would later use Tycho's astronomical information to develop his own theories of astronomy.

He purchased an ephemeris and books such as Sacrobosco's Tractatus de Sphaera. Tycho's naked eye measurements of planetary parallax were accurate to the arcminute. These jealously guarded measurements became the possessions of Kepler following his death. Tycho was the last major astronomer to work without the aid of a telescope, soon to be turned toward the sky by Galileo.

 On November 11, 1572, Tycho observed a supernova for a ' new 'star.' This discovery was decisive for his choice of astronomy as a profession.

 

Kepler tried, but was unable, to persuade Tycho to adopt the heliocentric model of the solar system. Tycho believed in a modified geocentric model known as the Tychonic system, for the same reasons that he argued that the supernova of 1572 was not near the Earth. He argued that if the Earth were in motion, then nearby stars should appear to shift their positions with respect to  background stars. In fact, this effect of parallax does exist; it could not be observed with the naked eye, or even with the telescopes of the next two hundred years, because even the nearest stars are much more distant than most astronomers of the time believed possible. The Tychonic

 system is very similar to the Copernican one, except that it has a static earth instead of a static sun.

 

Tycho Brahe 's religiosity and firm belief in astrology including the desires to reform astronomy

 by these methods inspires to calculate more accurate positions for the planets. He uses strict

 and constant physical observation. He divides the universe again, with the induction of what are

 comets and paths of the tail. Therefore the unchanging and changing are trimmed down a little as

 comments link the corruptible to the incorruptible - the changing and the unchanging.

 

• Tycho Brahe, most detailed observations up until his time. Whittled down the variable of arc to somewhat a minuet in some cases. Tycho improved the accuracy of arc by a factor of 10 over Copernicus.

• Tycho became the first astronomer to complete the great set of planetary data for motion around the sun. His constant measurements coupled with his new techniques credited himself with the most accurate astronomical observations of his time.

Trivia:

 

Tycho had trouble with his water clock, apparently the mercury flux, so this little problem helped him create new methods to calculate the 'natural positions' of the stars.

                    Problem sets for Tycho

      Problem? A lousy clock at Uraniborg.

        Significance: This problem' solution created the Following technique to correct this problem.

1. Couldn't do Right Ascension.

2. Most reliant on the sun.

3. Measured from the sun.

4. Needed intermediate marker because one cannot look at the sun and keep one's eyes in tact.

5. Used Venus. In 1582 he had a good view of Venus.

6. Then he measured the angular distance before Venus to the sun.

7. Now he has a reference point.

8. After sun goes down, he uses Venus to make another reference point. This created a good solution.

9. Then once he gets a star for a reference point, he later used a Ran star, in the zodiac constellation of Aries.

10. This technique created a good calculation of arc. In fact, down to seconds of arc. Ptolemy would be happy with four-minuets of arc.

                    Now a problem with the tilt of the earth

                        How do I figure in the variables for the tilt of axis of earth?

                    Tycho had to revert to trigonometry.
 

Finding the moving sun.

 

α_ws     =  90  -  Ф   - Є

 

α_ss     =  90  -  Ф  + Є

 

α_np   +   α_ss  = 180° - 2Ф

 

Ф (sun)  ≠ Ф (north pole).

To get around the seven digit calculations, a tedious process, he decided to cheat. Here is the formula.

cosA cosB  =  ½  [ cos (A-B) - cos (A+B) ]

This is the way he got around the long seven digit calculations!

            Atmospheric Problems

Refraction: Refraction of light raises the star from the earth's vantage-point because of the atmospheric disturbance. However, to make correct observations, the stars at the zenith remained with a radius of strait-light, i.e. no refraction. Therefore, Tycho's refraction tables remain pretty good.

However, his parallax measurements were not so good.

Parallax: Parallax depresses the starlight.

Comets and Tycho

Tycho was the first to put a circle trajectory with a comet that goes around the sun and the tail opposite of the sun. Comet of 1577, corresponded to his parallax studies. Tycho's system of comets helped his theory of orbits of planets around the sun.

Planets go around the sun but the earth is still the center of the universe. Jesuits  adopted this until the 1700s.

The earth was the supreme existence in the universe, to place it not at the center of the universe created a heretical issue.

Tycho improved the accuracy of arc by a factor of 10 over Copernicus. Still small discrepancies needed addressing in which Kepler will struggle with.

Johannes Kepler

True founder of the New Astronomy.

Significance: True founder of the New Astronomy. A new celestial geometry.  Three laws of planetary motion.

Keyword: elliptical.

Expertise: Mathematics.

Most important fact: Used the true sun and not the mean sun.

Very important cultural significance on schools: He entered school on a pilot program to give under representative students a chance at an education.

This was a limited experiment, but an early example of public programs to represent good students who studied but were otherwise too poor to pay for school.

Personal-cultural significance: He is a product of the Reformation.

The three laws:

Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution and the true founder of the New Astronomy, was a German

 Lutheran mathematician, astrologer and imperial astronoger, and astronomer. He is best known for his laws of planetary motion, based on his Astronomia nova,

 Harmonice Mundi and the textbook Epitome of Copernican Astronomy.

Through his career Kepler was a mathematics teacher at a Graz (Southern Austria and heavily Catholic) seminary school (later the University of Graz), an assistant to

 Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, and court astrologer to General Wallenstein. He also did fundamental work in

 the field of optics and helped to legitimate the telescopic discoveries of his contemporary Galileo Galilei.

In 1587, after moving through grammar school, Latin school, and lower and higher seminary in the Lutheran education system, Kepler began attending the University

 of Tübingen as a theology student, where he proved himself to be a superb mathematician and earned a reputation as a skillful astrologer. It is here he met and then

 learned under a teacher he says profoundly influenced him. Michael Maestlin (1550-1631) was a German astronomer and mathematician. He was a professor at the

 University of Tübingen for 47 years. The two corresponded for some. Although he primarily taught the Ptolemaic view solar system, he was also one of the first to

 accept and teach the Copernican view in which Kepler fell in love with astronomy.

Kepler left Graz after a new leader decided to kick-out all protestants, and gave him one week to leave. He had roots and a family. He left for Prague. About the same

 time Tycho had a similar fate, in that his King patron died and his King's son had a different attitude about Tycho's investigations and its significance. So Tycho

 packed up his equipment and moved to Prague. This is were the two met. Kepler needed to win Tycho's trust and finally was given the job of finishing the Tychonic

 Universe before Tycho passes away from a cerebral hemorrhage.

As Imperial Mathematician, Kepler inherited Tycho's responsibility for the Emperor's horoscopes as well as the commission to produce the Rudolphine Tables (1627). The Rudolphine Tables, his  final project and dedicated to his former patron King Rudolpho II,   proved the greatest measurements of planet’s true positions up till this time. He manages to predict them within one second of arc. Ptolemy would be happy with  a few minuets of arc.

His most significant achievements came from the realization that the planets moved in elliptical, not circular, orbits. This realization was a direct consequence of his failed attempt to fit the planetary orbits within polyhedra. Kepler's willingness to abandon his most cherished theory in the face of precise observational evidence also indicates that he had a very modern attitude to scientific research. Kepler also made great steps in trying to describe the motion of the planets by appealing to a force which resembled magnetism, which he believed emanated from the sun. Although he did not discover gravity, he seems to have attempted to invoke the first empirical example of a universal law to explain the behaviour of both earthly and heavenly bodies.

Kepler trivia:

1. Kepler: Architectonics come to him after contemplating on the musical theory and his old teachings of the astrology phenomenon called the trigon cycle.

2. True founder of the New Astronomy.

3. He opposed an infinite Universe.

4. Thought the universe contained, or more aptly, composed of ice.

5. Philosophy: Cosmos not formed by chance. God had a reason and a order. God had a plan, not random. No chaos.

6. Sought underlying reason.

7.  

8. Concerned with finding the constructional laws, also referred too as 'Architectonics'.

9. Kepler was the first, and the only one up to the time of Descartes, to ask for a physical description. (Koyré  121).

10. Kepler always raised questions that others never thought of or were to scared to ask. This significance brought progress and a ' New Astronomy'.

11. He was the first to conceive and conscientiously follow, if not realize, a programme directed to the scientific unification of the starry Universe and the

12.  sublunary Universe, and substitute celestial dynamics for kinematics of circles and the metaphysics of spheres in traditional astronomy (Koyré  119).

13. He based his dynamics on solar astro-biology by ascribing to the Sun a spiritual, perhaps even intelletive, driving force, and to the Earth not only a spiritual,

14.  but also a sensory driving force (Koyré  119).

• Kepler, placed the true sun instead of the mean sun as the measurement.

  • 'Vicarious hypotenuse,' in trying a simple Ptolemy's scheme is his first approach to mars.

  • Three laws of motion.

  • 1596 Kepler is called the calculator. He pursues his seven year war with Mars. 

• Copernicus's center is the mean sun, not the true sun. The mean sun is our clock, but everyone knows out clock is not accurate.

• Aristotle became the first person to discuss and write upon the universe and astronomy. His cosmology would gain much acceptance up until the 1500’s, where

•  Copernicus and Galileo began to figure out that Europe is not the center of the universe. As a young man it is astrology that starts Kepler in the direction of his

•  life, First question was the calendar reform for him.

He is sometimes referred to as "the first theoretical astrophysicist", although Carl Sagan also referred to him as the last scientific astrologer.

Johannes Kepler Trivia:

1. He had a miserable beginning.

2. His mother was a witch and tried, and he represented her in court and won.

3. His father was a vagabond, soldier.

4. He was premature at birth and sickly.

5. Poor and needed affirmative action to get into school. The program recognized his brilliance.

6. He did his horoscope to make sure he came from a bad sign - he wanted to understand why he was the way he was.

7. He said when he was little he gnawed on bones and ate hardened bread -crust.

8. He received a good education, mainly in Greek as the northern schools curriculum.

9. He, of course, trained as an astrologer as was the curriculum which helped him calendar problems.

10. Kepler began attending the University of Tübingen as a theology student, where he proved himself to be a superb mathematician and earned a reputation as a skillful astrologer.

11. He learned Copernicus in school, at this time a few places were beginning to use his work. He fell in love with the ideas of Copernicus' universe.

12. He said this is religion. The Sun is a better proxy of God, then the earth.

13. We need a triune God. The Sun, the Earth and the emptiness is the holy ghost.

14. Kepler wanted to become a pastor.

                   Kepler's motivation.

Why are there six planets in Kepler's universe when Ptolemy had seven?

He dropped the moon because it couldn't fit into his new geometrical system of distance.

Graz Sothern Austria was heavily Catholic, but for  a brief period of time in certain regions Protestants moved in while the king allowed this due to peace. He was a

 teacher, lecturer of astrology and math. He took Stadius' position. Then the king died and his son reverting back to the old norms asked for everyone to get out. He

 gave Kepler seven days to move, although, Kepler had roots with family and kids in Graz. Kepler noted the astrology charts of bad omens for Protestants in Graz.  At

 Graz, Kepler publishes prophesying almanacs and predicts Turkish events. The archduke kicked him out of Graz because he wanted to impose his religion on the

 region. He left to Prague. Tycho left for Prague after being kicked out/ or leaving his island observatory do to political changes too. His patron, the king died and the

 new one decided the island was more useful for other reasons.  They met up in Prague. They argued together because instantly Kepler wanted to see all the data that

 Tycho had compiled, but Tycho was apprehensive and a little secretive. Rudolpho II (1552-1612, Roman emperor, son of the emperor Maximilian II )  employed Tycho

 and possibly the astronomer/astrologer didn't want his data open to others to get work. However, Kepler finally signed on to Tycho's work and after the Danish

 nobleman died he retained all his data and Rudolpho hired him. Rudolpho was fond of astrology.

Rudolpho died and he went back to Tubingen and talked for a commission of his architectonic structure in a cup of silver, but it didn't happen. This why we only have a

 written record of this.

• Tycho not wanting to be rude at state function and get up from the table and go to the bathroom, split his gut and died.

• As now an Imperial Mathematician, Kepler took responsibility for the Emperor's horoscopes as well as the commission to produce the Rudolphine Tables.

The Jesuits, ironically, noticed his brilliance and asked him to return, in which he did.

He no longer taught at a school, but was used as a computer to calculate mathematical sets. This experience gave him confidence and expertise along with speed.

 Kepler and Religious references to his work.  Kepler wanted to become a pastor before he was side tracked by the love of Copernicus' universe. But his

 theories remain cemented in religion.  Kepler: two universes the 'mobile', now only five bodies and the unmovable (stars). There are six bodies in motion around the

 Sun. The Sun is motionless and the driving force.

His trinity theory appears inspired by Nicholas of Cusa - architectonic...( Koyré,  154). The reasoning mearly points out that Copernicus' universe is not heliocentric, but heliostatic. Therefore the Sun plays a larger role and the earth plays a less significant role in the Universe contrary to  Ptolemy as it does in Copernicus' work. (Koyré, 155).

God the father was the Sun. The empty space the holy ghost. Earth was imperfect and so it goes was hell. The sun was paradise.

Johannes Kepler's Polyhedra

Trigon of astrology

mysticism

Revelation

"I see the secret of the Universe"  - Kepler.

The Trigon of astrology leads him in a direction to explains why there are only six planets. Kepler creates charts of trigon conjunctions of Jupiter and Saturn and he

 notices something that becomes somewhat of a satori for him. When the table is complete he noticed a circle in the center absent of anything but space. He therefore

 knew that this was the difference in diameter of Jupiter and Saturn. Therefore, he quickly decided to understand if all the planets could be geometrified so that they

 would all relate to each other and in a form and also correspond to a geometric structure. However, this proved inaccurate or a point that beauty and nature are not

 always complimentary. It is not perfect, but close, and he eventually let it go which was a first example of how modern astronomers should work. If it doesn't work,

 then move on. However, it had significant meaning to his life and his discoveries.

 Astrology leads to his great discovery on 'Architectonics'

The Platonic Solids

These five essential shapes are the only conceivable solids whose sides and edges are exactly the same. They are also the only perfectly symmetrical three-dimensional polygons.

Icosahedron (20 sides) :Dodecahedron (12 sides) :Octahedron (8 sides) :Hexahedron,cube (6 sides) :Tetrahedron (4 sides) (Wilson).

Kepler  lectured on trigons and he no doubt became an excellent intellect upon its mysteries. This, of course, fitted into astrology as he discussed fiery trigons and repeatable element conjunctions.  When he came upon his satori, and decided to place a triangle into the orbit of schemes, he came across his brilliant discovery.  What he needed to do he did and dropped our moon as a planet to a mere satellite (Copernicus was the first to drop the moon as a planet)  and implanted the six remaining spheres into these above geometric perfect shapes. Then calculated their radiuses.

• Take 1000. Calculations below.
• The relationship between the radii and the sphere.

"The agreement is decidedly not perfect, but it is too significant to be accidental. The discrepancy between the data and calculation is considerable in the case of

 Jupiter and Mercury, though we ought not to be surprised thereat in view of the enormous distance which separates us from the former, and the difficulty in providing

 a satisfactory theory of the motion of Mercury" ( Koyré 147).

"This geometric relationship served Kepler as a mental model of the solar system" (1.1 Kepler). The crack-pot theory proved genius. Copernicus, Kepler concluded had

 made the mistakes. This was a confident move, knowing that Copernicus' intellect remained admired. " Kepler said. " This crackpot idea is better than Copernicus." "

 He must be wrong, not me." This initial astrology knowledge led to the three laws of motion. Copernicus happened to use the mean sun instead of the true sun,' " in

 order to shorten the calculations' and ' in order not to alarm his readers by departing too much from Ptolemy' "(Koyré  147).

The motion of the planets believed in ancient times to have the same velocity. Kepler disagreed. "  The periods of orbits are 'not' directly proportional to the distance, but differ considerably from therefrom..." (Koyré 150-1). See the soul and the further away from the sun the weaker the 'motive.' For animal Motive and material motive and adoption then late change of mind, see Joseph Scaliger influence with Kepler. The further away from the sun the slower the souls ( Force) moved the 'movables.' The planets had moderate activity and the stars no activity, but the Sun had the greatest because it was the force that moved the planets around itself, just as God is the prime mover of life.

The trinity. God is the trinity. Therefore,

God's image is revealed in the sphere by God the Father at the center, God the Son on the surface, God the holy Ghost mid-way ( emptiness)  between the center point  and the circumference (Koyré 143), ( See satori of trigon, above) .

                    Why did the ancients think the Earth was the center of the Universe?

They didn't understand that the Earth spun on its axis. Therefore, if the earth went all the way around the sun in one day - it would fling-it-out into the universe in a violent motion. Thus, the earth was the center of the Universe.

" 'If the Earth revolved?, said Ptolemy of Alexandria, that is to say made one revolution in a
 day.... this motion, which is twenty-four hours passes over the whole circumference of the Earth, would-be extremely violent and of unsurpassable speed.. Now things move in a violent motion seem to be totally unable to gather together, but rather to disperse...'" (Koyré 112).
 

The 7 Year War on Mars

'Vicarious hypotenuse,' in trying a simple Ptolemy's scheme is his first approach to mars. 900

 pages of calculations comes to naught, but the process leads to understandings that will factor into his discoveries about motion. Kepler talked to Tycho on his deathbed and was giving Tycho's works with promises that he finish the Tycho universe.

Three laws of planetary motion.

Chronology:

Architectonics, the platonic solids and the placements of the planets and their orbits.

Music of the spheres (two day's before the publication of Harmonice Mundi, Rudolpho paid for the publishing but not Kepler's salars ): Kepler, decided to take a break from his long war with mars and picked up a book on Music theory. He needed a change of pace from his 900 logins from his pages on Mars. Chronologically this is his second law. By using notes visualized on a music sheet he maps out the eccentricities of the planets. Therefore, Mars has the greatest musical variation while Venus has the least. The object was to get the pitch ( tone) of the  planets. Venus is almost a circle in its orbit therefore the notes on the music page are all the same notes.

 This would indicate a pitch for Venus.

Area/time = constant= total area/period is chronologically his third law, but it is called his second law.

The Titius-Bode Law is rough rule that predicts the spacing of the planets in the Solar System. The relationship was first pointed out by Johann Titius in 1766 and was formulated as a mathematical expression by J.E. Bode in 1778. It lead Bode to predict the existence of another planet between Mars and Jupiter in what we now recognize as the asteroid belt [2.8] (Bode).

Hershel discovered Uranus, but he knew it was out there by this law. Somehow this law skips Neptune and the next value is right smack on Pluto, but many do not

 believe it is an actual planet.

"The amount of "flattening" of the ellipse is termed the eccentricity". (The Online ).

          Law one is actually his second law in regards to chronology of events:

I. Elliptical orbit law.

The orbits of the planets are ellipses,
with the Sun at one focus of the ellipse.
 

"Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this" (The Online ).

Vicarious Hypothesis ( eccentric circle, bi-sect and equant point). He knew it wasn't true, but still leads him in a correct direction. 

Brought back out of his failure for Mars a rule. Equal times sweep out equal distances: arcs = areas . He gets rid of the eqaunt point.

This leads to the Area law ( the absence of an equant). He asks what is the shape of this stuff ( the area)? He uses computations from Tycho, and now defines the shapes of the sectors in a circumference. XC=CS, as X increases with time. He wrote 900 pages on this. The eccentric circle area doesn't work either, but it gives him a clue. What about equal quadrants of time. Therefore = arcs= areas. The Area Law is now absent of all equants. He asks "what is the shape of this stuff?" ( that is the quadrants?). After he defines the shape.

 

In a letter of a friend in playfulness says. " Circles are the definitions of Astronomy," his friend says, " you destroyed Astronomy."

Kepler writes back: "if matter were more tractable, then of course, stars and planets would travel in circles." He is saying that if the planets and stars traveled in a circle Astronomy would have been solved many millennia ago, in which he is correct.

 

Astronomia nova (1609)  has the presentation of the law.

Equal areas in equal times. Chronology this is his first law.

II. The Area Law.
 
The line joining the planet to the Sun
sweeps out equal areas in equal time
as the planet travels around the ellipse.

Arcs = Area
"Kepler's second law is illustrated in the preceding figure. The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it

 is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest approach of the

 planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. Hence, by Kepler's second law, the planet moves fastest when it is near

 perihelion and slowest when it is near aphelion" (The Online ).


III.  The Harmonic law - Constant Proportionality.

The time required for a planet to orbit
the sun, called its period, is proportional
to the long axis of the ellipse raised to
the 3/2 power. The constant of
proportionality is the same for all the
planets. (more in-depth below).
 

 

Kepler's 3^rd Law: P² = a³

 

P₁² / P₂²

―――

P₂² / R₂³

Area/time = total area/period= (t²/r³⁾

One can see that column one is off and  Mercury clearly must be downsized and Saturn must be increased. So Kepler, took t²/r³ and found out all the r=1.0.This became his third law.

Kepler's 3^rd law is a mathematical formula. It means that if you know the period of a planet's orbit (P = how long it takes the planet to go around the Sun), then you can determine that planet's distance from the sun (a = the semimajor axis of the planet's orbit), because P² = a³. t also tells us that planets which are far away from the Sun have longer periods than those close to the Sun. They move more slowly around the Sun. Look at the formula: if you make the distance, a, large, then P must also be large (University).

" Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same" (The Online ).   

Kepler's Universe is 4 million Solar diameters. He makes it three times larger than Tycho's.

r _saturn /d _saturn =  d _saturn  /r _star

Kepler, after Rudolpho dies, dedicated to his parton's name, his planetary tables, called the  Rudolpho tables.

Kepler's final work is the Epitome of Copernican Astronomy. He tried to derive something that Newton about 70 years later get on too. The magnification attraction of the planets and the sun. This was mathematical physics applied to astronomy, which would end up as a new field with Newton.

To Kepler this was the essence of God's thoughts on creating the universe. He moved after a Protestant crackdown and hooked up with Wallenstein who gave him shelter.

Kepler disdained astrologers who pandered to the tastes of the common man without knowledge of the abstract and general rules, but he saw compiling prognostications as a justified means of supplementing his meager income. At least 800 horoscopes and natal charts drawn up by Kepler are still extant, several of himself and his family, accompanied by some unflattering remarks.

Work Cited:

Bode's Law Department of Astronomy. ( Ithaca: NY Cornell University, 2006),  <http://astrosun2.astro.cornell.edu/academics/courses/astro201/bodes_law.htm> 2006.

 

Khun S. Thomas. The Copernican Revolution 'Planetary Astronomy in the Development of Western Thought'. by the President of Fellows of Harvard. 1957. r. 1985. 30,32.

 

Heilbron, J. L. Astronomy and Astrology in Medieval and Early Modern Europe. Represented in University of California Berkeley Reader. History 181A. January 2006.

( Astrolabe/23 J. L. Heilbron).

 

1.1 Kepler. <http://cti.itc.virginia.edu/~meg3c/classes/tcc313_inuse/Book/chap1/kepler.html>2006.

 

Koyré, Alexander , The Astronomical Revolution: Copernicus-Kepler- Borelli 2nd ed. ( Ithaca: Dover Publications, Inc, New York),1992.

 

Wilson, Jonathan . Symmetry:two sides to every story  <http://www.geocities.com/Omegaman_UK/symmetry.html> 2006.

 

The Online Journey Through Astronomy, Johannes Kepler: The Laws of Planetary Motion , (www.onlineastronomy.com), <http://csep10.phys.utk.edu/astr161/lect/history/kepler.html> 2006.
 

University Corporation for Atmospheric Research, Windows to the Universe, , 1995-1999, 2000, <http://www.windows.ucar.edu/tour/link=/the_universe/uts/orbits_data.html

&INPUT_STRING=Kepler&SEARCH_TYPE=phrase&USER_LEVEL=mid&edu=high> 2006.

Notes: Trigon and history with it.

Orbital Data for the Planets

March 10th, 2006

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