discussion summary practice problems




Aristarchus of Samos (310–230 BCE) Greece

  • the earth is bigger than the moon (~3 times larger – 8:3 to be more exact) can be seen from the shadow of the Earth on the moon during a lunar eclipse
  • the sun is very much farther away than the moon, as can be seen by the fact the time separating first quarter and third quarter moons is the same as the time between the third quarter and the first quarter
  • the sun is also very much larger than the moon as can be seen during a solar eclipse
  • the sun must be the center of the universe, since it is so very, very large
  • the stars are very far away since there is no apparent parallax

However, Aristarchus’ only remaining work on the topic, On the Sizes and Distances of the Sun and Moon, is geocentric!

Pythagoras of Samos (582–496 BCE) Greece (Ionia)


Nicolaus Copernicus (1473–1543) Poland (latinized version of Mikołaj Kopernik). De Revolutionibus Orbium Cœlestium (On the Revolutions of Heavenly Spheres) 1543.

In medio uero omnium residet Sol. In the center of all rests the sun.


  • People gave ear to an upstart astrologer who strove to show that the Earth revolves, not the heavens or the firmament, the sun and the moon…. This fool wishes to reverse the entire science of astronomy; but sacred scripture tells us that Joshua commanded the sun to stand still, and not the Earth. Martin Luther, referring to Joshua 10:13, in his series of “Table Talks” (1539)
  • The eyes are witnesses that the heavens revolve in the space of twenty-four hours. But certain men, either from the love of novelty, or to make a display of ingenuity, have concluded that the Earth moves; and they maintain that neither the eighth sphere nor the sun revolves…. Now, it is a want of honesty and decency to assert such notions publicly, and the example is pernicious. It is the part of a good mind to accept the truth as revealed by God and to acquiesce in it. Melanchthon, emphasizing Ecclesiastes 1:4-5
  • Who will venture to place the authority of Copernicus above that of the Holy Spirit? John Calvin, citing Psalm 93:1 in his Commentary on Genesis
  • … And whereas it has also come to the knowledge of the said Congregation that the Pythagorean doctrine — which is false and altogether opposed to the Holy Scripture — of the motion of the Earth and the immobility of the Sun, which is also taught by Nicolaus Copernicus in De Revolutionibus Orbium Cœlestium, and by Diego de Zuñiga On Job, is now being spread abroad and accepted by many… Therefore, in order that this opinion may not insinuate itself any further to the prejudice of Catholic truth, the Holy Congregation has decreed that the said Nicolaus Copernicus, De Revolutionibus Orbium, and Diego de Zuñiga, On Job, be suspended until they are corrected. The Roman Catholic Church, from The Decree of the Roman Catholic Congregation of the Index which condemned De Revolutionibus on March 5, 1616


  • heliocentric with circular orbits and a few (48?) epicycles
  • The outermost sphere of the fixed stars is made immobile. Previously it was the prime mover?
  • Moving the sun to the center made calculations easier.
  • It did not work as well as Ptolemy’s geocentric system, which was at the time, more accurate.
  • No observational evidence that the Earth moved, therefore it was not taken seriously. (Before you laugh at these “foolish people”, look at the ground. Do you see it moving? Prove to me that the Earth is not fixed.)

How heliocentrism deals with retrograde motion.


Giordano Bruno (1548–1600) Italy (Naples)

  • La Cena de le Ceneri (The Ash Wednesday Supper). 1584
  • De l’Infinito, Universo, e Mondi (On the Infinite Universe and Worlds). 1584
    • In space there are countless constellations, suns and planets; we see only the suns because they give light; the planets remain invisible, for they are small and dark. There are also numberless earths circling around their suns, no worse and no less than this globe of ours. For no reasonable mind can assume that heavenly bodies that may be far more magnificent than ours would not bear upon them creatures similar or even superior to those upon our human earth.
  • De Immenso (On Immensity). 1591
  • unknown
    • Open wide the door for us, so that we may look out into the immeasurable starry universe; show us that other worlds like ours occupy the ethereal realms.


Galileo Galilei (1564–1642) Italy

three big (as in important) books

  1. Siderius Nuncius (Starry Messenger). 1610
    telescopic evidence for the heliocentric model, the first printing sold out within days (March 12), four major observations

    1. mountains on the moon — moon was supposed to be perfect”From observations… repeated many times I have been led to the opinion and conviction that the surface of the moon is not smooth, uniform, and precisely spherical as a great number of philosophers believe it (and other heavenly bodies) to be, but is uneven, rough, and full of cavities and prominence’s, being not unlike the face of the Earth, relieved by chains of mountains and deep valleys.”
    2. planets look like disks, stars still look like dots
    3. milky way made of stars — there are things invisible to the naked eye”I have observed the nature and the material of the Milky Way. With the aid of a telescope this has been scrutinized so directly and with such ocular certainty that all disputes which have vexed philosophers through so many ages have been resolved, and we are at last freed from worldly debates about it The galaxy is, in fact, nothing but a congress of innumerable stars grouped together in clusters. Upon whatever part of it the telescope is directed, a vast crowd of stars is immediately presented to view. Many of them are rather large and quite bright, while the number of smaller ones is quite beyond calculation.”
    4. moons of jupiter — was thought that moon could not keep up with moving earth (first sighted January 7, realized they were moons on January 15), Jupiter shares something in common with the Earth, both are satellites of the sun” On the seventh day of January in this present year 1610, at the first hour of night, when I was viewing the heavenly bodies with a telescope, Jupiter presented itself to me; and because I had prepared a very excellent instrument for myself, I perceived (as I had not before, on account of the weakness of my previous instrument) that beside the planet there were three starlets, small indeed, but very bright. Though I believed them to be among a host of fixed stars, they aroused my curiosity somewhat by appearing to lie in an exact straight line parallel to the ecliptic, and by their being more splendid than others of their size. Their arrangement with respect to Jupiter and each other was the following:” [Descriptive text removed between diagrams. Galileo uses a tipped-over zero or and “O” to represent Jupiter and asterisks to represent the “starlets”.]

      “Such are the observations concerning the four Medicean planets recently first discovered by me….”

      Galileo then goes on to describe how these “starlets” wander around the bright dot that is Jupiter. First appearing ahead then behind the planet, they follow Jupiter no matter which way it moves across the sky, be it in direct or retrograde motion. Like a miniature solar system (a concept that did not really exist at the time), each dot shifts from side to side within a restricted orbit, the closest one in having the shortest period and the farthest one out having the longest period. Galileo named them the “Medicean planets” after his benefactor Cosimo II de’ Medici, the Grand Duke of Tuscany. He is using the word “planet” with its original meaning of “wandering star”, which is what these objects look like when viewed from the Earth — bright flecks of light that move relative to the other “fixed stars”. The meaning of the word has shifted, so that nowadays the word “planet” refers only to the eight or nine major bodies orbiting the sun — a shift in meaning almost entirely due to the observations of Galileo. The “Medicean planets” are now the “Galilean moons”.

      Galileo discovered something new and hugely important — satellites of a satellite; bodies orbiting a body orbiting another, more significant, body. Moons! The comparison is earth-shaking. Jupiter is a planet (in current language, a major body orbiting the sun). It has bodies that orbit it. The earth has a body that orbits it. We call it the moon. The earth has one moon. Jupiter has many moons. Galileo saw four, but that was just the beginning. Jupiter is now known to have nearly sixty moons. This means that Jupiter and the Earth are similar and maybe even equal in stature. Jupiter is a planet and so is the Earth. (Gasp!)

      “Here we have a fine and elegant argument for quieting the doubts of those who, while accepting with tranquil mind the revolutions of the planets about the sun in the Copernican system, are mightily disturbed to have the moon alone revolve about the Earth and accompany it in an annual rotation about the sun.”

      Given what we now know about the cosmos, it gets even worse for the geocentrists. The earth is certainly something, but now Jupiter is more than the Earth, and the sun is even more than all of them combined. The earth has one moon, one satellite, one thing that orbits it. Jupiter had four in Galileo’s day and about sixty in our day. The sun had six in Galileo’s day — Mercury, Venus, Mars, Jupiter, Saturn, and now the Earth too. When you include all the objects known today — planets, asteroids, comets, Kuiper belt objects, and the Oort cloud — you up the number to about a thousand. The earth goes from the center of all things to just one rock among many orbiting a bigger ball of fire.

      “[N]ow we have not just one planet rotating about another while both run through a great orbit around the sun; our own eyes show us four stars which wander around Jupiter as does the moon around the Earth, while all together trace out a grand revolution about the sun….”

  2. Istoria e dimostrazioni intorno alle Macchie Solari e loro accidenti. (History and Demonstrations Concerning Sunspots and Their Phenomena.) 1613 — commonly called The Letters on Sunspots, further evidence for a heliocentric system
    1. sunspots — sun is not perfect and it rotates
    2. phases of venus — verifies heliocentric geometry
  3. Dialogo sopra i due massimi sistemi del mondo, Tolemaico, e Copernicano. (Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican.) 1632.
    1. Salviati argues for the Copernican position and presents some of Galileo’s views directly, calling him the “Academician” in honor of Galileo’s membership in the Academia dei Lincei. He is named after Galileo’s friend Filipo Salviati (1582-1614).
    2. Sagredo is an intelligent layman who is initially neutral. He is named after Galileo’s friend Giovanfrancesco Sagredo (1571-1620).
    3. Simplicio is a dedicated follower of Ptolemy and Aristotle, who presents the traditional views and the arguments against the Copernican position. He is modeled after Ludovico delle Colombe (1565-1616) and Cesare Cremonini (1550-1631), both of whom were conservative philosophers. The character’s name is not “Simpleton”, but is taken from the sixth-century philosopher Simplicius, who wrote notable commentaries on Aristotle. [Maffeo Barberini – Pope Urban VII (1623-1644)?]
    • on the Index of Prohibited Books from 1616 to 1835 (with De Revolutionibus)
    • held under house for eight years arrest until his death
    • condemned in 1633, condemnation reversed gradually (last word in 1992)

Just Quotes

  • If the sacred scribes had meant to teach men astronomy, then why did they leave it out? Galileo
  • The Galileo affair was unique and contrary to the history of the Catholic Church, which has always supported scholarship, and founded universities… God can do anything God wants…. Guy Consolmagno, Vatican Astronomer. Hazel Muir, Heaven’s Observer, New Scientist 2231, 23 March 2000.


Tycho Brahe (1546–1601) Denmark


  • Compromise system
  • Uraniborg (Castle of the Heavens) 1576-1597
  • moved to Prague in 1597 at the request of Emperor Rudolph of Bohemia
  • Rightly determined that comets are beyond the moon, must have an orbit of some strange sort.
  • There are no spheres that carry the planets.
    • Since comets travel on an orbit that crosses the orbits of the planets, they could not be affixed to solid, crystalline spheres. Either the comet would stop or the spheres would shatter.
    • Crystalline spheres would refract light, but we do not see the position of the fixed stars shift.
    • Sometimes the planet Mars is nearer to the Earth than the sun. Thus the sphere of Mars and the sphere of the sun must overlap — something that is not possible were they solid.


  • Now it is quite clear to me that there are no solid spheres in the heavens, and those that have been devised by authors to save the appearances, exist only in their imagination, for the purpose of permitting the mind to conceive the motion which the heavenly bodies trace in their courses. Quoted in A L Mackay, Dictionary of Scientific Quotations (London 1994)
  • Those who study the stars have God for a teacher.


Johannes Kepler (1571–1630) Holy Roman Empire (now Austria)

Kepler was a terrible high school teacher.

Kepler’s first attempt fails

Mysterium Cosmographicum (Cosmic Mystery) 1596.

Before the universe was created, there were no numbers except the Trinity, which is God himself… For, the line and the plane imply no numbers: here infinitude itself reigns. Let us consider, therefore, the solids. We must first eliminate the irregular solids, because we are only concerned with orderly creation. There remain six bodies, the sphere and the five regular polyhedra. To the sphere corresponds the heaven. On the other hand, the dynamic world is represented by the flat-faces solids. Of these there are five: when viewed as boundaries, however, these five determine six distinct things: hence the six planets that revolve about the sun. This is also the reason why there are but six planets.

I have further shown that the regular solids fall into two groups: three in one, and two in the other. To the larger group belongs, first of all, the Cube, then the Pyramid, and finally the Dodecahedron. To the second group belongs, first, the Octahedron, and second, the Icosahedron. That is why the most important portion of the universe, the Earth — where God’s image is reflected in man — separates the two groups. For, as I have proved next, the solids of the first group must lie beyond the Earth’s orbit, and those of the second group within… Thus I was led to assign the Cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and the Octahedron to Mercury.

radius of sphere ratio R/r
platonic solid outer (R) inner (r) deviation
9.53876 au
5.20283 au
√3 = 1.732
5.20283 au
1.52369 au
√15 + √3
1.52369 au
√(250 + 110√5)
1 au
√(10 + 2√5)
1 au
3√3 + √15
0.72333 au
0.72333 au
0.38710 au
√3 = 1.732
Kepler’s Platonic solid model of the solar system

Kepler decided the observations of the planets were wrong, not his model of nested platonic solids.

This model was rendered useless when Uranus (the seventh planet) was discovered in 1781, Neptune (the eighth planet) in 1846, Ceres (the first asteroid) in 1801, and Pluto (the first Kuiper belt object) in 1930. More than 10,000 objects orbiting the sun have been identified.

The counter reformation steps in and Kepler is driven out of Graz. Philip III (1598-1621) Son of Philip II becomes leader of Spanish house of Hapsburg says to Pope: “I would rather lose a hundred lives, if I had them, than consent to rule over heretics.” This rebellion would drag on until 1648, become part of the wider European struggle known as the Thirty Years War (1618-1648)

Kepler’s Good Stuff

  • Astronomia Nova seu Physica Cœlestis (New Astronomy or Celestial Physics) 1609
    1. Law of Elliptical Orbits: The orbit of a planet about the sun is an ellipse with the sun at one focus.
    2. Law of Equal Areas: A line joining a planet and the sun sweeps out equal areas during equal intervals of time.
    • Planets move with variable speed.
    • Added the recently discovered Galilean moons of Jupiter to his system (and the “supporting stars” of Saturn, which we now recognize as the rings). There are the six “primary” planets that orbit the sun and the “secondary” planets that orbit the primary planets and move along with them. We would call these secondary planets satellites.
    • Laws also apply to satellite orbits (constant depends on mass of object being orbited)
  • Harmonices Mundi (Harmonies of the World) 1619
    1. Harmonic Law: The square of the sidereal period of a planet is directly proportional to the cube of the orbit’s semimajor axis (r3 ∝ T2).
    • Empirical not theoretical in origin (also included “music of the planets”and platonic solids)
    • Originally tried a geometric description, 5 nested platonic solids give 6 planetary orbits, model fails since there are many more than 6 objects orbiting the sun
    • On how he discovered his Third law:… and if you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances… Harmonices mundi(Linz, 1619) Book 5, Chapter 3, trans. Aiton, Duncan and Field, p. 411. Quoted in J. Koenderink, Solid Shape, Cambridge MA: MIT Press, 1990.
    • Dritte Keplersche Gesetz: “Die Quadrate der Umlaufszeiten zweier Planeten verhalten sich wie die dritten Potenzen ihrer mittleren Abstände.” [The squares of the rotating times of two planets behave like the third powers of their middle distances.]
  • Epitome Astronomiæ Copernicanæ (Summary of Copernican Astronomy) 1618-1621

Additional Quotes

Ubi materia, ibi geometria. Where there is matter, there is geometry.
Geometria una et æterna est in mente Dei refulgens: cuius consortium hominibus tributum inter causas est, cur homo sit imago Dei. Geometry is one and eternal shining in the mind of God: that share in it accorded to men is one of the reasons that Man is the image of God.
Mensus eram cœlos, nunc Terræ metior umbras. Mens cœlestis erat, corporis umbra jacet. I used to measure the Heavens, now I measure the shadows of Earth. The mind belonged to Heaven, the body’s shadow lies here.
Geometry existed before the Creation. It is co-eternal with the mind of God. Geometry provided God with a model for the Creation. Geometry is God Himself.
Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?), supplied God with patterns for the creation of the world, and passed over to Man along with the image of God.
My brain gets tired when I try to understand what I wrote, and I find it hard to rediscover the connexion between the figures and the text, that I established myself.

Kepler wrote the first work of science fiction — the Somnium (1634) — published by Kepler’s son Ludwig, four years after his death.

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