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Natural Science - Year I

Unit 10: Aristotle, Ptolemy, and the Planets

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Science Web Assignment for Unit 10

This Unit's Homework Page History Lecture Science Lecture Lab Parents' Notes

Science Lecture for Unit 10:
Aristotle, Ptolemy, and the Planets

For Class

Explaining How Planets Move

In most of our science units, we look at the current version of the theories that have their origins in the history topic for the unit. But this time, we will concentrate on the actual theories of Ptolemy and the measurements used to establish his system.

Aristotle's theory tied together astronomy, physics, medicine, and even psychology by providing a physical model for the interactions of celestial and terrestrial matter. But it became clear through observations made by the time of Jesus that the spheres of Eudoxus, even with Aristotle's adjustments, could not account accurately enough for all observed planetary positions. Another tradition of calculations based on direct observation developed, until around 160 ce, Claudius Ptolemy of Alexandria used measurements by Eratosthenes, Anaxagoras, Aristarchus, and Hipparchus to produce a new theory of planetary motions.

The Calculation of Eclipses

Like all other astronomers before and after them, the Greeks were looking for patterns of behavior. They discovered that many astronomical events, like lunar and solar eclipses, conjunctions of the planets with the sun and with each other, recur in set periods. The pattern of eclipses is called the Saros cycle, and lasts 18 years, 11 and 1/3 days. If a total solar eclipse occurs like the one on August 11, 1999, a precisely similar eclipse will occur 18 years, 11 and 1/3 days later, or on August 22, 2017, but 8 hours later, sun time. The 1999 eclipse was visible in England, Europe, and the Middle East. There is an 8 hour difference between London, England, and Seattle, Washington, so the 2017 eclipse was visible from Seattle eastward about 3000 miles—in other words, over most of the North American continent.

The reason a solar eclipse doesn't occur every new moon or a lunar eclipse every full moon was Anaxagoras' great discovery.

We have already looked at the standard lunar orbit in terms of its phases. What Anaxagoras realized was that the plane of the sun's orbit (the ecliptic, as seen from earth) and the plane of the moon's orbit are close but not the same. The moon's orbit is inclined to the ecliptic about 5°. If we imagine the moon's orbit as a circle tilted and intersecting the plane containing the sun's orbit, the only time the moon can be directly in front of the sun is when it is on the points of its orbit that intersect the plane. These points are the nodes of the moon's orbit.

This means that if we draw a line from the earth to the sun, the moon is above that line for half its orbit, and below the line for the other half of the orbit. Because the moon is small relative to the sun, the moon's shadow falls on the earth only when the moon is on the line between the earth and the sun.

Moon Nodes 1

The orbit of the moon wobbles, so that the nodes (where the thin black line intersects the sun's orbit) changes position over time. The diagram below shows the "new moon" position above and below the ecliptic plane (the plane of the sun's orbit as seen from the earth) as the highest point of the lunar orbit above the ecliptic plane rotates around.

Moon Nodes 2

When the nodes line up with the sun and the moon is in new phase, a solar eclipse will occur. If the moon is full, a lunar eclipse will occur. You should note that the August 11 solar eclipse was preceded two weeks earlier by a full lunar eclipse (also visible from the East Coast and most of Europe). So our solar eclipse of 2017 was likewise be preceded by a total lunar eclipse.

The later advancements, and in particular, calculations of distances to the sun, moon, and planets, rested on the development of mathematical tools by Euclid, Apollonius, and Archimedes.

The Measure of the Earth

Now we look at the advances in the measurement of the earth and the heavens by the Hellenists Eratosthenes and Aristarchus. The methods used by these two Greek mathematicians are perfectly sound, and Eratosthenes' calculation of the circumference of the earth is easily within his experimental accuracy--that is, his ability to determine the angles he was measuring. Unfortunately for Aristarchus, accurate measurement of the angles involved in his calculations was far more crucial. He was able to determine the distance to the moon within a few percent, but because of a mis-measurement of the angles used for the solar distance calculations, he got an answer that was only about 5% of the actual value.

Now we will look at some web presentations. You may also use encyclopedias and look up information on other websites if you don't find the information below answers all your questions.

Read about Earth measurements (this is actually a link to the next lecture in Michael Fowler's series). [1 long Web page -- click on "All", several diagrams]

  •  Be sure that you understand Eratosthenes method for measuring the earth's diameter.
  • How did Aristarchus determine the distance to the moon?
  • How accurate was his answer?
  • Why did Aristarchus have a problem determining the correct distance to the sun?

These quantitative measurements provided the inspiration which drove Claudius Ptolemy to attempt to create a system which would produce accurate predictions of planetary positions. Based on the earlier theories, Ptolemy adopted and tried to work within the following constraints:

  1. All celestial bodies move with uniform circular motion. This means they must travel in perfect circles without speeding up or slowing down.
  2. All celestial bodies are made of perfect (incorruptible) materials and are perfect spheres.
  3. The spherical earth is at the center of the universe, and small compared to the size of the universe.

See Dr. Fowler's lecture on Ptolemy and the epicycle and eccentric system. [1 long Web page; -- click on "All"]

  • What are each of the following motions, visible with naked-eye observations, that any planetary theory must address: daily motion, annual motion, prograde motion, retrograde motion, stationary points.
  • How many spheres did Eudoxus need to explain the sun's motion? What did each do?
  • What was the problem with Eudoxus' model of solar motion?
  • What major observation was Eudoxus unable to explain with his fixed -distance spheres?
  • What were Hipparchus' major contributions to the theory of the planetary motions?
  • What is an epicycle?
  • How does Ptolemy account for the observation that Mercury and Venus are always near the sun?
  • What proof did Ptolemy offer for his claim that the earth was spherical?


As you study Ptolemy's system, keep in mind that it was used for nearly 1600 years not because an institution like the church dictated it (the Arabs used it, and they vigorously rejected any claims of Papal domination!), not because it was particularly simple (it required the most advanced mathematical tools then available, much as our cosmological theories do today), but because it worked. It passed the supreme test of science. If you did the calculations correctly, you could determine the positions of the planets against the background stars within 0.5° —about the width of the moon—for up to 50 years based on a single set of starting observations. A half-degree error was well within the accuracy of all the instruments the Greeks, Romans, Arabs, and medieval European astronomers had.

Ptolemy was concerned with providing a method of producing accurate predictions. His construction is a mathematical marvel, but it is not clear that Ptolemy or any of those who used his calculation model really believed that planets moved in such odd ways.

The original system proposed by Hipparchus used epicycles which were able to model retrograde motion and to explain the dimming and brightening of the planets by placing them at times closer, then farther away from earth.

Ptolemaic Devices
Ptolemaic Epicycle Animation

But Ptolemy had to introduce variations into this simple model in order to account for variations in the motions of the planets. For Mars, the apparent speeding up and slowing down of Mars on its orbit could not be calculated properly by using an earth-centered deferent circle. The epicycle needed to move on a deferent which was off-center, or eccentric. A planet moving on such an epicycle actually traces out an ellipse over time. [It turns out that by carefully choosing the relative sizes of the deferent or eccentric circle and the epicycle circle, and by setting the speed of the epicycle on the deferent and the speed and direction of the planet on the epicycle, it is possible to obtain even angular motion--planets that move in triangles, squares, and curly-que patterns, as well as the ellipse which Kepler eventually derived from observations of the motions of Mars.

Ptolemy's system was complex, certainly. It required (depending on how you count them) between 60 and 80 epicycles and deferents to account for the motions of the moon, sun, Mercury, Venus, Mars, Jupiter, and Saturn. But Ptolemy's epicycles followed strict rules of geometry, so anyone who was determined enough could use them to calculate and predict the positions of the planets to within observational accuracy. For nearly 1300 years, Ptolemy's system served European and Arab astronomers, mathematicians, and medical doctors who relied on natal horoscopes for determining the proper diagnosis and treatment of diseases.

Classical Astronomy

The stage was set for a classical confrontation. On the one hand was a theory that explained the physical characteristics of certain objects, and gave coherent reasons for their motion and behavior. On the other hand was a theory that used mathematical models with no direct ties to physical entities, but that could accurately predict behavior. The conflict in astronomy was not resolved until the scientific revolution of the 16th and 17th centuries. Similar conflicts still exist in quantum mechanics, where we must use both wave and particle theories to account for all the phenomena we observe.

One sometimes runs across a lament that the Hellenists and medieval astronomers were not intelligent enough to adopt the proposal of Archimedes' contemporary, Aristarchus, that the planetary system was really heliocentric (sun centered) and not geocentric (earth-centered). When we get to the sixteenth century, we'll discuss Copernicus' adoption of this theory in detail, but it is import to keep in mind that even when Copernicus published his heliocentric theory in 1543, there was no observational proof for the annual motion of the earth -- in fact, such proof wasn't supplied until Bessel managed to detect stellar aberration in 1838. Nor was there any physical explanation as to why the sun should be at the center -- that wouldn't be available until Newton's work on gravity was published. Moreover, the Ptolemaic system provided more accurate predictions for some years, based on the available data, than did Copernicus' theory. Finally, because he still had to use epicycles to account for retrograde and irregular motions, Copernicus' planetary system was no easier to use in making predictions than Ptolemy's. In turning down Aristarchus' heliocentric system for the fifteen centuries before Copernicus, the Hellenists and the medieval European and Arab astronomers who adopted Ptolemy's system were not particularly more stupid than later philosophers and scientists; they simply refused—as a good scientist should—to change a theory which works for one which isn't any better at explaining the phenomena.

The Greeks and medieval philosophers worked within cultural frameworks which gave them a different approach to the universe than ours, but they also had to work within the limits of their instruments. Ptolemy could be up to a degree off (about twice the diameter of the moon); he could not trust his measurements to be any better than that. One of the reasons his system survived is because it did predict the planetary positions within this observational accuracy. This was enough to get people to use it even though it was clumsy and highly criticized for its complexity by a number of people through the centuries.

Study/Discussion Questions

Further Study On your Own (Optional)