Models of Planetary Motion
from Antiquity to the Renaissance


Craig Sean McConnell
Assistant Professor of Liberal Studies
California State University, Fullerton

Contents
         Introduction
         Retrograde Motion of the Planets
         The Eudoxan Solution
         From Hippopede to Retrograde Loops
         The Aristotelian Cosmos
         Problems with the Eudoxan Solution
         The Ptolemaic Solution
            Epicycle on Deferent
            Strictly Uniform Motion
            The Eccentric Circle
            The Equant Circle
            UCM, Eccentric, and Equant Compared
         The Copernican Solution
         Mathematical Equivalence of Ptolemaic and Copernican Models
         For Further Study

Introduction
Since antiquity, astronomers have attempted to explain the motions they observed in the heavens with geometrical models. This web site has been designed to help students in history of astronomy courses who are encountering these models for the first time. Often, students struggle to visualize how the static drawings in their textbook relate to the complex motions of the planets. By animating these images, I hope students will be able to more completely "see" how combinations of circles and spheres produced the distinctive retrograde motions exhibited by the planets. These images are not drawn to scale; they are meant only to serve as an aid to understanding how these models account for the motions in the heavens.

Though this site includes a narrative description of the elements of these astronomical models, it is not intended to serve as a complete introduction to the history of ancient astronomy. For such an introduction, please consult one of the texts recommended below.

Retrograde Motion of the Planets
Observations of Mars spaced a few weeks apart reveal motion relative to constellations in the zodiac. Mars usually moves from the West to the East compared to these "fixed stars," but occasionally changes direction, moving from East to West for a few weeks before returning to its more normal motion:




Since antiquity, astronomers have observed all of the planets (Mercury, Venus, Mars, Jupiter, and Saturn) undertake similar motions.

Some sought a sensible explanation for this intriguing behavior.  The daily rising and setting of the stars suggested to many that motions in the heavens are uniform, circular, and eternal.  Could it be that the complicated motions of the planets are the result of combinations of uniform circular motion

The Eudoxan Solution
Eudoxus of Cnidus (c. 390 - c. 337 B.C.) envisioned a system of spheres whose combined uniform motion would resemble a "hippopede," a figure eight. The Eudoxan spheres share a common center, occupied by the Earth, but do not rotate around a common axis:

Two "nested" spheres that share a common center.  The outer sphere's axis and the inner sphere's axis are offset.  A planet is embedded in the equator of the inner sphere.   The axis of the inner sphere is embedded in the outer sphere, so that the inner sphere will share the outer sphere's motion.

(The gray polar caps have been added to help the viewer see these diagrams as spheres).

 

By rotating the outer sphere and the inner sphere in opposite directions but with the same period, the planet embedded in the inner sphere will remain almost stationary in longitude, but its latitude will oscillate by an angle twice that of the offset between the axes of the spheres.  The combined motion will be a figure eight as the equator of the inner sphere is continuously re-oriented by the motion of the outer sphere:

The outer sphere is rotating counter-clockwise as seen from above.

The inner sphere is rotating clockwise as seen from above.


From Hippopede to Retrograde Loops
Eudoxus added yet another sphere to make this hippopede rise and set daily.  The above two spheres are embedded in the third sphere, whose axis is roughly perpendicular to that of the outer sphere.  The result is that the hippopede is smeared across the sky--the motion is predominantly West to East, but occasionally East to West, as had been observed:



Note the similarities between this figure and the sky map at the top of this page.

The Aristotelian Cosmos
    Forthcoming


Problems with the Eudoxan Solution
    Forthcoming


The Ptolemaic Solution

Claudius Ptolemy (fl. 150 A.D.) approached the problem of modelling the motions of the planets with a greater emphasis on accurate prediction. He utilized an entirely different geometric model to account for the retrograde motion of planets, a device known as an epicycle on deferent. In the figure below, the white circle is the deferent, which carries the red circle (the epicycle) around the earth. The planet Mars (the small closed red circle) thus tumbles around the sky according to the combination of two uniform circular motions. The resultant path of the planet as seen from the ground periodically moves backwards compared to the overall motion of the planet.


The epicycle on deferent model is quite flexible. By adjusting the size of the epicycle, Ptolemy could fit his model to the size of the retrograde loops observed in the heavens. By adjusting the rates of rotation, he could adjust the frequency of retrograde motion.

Ptolemy's model was made even more powerful by the inclusion of devices that allowed him to vary the amount of time a heavenly body spent in each part of the sky. To do this, he had to be more flexible in his sense of what exactly counted as "uniform circular motion".

There are three senses in which a body can be said to move along a circular path with some manner of uniformity:

Strictly Uniform Circular Motion

This motion is uniform in every way--the speed of the yellow object along the circle is uniform, the circle shares a center with the earth, the object sweeps out equal angles in equal times.


The Eccentric Circle

The green object sweeps out equal angles in equal times, maintaining a constant speed in its perfectly circular path. But note that from the earth, the object appears to move faster in the bottom half of its motion than in the top half.

The motion of the green object is perfectly uniform, though it will appear nonuniform to an observer on earth because the earth does not coincide with the center of the motion. 



The Equant Circle

The equant model has a body in motion on a circular path that does not share a center with the earth. Further, the object's speed actually varies in its orbit around that circle--it moves faster in the bottom half, slower on the top half.

This motion is uniform only in that the red object sweeps out equal angles in equal times from a reference point inside the circle. This point, the equant point, is not at the center of the circle nor at the center of the earth. The speed of the object is nonuniform from the perspective of the equant point, the center of the equant circle, and the center of the earth. 

 




In the following diagram, all three of these models are in motion at once.  The Yellow object (Strictly Uniform Circular Motion), the Green object (Eccentric Circular Motion) and the Red object (Equant Circular Motion) all complete one circuit in the same amount of time.  Compared to Strictly UCM, the objects on the eccentric and the equant move more rapidly in the bottom half of their motion and less rapidly in the top half of their motion.  In fact, from the earth perspective, the speeds of the Yellow and Red objects are always changing:

UCM, Eccentric, And Equant Compared

Compared to UCM, the eccentric lags behind UCM on the left side of the circle and speeds past UCM on the right side.  Equant motion is even more disuniform, lagging behind the eccentric on the left side of the circle, and speeding past it on the right side.   

An earth observer would project all of these motions against the background stars.  To compare the motions, follow the trace lines on the lower of the two white circles.  

The predictive power of Ptolemy's Almagest was the result of combinations of these models--he envisioned epicylces on equants, epicycles on epicycles, and so on.

Ptolemy has been admired for the accuracy of his predictions, but he has also been criticised for deviating so far from the ideals of strictly uniform circular motion.



The Copernican Solution
In 1543, Nicolaus Copernicus (1473 - 1543 A.D.) proposed that the earth travels around the sun, as do the other planets. For Copernicus, the apparent retrograde motion of the planets is a result of the relative speeds of the earth and the planets as observed from the earth:

The path of Mars, as viewed from the Earth The path of Venus, as viewed from the Earth.





Mathematical Equivalence of Ptolemaic and Copernican Models

Ptolemy's Model for the planet Mars. Copernicus' Model for the planet Mars.

As can be seen here, both models predict the same sort of motion. Just as Ptolemy was able to calibrate his model by adjusting the size and speed of the epicycles and deferents, Copernicus was able to calibrate his model by adjusting the size and speed of a planet's orbit.  Despite his displeasure with the equant model, which he found to be a deviation from the goal of uniform circular motion, Copernicus retained the epicylce on deferent model, which allowed him to make predictions of the future positions of the planets with accuracy comparable to Ptolemy.  

From a strictly mathematical point of view, the two models are equivalent--both can be used to predict the motions of the planets with great accuracy. The decision to endorse the Copernican model and reject the Ptolemaic model could not be made strictly on the criterion of accuracy.


For Further Study
Introductory
    David C. Lindberg, The Beginnings of Western Science. University of Chicago Press, 1992.
    G.E.R. Lloyd, Early Greek Science: Thales to Aristotle. W.W. Norton & Company, 1970.
    G.E.R. Lloyd, Greek Science After Aristotle. W.W. Norton & Company, 1973.
Advanced
    Michael J. Crowe, Theories of the World from Antiquity to the Renaissance. Dover, 1990.
    Thomas S. Kuhn, The Copernican Revolution. Harvard, 1957.
    G.E.R. Lloyd, Magic, Reason, and Experience. Cambridge University Press, 1979.
    O. Neugebauer, The Exact Sciences in Antiquity. Harper, 1962.


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Text, Design, and Layout by Craig Sean McConnell

Images and Animated Sequences Prepared by
Tommy Huerta and Craig McConnell

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