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 reoriented by
the motion of the outer sphere:

The outer sphere is rotating counterclockwise 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 skythe 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 waythe 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 circleit 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 modelshe 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 equivalentboth 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.
Text, Design, and Layout by Craig Sean McConnell
Images and Animated Sequences Prepared by
Tommy Huerta and Craig McConnell
Feedback is appreciatedsend email here.