The Mathematical Models of the Greeks

On this page I will explain the mathematical models devised by Greek astronomers and mathematicians to account for (and ultimately to be able to predict) the motions of the celestial bodies (the stars, sun, moon and five visible planets). As you will see, these models are all based on combinations of uniform circular motion. Remember that the Greeks believed that the heavens were perfect and orderly, and that the sphere was the most perfect and harmonious shape. They further believed that uniform circular motion, which has no beginning and no end, was the most perfect kind of motion. So, the perfect and orderly heavens were spherical, and they were composed of numerous spheres. And all the observed motions of heavenly bodies were uniform and circular, or were combinations of uniform circular motions.

Unless you have prodigious mental visualization skills (which I certainly do not!), these models are nearly impossible to grasp without diagrams. And an even better aid to understanding is animated diagrams. Fortunately, some very clever people have produced animated diagrams of Greek astronomical models and put them on the web. I have included links to these resources. These links are part of your required reading, so go over them carefully.

The biggest challenge for the ancient Greeks was figuring out how the order and pattern underlying the seemingly irregular motions of the planets (recall the phenomenon of retrograde motion). Eudoxus of Cnidus (408 –  355 B.C.), a philosopher and mathematician who studied with Plato, devised the first important model of the cosmos that accounted for the complexities of planetary motion. He devised a mathematical model of the universe that accounted for the motions of the sun, moon, and planets by means of concentric spheres. Here is a diagram of his model for one planet.

eudoxus

Eudoxus’ model of planetary motion. By Professor Richard Pogge.

The planet’s motion is the combined motion of four concentric and interlocking spheres all centered on the earth. The outermost sphere rotates east to west once every twenty-four hours. This gives you the planet’s daily rising and setting. The second sphere rotates west to east on an axis tilted somewhat from that of the outermost sphere. This accounts for the planet’s movement through the zodiac. The inner two spheres rotate on different axes and at different speeds and they account for the planet’s retrograde motion. Each sphere conveys it motion to the sphere underneath it. The planet is fixed on the innermost sphere, so its motion is the combined motion of all four spheres. Watch this video clip to see how this works:

Eudoxus video

OnionbellyEudoxus appears to have used this system of concentric spheres only as a computational device. That is, he did not argue that the heavens were actually made up of solid concentric spheres with planets embedded in these spheres. Eudoxus’ system was enormously influential because Aristotle adopted it. Aristotle changed the system in a fundamental way – he argued that it represented the real physical structure of the cosmos. So Aristotle’s celestial realm is more complicated than I described in my discussion of Aristotelian physics.  Each planet has multiple concentric spheres (made of ether) associated with it.  Aristotle’s heavens are actually made up of 56 concentric spheres.  These spheres are all nested inside each other with no “space” in between them.  I like to think of Aristotle’s universe as a giant cosmic onion, with layer after layer of spheres made of ether.

 

However, as the video explains, the Eudoxian model is not completely accurate for the outer planets. Aristotle never worked out the mathematics of this system, and it was not used to make predictions about where celestial bodies would be at particular times.

Let me turn back to mathematical models of the cosmos, that it, models that allowed the ancient Greeks to accurately calculate where a planet or the moon or sun would be at any given time. Greek astronomers after Aristotle developed two powerful models to account for celestial motions. Unlike Aristotle, they did not argue that these models were physically real. These were computational devices.

Eccentric

Eccentric model. Wikimedia Commons.

The first mathematical device was the eccentric. In the eccentric model, the planet moves with uniform circular motion, but this motion is not centered on the earth. Instead, the earth is offset from the center of the circle by some distance – called the eccentricity. In the diagram on the left, C is the center of the circle and E is the earth.  The distance between C and the center of E is the eccentricity.  Although the planet is moving with uniform circular motion, because the earth is not at the center of this rotation, to an observer on earth the planet appears to move with varying speed. This was an important model because no celestial body (except the stars) was observed to move around the earth uniformly. The sun, moon and planets all appeared to move faster at some times and slower at others.

 

 

 

 

 

Epicycle-Ptolemy

Epicycle on deferent model. Wikimedia

 

The second mathematical device was the epicycle on deferent. In this model, the planet is carried on a small circle (the epicycle) that rotates with constant speed. The center of this small circle in turn rotates around a large circle (the deferent). This model can account for retrograde motion, because the combination of these two uniform circular motions can produce backward loops.

 

 

 

 

 

 

 

 

Equant

Equant model. Wikimedia Commons.

In the second century A.D., Claudius Ptolemy of Alexandria (ca. 100 – ca. 170) produced a work of mathematical astronomy that synthesized and advanced the work of his predecessors. Ptolemy produced the most comprehensive and accurate models of the motions of celestial bodies, and his work was read and used for centuries. In Greek, this work was titled the Mathematical Compilation or Syntaxis. It was translated into Arabic, and then into Latin, and medieval and early modern Europeans knew it by its Arabic title, the Almagest. In the Almagest, Ptolemy used combinations of eccentric circles and epicycles to work out accurate mathematical models for the motions of each planet, the moon, sun and stars. In addition, Ptolemy devised another mathematical model, the equant, which he incorporated into his models for the motions of celestial bodies. In the equant model, the planet moves around a circle, and it sweeps out equal angles in equal times as measured from an equant point (Q in the diagram on the left), which is different than the center of the circle (C in the diagram). (If the planet swept out equal areas in equal times as measured from the center of the circle, it would be moving with uniform circular motion. As in the eccentric model, an observer on the earth (E) will see the planet moving with non-uniform speed.

 

 

To see animated versions of these three models, look at the website Models of Planetary Motion from Antiquity to the Renaissance, put together by Craig Sean McConnell.  Another excellent resource, although this is OPTIONAL, is Dennis Duke’s Ancient Planetary Model Animations.

 

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