How the Greeks Used Geometry to Understand the Stars

Michael Fowler, University of Virginia

Crystal Spheres: Plato, Eudoxus, Aristotle

Plato, with his belief that the world was constructed with geometric simplicity and elegance, felt certain that the sun, moon and planets, being made of aither, would have a natural circular motion, since that is the simplest uniform motion that repeats itself endlessly, as their motion did.  However, although the “fixed stars” did in fact move in simple circles about the North star, the sun, moon and planets traced out much more complicated paths across the sky.  These paths had been followed closely and recorded since early Babylonian civilization, so were very well known.  Plato suggested that perhaps these complicated paths were actually combinations of simple circular motions, and challenged his Athenian colleagues to prove it.

The first real progress on the problem was made by Eudoxus, at Plato’s academy.  Eudoxus placed all the fixed stars on a huge sphere, the earth itself a much smaller sphere fixed at the center.  The huge sphere rotated about the earth once every twenty-four hours.  So far, this is the standard “starry vault” picture.  Then Eudoxus assumed the sun to be attached to another sphere, concentric with the fixed stars’ sphere, that is, it was also centered on the earth.  This new sphere, lying entirely inside the sphere carrying the fixed stars, had to be transparent, since the fixed stars are very visible.  The new sphere was attached to the fixed stars’ sphere so that it, too, went around every twenty-four hours, but in addition it rotated slowly about the two axis points where it was attached to the big sphere, and this extra rotation was once a year.  This meant that the sun, viewed against the backdrop of the fixed stars, traced out a big circular path which it covered in a year.  This path is the ecliptic.  To get it all right, the ecliptic has to be tilted at 23½ degrees to the “equator” line of the fixed stars, taking the North star as the “north pole”.

This gives a pretty accurate representation of the sun’s motion, but it didn’t quite account for all the known observations at that time.  For one thing, if the sun goes around the ecliptic at an exactly uniform rate, the time intervals between the solstices and the equinoxes will all be equal.  In fact, they’re not-so the sun moves a little faster around some parts of its yearly journey through the ecliptic than other parts.  This, and other considerations, led to the introduction of three more spheres to describe the sun’s motion.  Of course, to actually show that the combination of these motions gave an accurate representation of the sun’s observed motion required considerable geometric skill! Aristotle wrote a summary of the “state of the art” in accounting for all the observed planetary motions, and also those of the sun and the moon.  This required the introduction of fifty-five concentric transparent spheres.  Still, it did account for everything observed in terms of simple circular motion, the only kind of motion thought to be allowed for aither.  Aristotle himself believed the crystal spheres existed as physical entities, although Eudoxus may have viewed them as simply a computational device.

It is interesting to note that, despite our earlier claim that the Greeks “discovered nature”, Plato believed the planets to be animate beings.  He argued that it was not possible that they should accurately describe their orbits year after year if they didn’t know what they were doing—that is, if they had no soul attached.