The Eudoxan model of the heavens: A brief exposition

Let’s first do the model for ONE planet. There are four spheres. We number them, starting with the outermost, as 1, 2, 3, and 4. Sphere 1, with its north pole on the north celestial pole (essentially the North Star), rotates westward, once a day. Sphere 2 has its poles on sphere 1, but is tilted, so that its equator is on the ecliptic. It rotates eastward: once a year for Mercury or Venus, and at varying periods (approximately 2, 12, or 30 years, respectively) for Mars, Jupiter, or Saturn. Sphere 3 has its poles attached to the equator of sphere 2. Sphere 4 has its poles fixed to the inside of sphere 3. Sphere 4’s poles are a little out of alignment with sphere 3’s (see diagram above). Spheres 3 and 4 rotate with the same period BUT IN OPPOSITE DIRECTIONS. The PLANET is attached to the equator of sphere 4.

The combined motions of spheres 3 and 4 cause the planet to move in a figure-eight loop called a hippopede (“horse fetter”), which is supposed to model retrograde motion. How the hippopede motion arises is shown below (the planet is marked with a P):

To see how this fits in with the first picture, imagine it lying down (rotated by about 90 degrees) and inserted inside sphere 2.

Another (possibly more convincing) way to see hippopede motion is to take a world globe and spin it slowly about its axis in one direction while turning it slowly on its base in the other direction. Any fixed point on the equator (e.g., zero degrees longitude) will trace out a hippopede.

The models for the sun and moon were simpler, having only THREE spheres. Sphere 1 was the same as for the planets. Sphere 2 had its equator on the ecliptic, with a period of one month for the moon and one year for the sun. Sphere 3 had its poles a little out of alignment with sphere 2’s. It accounted for the (apparent) movement of the moon and sun out of the ecliptic. This apparent movement is real for the moon but not for the sun (i.e., Eudoxus blew it here).

This model had FOUR MAJOR PROBLEMS.

  1. Since it was based on uniform circular motion, it predicted that the seasons would all have the same length; and they don’t.
  2. Again, because it was based on uniform circular motion, the model predicted that a given planet’s retrograde loops would all have the same shape; and they don’t.
  3. Under this model, any given celestial body will always keep the same distance from Earth. But even the ancient Greeks could see that the real angular width of the moon changed, which meant that it was sometimes closer and sometimes farther off. Also, the brightness of the planets changed, and the ancients interpreted this to mean that their distances were not always the same.
  4. The model had a serious technical problem related to Mars and Venus. If you adjusted the rotational periods of the spheres so that Mars (or Venus) had the correct synodical year (the time between successive passages behind the sun), there was no retrograde motion, and if you fixed the retrograde motions, the synodical year would be wrong.