On Tuesday my class began to move beyond the two sphere model of the universe and started focusing on explaining the motions of the planets. We examined two theories regarding planetary motions that were devised in antiquity.
The first theory we looked at is due to Eudoxus of Cnidus. Eudoxus' basic idea was that each planet had associated with it a set of nested spheres. All of these spheres were centered on Earth. The planet itself lay on the surface of the innermost sphere (which we will call sphere 1). That innermost sphere spins about an axis. The ends of this axis are fixed to the inner surface of the next sphere out (we will call it sphere 2). Sphere 2 also spins about an axis, and it turns out that it MUST spin at the same rate as sphere 1, but in the opposite sense (we'll talk about why in a bit). If the axis of sphere 1 is aligned with the axis of sphere 2 then the planet would just stay motionless, because these two spheres would cancel each other. However, if you tilt the axis of sphere 1 relative to the axis of sphere 2, the planet executes a little figure-8 shaped jaunt. It doesn't really get anywhere, it just does figure-eights (actually a shape known as a hippopede) in one part of the sky. But for now we don't want it to get anywhere. That's why the two spheres spin at the same rate in opposite senses, so that the planet wiggles around a bit but doesn't actually go to a new part of the sky.
Now we connect the axis of sphere 2 to the equator of sphere 3 (the next one out). Sphere 3 spins around so that in addition to the figure-8 motion we get a motion of the whole assembly on a big circle around the sky. You set this up so that this motion is eastward, and the big circle is the Ecliptic. The result is pretty amazing: the planet has a general eastward drift, but occasionally it stops and goes west for a bit before returning to its eastward motion. In other words, the first three spheres qualitatively reproduce the retrograde motion of the planets. Sphere 3 then attaches to sphere 4 at a 23.5 degree tilt, so that sphere 4 has its axis aligned with the axis of the Celestial Sphere, so that sphere 4 can produce the daily motion of the stars around our sky.
The whole thing is pretty amazing. It is hard to visualize how it works from a description. You really have to see it in action, so my student spend some time playing around with my Spheres of Eudoxus simulation. How Eudoxus figured out that these motions would produce the retrograde behavior is just beyond me.
But then I lead my students to see one of the big problems with this model. The planet lies on a sphere that is centered on Earth. So its distance from Earth never changes. So there is no explanation for the changes in brightness that are observed in the planets (particularly Mars). It also turns out that Eudoxus theory fails at the level of quantitative prediction - you just can't build a version of this theory that will accurately determine the future locations of the planets in the sky. But it gets a major qualitative feature right, and it fits in beautifully with the Aristotelian cosmos.
Moving on from Eudoxus we investigate the deferent-epicycle model originally formulated by Apollonius of Perga. The pure version of the model consists of a large circle, known as the deferent, centered on Earth. Another smaller circle, known as the epicycle, has its center on the deferent circle. The deferent spins around Earth, and the epicycle spins around its (now moving) center. This combination of two circular motions produces a planetary orbit that is akin to a spirograph, but the Ancient Greeks didn't really think in terms of the planets moving in orbits through space. They thought of rotating circles (which may or may not be the equators of rotating spheres). So the spirograph picture wasn't really discussed until much later (by Kepler).
Again, this idea is hard to picture. My students explore the idea with my SuperiorPtolemaic simulation. Again, this model is pretty amazing, although for me it seems easier to figure out than Eudoxus' model. The cool thing is that it not only reproduces the retrograde motion of a superior planet, it also brings the planet closest to Earth during retrograde. That explains why superior planets, like Mars, are brightest when they retrograde. It is an automatic feature of the model. It occurs naturally from the basic premises of the deferent and epicycle. Pretty neat.
Not only that, but you can set things up so that the planet does its retrograde/brightness thing when it is in opposition to the Sun, as happens with real (superior) planets. In order to do this you must synchronize the epicycle motion with the motion of the Sun around Earth (we are still geocentric so far). This is not at all a natural thing to do in this model, although there is also no reason why you shouldn't do it. It's a bit ad hoc, but it makes it all work.
So on a qualitative level, the deferent-epicycle model was a huge success and it quickly supplanted Eudoxus' homocentric spheres (although homocentric spheres made a comeback among Arabic speaking astronomers during the Middle Ages, and again among Europeans during the Renaissance). But the basic model was not quite sufficient to give accurate quantitative predictions. It still needed to be tweaked, and it was Ptolemy (following some important work by Hipparchus) who would do the necessary tweaking.
This activity really makes me think about the underdetermination thesis (sometimes called the Duhem-Quine thesis), which states that for any set of observational facts there are many theories (perhaps an infinite number) that will fit those facts. From a Logical Point of View* that is certainly true, but most of those theories would be immediately dismissed as unreasonably complicated or arbitrary. In practice, it is really hard to come up with EVEN ONE reasonable theory that fits even just the qualitative features of the observations. Eudoxus got part of it right with his theory, but Apollonius really nailed it. To some extent, all subsequent theories about planetary motions were variations on the theme of the deferent-epicycle model. Yes, there were many different versions (which would seem to fit the Duhem-Quine thesis) but they employ almost the same geometry. I guess my point is just that coming up with even one good theory is hard, and we should all be very impressed by the ideas of Eudoxus and Apollonius even if we no longer think they were quite right.
*inside joke for logicians (see the works of WVO Quine if you want to get the joke)
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