Today my class reached the pinnacle of Ancient Greek astronomy: the Mathematical Syntaxis of Ptolemy (later referred to as the Almagest because of its Arabic name, meaning the "the greatest"). We had already looked at the basic epicycle/deferent model developed by Apollonius, and we saw that it could produce retrograde motion and link that motion with increased brightness in a natural way. We also saw that retrograde/brightness could be linked to opposition (for superior planets) or conjunction (for inferior) by synchronizing the motion of the epicycle (for superior planets) or the deferent (for inferior) to the motion of the Sun. We examined all of these things using my Superior Ptolemaic and Inferior Ptolemaic simulations.
Today we looked at some modification to the basic deferent/epicycle model. First we examined Hipparchus' idea of using an eccentric circular orbit for the Sun. The Sun moves uniformly along a circular path, but the center of this circle is located away from Earth. This causes an apparent change in the speed of the Sun as seen against the background stars: faster when the Sun is closer to Earth, slower when the Sun is farther away. This is particularly interesting because we had learned earlier that the Sun moves fastest relative to the stars in WINTER. So this implies that the Sun is closest to Earth in winter, which contradicts a commonly held misconception about the seasons (that they are related to the distance of the Sun from Earth - they aren't, they are determined by the inclination of Earth's rotational axis relative to its orbital plane).
One thing I had my students do was derive the formula that gives the eccentricity of the Sun's orbit from the maximum and minimum angular speeds of the Sun. The algebra is not bad at all, but it is entirely symbolic. Many of my students just got wide-eyed and didn't want to try it. In the end I got them all through with a few hints, and hopefully they realized it was not as scary as they were making it out to be. Math phobia is a terrible thing. I wish I could banish it from the world.
After the eccentric we moved on to the equant. This is the hardest part of Ptolemy to understand. The equant is a point that is off from the center of a circular orbit. A body moving along an orbit with an equant changes speed in such a way that the angular speeds as seen from the equant point is constant. So ti moves faster when it is far from the equant, slower when it is close to the equant. I have (unpublished) simulations to illustrate the eccentric and equant mechanisms by themselves, but they are also built into the Superior Ptolemaic and Inferior Ptolemaic simulations mentioned above.
Finally, we see how Ptolemy determined the periods of motion for the deferents and epicycles. The period of the epicycle, measured relative to the deferent, is just the synodic period of the planet (the time between retrogrades). This measuring relative to the deferent seems weird to modern eyes - we want to measure everything relative to "absolute space" (ie, relative to the background stars). But the Ancient Greeks thought of these circles as being the equators of great transparent spheres that rotated around. The epicycle gets dragged around, and actually rotates, because of the motion of the deferent - but the Ancient Greeks didn't count that as a rotation of the epicycle itself. Only the extra bit of rotation that didn't come from being dragged around the deferent was considered as a true rotation of the epicycle. It is the period of this rotation which is equal to the planet's synodic period. The period of the deferent motion is pretty obviously (once you look at it) equal to the planet's zodiacal period (the time for the planet to go all the way around the Celestial Sphere, on average).
Next up we will work on the sizes of the epicycles relative to the deferents so that we can actually try to sketch out the scale model of the whole system of planets. In doing so, we will see just how poorly the whole Ptolemaic scheme fits with Aristotle's cosmology. In some ways the merger of Aristotelian cosmology and Ptolemaic astronomy represents one of the great compromises in the history of science. This compromise was probably only possible because of strong skepticism about whether or not astronomy could tell you what was really going on in the heavens. Some astronomers, at least, took the attitude that Aristotle had the basic structure right. Ptolemy didn't fit that structure, but Ptolemy was just giving us formulae for predicting planetary locations, not telling us what was really going on in the heavens. This attitude still existed (perhaps even more strongly) in the Renaissance. It was the opponents of this brand of skepticism who would ultimately revolutionize astronomy.
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