Today's activity involved determining the radii of the planetary orbits (in Astronomical Units) from observational data. It is pretty straightforward for inferior planets. You measure the maximum elongation of the planet from the Sun, and the sine of that angle gives you the radius of that planet's orbit in AU (assuming all orbits are circular and centered on the Sun, which is a decent approximation). For superior planets it is more difficult. It involves measuring the time from opposition to quadrature and then doing a few calculations before finally getting to the trigonometry that lets you find the radius of the planet's orbit.
The main point of this exercise is to show the students that in the Copernican system you can actually find the sizes of the planetary orbits, relative to the size of Earth's orbit (with a radius of 1AU), from observational data. This was not the case for Ptolemy. In the Ptolemaic system we can only determine the size of a planet's epicycle relative to the size of that same planet's deferent from observational data. There is no COMMON distance measure for the planets, nothing that they share that can allow us to connect the orbit of one planet to the orbit of another. But in the Copernican system all of our observations of planetary motion are a mixture of the motion of the planet and the motion of Earth. Once we learn how to disentangle those motions, the orbit of Earth becomes our common unit of measure. All other planetary orbits can be related, via observations, to the orbit of Earth. Everything hangs together.
This is yet another illustration of the remarkable coherence of the Copernican system. As Copernicus says, Ptolemy's theory is like taking a hand, a leg, a head, a torso, all from different individuals and sticking them together. The result is a monster, not a man. In the Copernican system everything is in just the right proportion to everything else. All the pieces fit neatly. And there is no freedom to have it any other way.
Perhaps this is the strongest aspect of the Copernican theory: necessity. Copernicus establishes lots of NECESSARY connections between things. There are necessary connections between the sizes of the planetary orbits. There is a necessary connection between whether a planet's orbit is smaller or larger than Earth's and a variety of observational characteristics for that planet. There is a necessary connection between the retrograde motion of a planet and that planet being in conjunction (for an inferior planet) or opposition (for a superior planet). Ptolemy has some necessary connections, particularly between retrograde and brightness (which Copernicus also has) - but Ptolemy doesn't have nearly as much necessity as Copernicus does.
Of course, necessity can be a weakness. If a certain connection is a necessary outcome of a theory, and then that connection is found to not exist, then the theory will be refuted or modified (or some auxiliary theory must be modified). But when the necessary outcomes of a theory are found to, in fact, exist then it speaks powerfully in favor of that theory. Saying that something MUST be true, and then finding it is true is much stronger than saying that it CAN be true and finding it is true. The latter is good, but the former is much better.
In today's activity we also discovered another beautiful feature of the Copernican theory. It is not a necessary outcome of the theory, but it is a feature of the theory that Copernicus would say is "pleasing to the mind." If we assume the Copernican theory, then we can show that the more distant a planet is from the Sun the longer its orbital period. Moreover, we can even show that the more distant the planet from the Sun the slower it moves along its orbit. There is a nice harmony here that Copernicus appreciated, even though he had no explanation for why it should be that way. Later, Kepler would seek to make this harmony a necessary outcome of his celestial dynamics. And even later, Newton would succeed in this venture.
I'll say more about the aesthetics of the Copernican theory in a later post. I'm reading Hallyn's The Poetic Structure of the World, and it is worth commenting on. But for now I'll just point out that these aesthetic features are really the main thing going for the Copernican theory in the 16th century. Without these features Copernicus offers nothing more than a theory that is observationally equivalent to Ptolemy, but which conflicts with Aristotelian physics and requires a gigantic Celestial Sphere with a tremendous void between Saturn and the stars. Most people didn't find the aesthetic features sufficiently appealing to overcome all of the negatives, but thankfully a few did.
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