Perhaps the elegance would be better illustrated if the planets were shown to move in elliptical rather than circular orbits - this of course was the primary historical difficulty in understanding planetary motion

My God, I just spent half of the last night debating this on Facebook -.-

My takeaway from that experience - when someone struggles with understanding why those two views are both equivalent and OK, make sure they understand this thing: heliocentric and geocentric models have only one purpose. They let you compute where planets will be at any given time. They do not try to answer why the planets are moving the way they are. For that, you need at least Newton's model of gravity, which actually doesn't differentiate between the Sun and planets at all.

Geocentric models still pop up in real life. When you're using a sky map to determine where some planet will be visible tonight, you're looking at a geocentric representation.

That was indeed the state of the art in Galileo's time. Heliocentrism wasn't necessarily more elegant, because it left open problems such as why people don't fly off the earth, and why the oceans don't slosh around uncontrollably. From what I've read, the Church was prepared to accept the model of Tycho Brahe, in which the sun goes around the earth, and the planets around the sun, which is yet another equivalent view within that conception of reality.

Newton gave us a couple of things. First, as you say, he gave us a model of gravity. Secondly, he gave us the framework within which to test whether the earth is rotating or not, e.g., through observations of things like Foucault's pendulum, and the Coriolis force.

Here's the problem with this: the planets' orbits aren't circles. And scientists had good enough measurements, that they knew circular orbits didn't match the observed behavior. So neither of those models are very good. It wasn't until people figured out that orbits are ellipses instead of circles, that we actually had a model that was simpler and more accurate than the cycles-and-epicycles of geocentrism.

Exactly this. Everyone says oh, it is so much easier to calculate with circles around the sun--this only true if you want wrong answers to the position of the planets. So wrong that you can see it with the naked eye. Nobody believed Copernicus because his model was not predictive--this gets left out when we want to tell stories about Medieval anti-scientific bias.

When Kepler came along and introduces an elliptical heliocentric model with the equal areas law, scientifically minded people switch to the heliocentric view. However, the math for this is really hard. We use elliptical curves in cryto today. You can calculate the position of a planet with Ptolemy's model on an abacus with nothing more complex than division.

I don't think that's a problem at all. The eccentricities are so small, you wouldn't even notice it.

The eccentricities are so small, you wouldn't even notice it.

They actually were. People had many years off accurate planetary positions, and circles... just didn't work.