It started from an actual theory based around the assumption that spherical motion was perfect. They needed 2 which did actually work for a while, eventually the most accurate model needed ~17 with people giving up on the underlying theory as the number of terms destroyed the initial idea.
Today with vastly more data and more accurate measurements you’d need effectively infinite terms, which makes it more obvious but you don’t need that level of absurdity to render judgment.
No it didn't. Epicycles were from the get go nothing but an attempt to fit a mathematical function to observed data to predict future positions of planets. It's a geometric method of curve fitting which is a weaker form of the fourier series, and the system was developed by greek mathematicians trying to improve upon Babylonian computations that didn't even have a geometric model. There is a reason that the moon, the only thing in the cosmos that does in fact orbit the earth, has the most complicated series of epicycles to describe its motion.
Ptolemy rejects Aristotle's cosmology which relied on perfect spherical motion. Ptolemy really did believe that the planets moved according to his model (ie it wasn't just a pure computational tool) but he was very clear that his model was based purely on mathematics. Not only did he not give a reason for why the cosmos should take this form, he openly speculates that the answer is unknowable, and works under the assumption "maybe they can move wherever they want and they just like moving this way."
Further, cycles were not added over time [1]. On day one there were 31 cycles and circles, and these were exactly the same ones being used at the time of Copernicus. You also don't need many epicycles to accurately produce a path identical to keplerian orbits. Completely arbitrary orbits can be described with finite epicycles. [2] Indeed the problem was that Ptolemy didn't fit the data by adding more epicycles, but instead through the Equant, which moved the positions of the centers of the epicycles, which meant adding more epicycles would not make it more accurate. The story of ever more epicycles being added to a bloated old theory that was streamlined by heliocentrism is a modern myth.
That’s a count of the total need to describe the motion of multiple celestial bodies.
I’m referring to the number of cycles needed to describe the motion of a single celestial body. There wasn’t enough data at high enough precision to need 17 cycles to describe the motion of a single celestial body until much later. At the time lesser precision was more common, but that someone really did go to such an extreme to create the best fit.
> Completely arbitrary orbits can be described with finite epicycles.
The number of points isn’t fixed with continuous observations. Your best fit for past data keeps needing new cycles over time unless you’re working backwards from a much better model. Even then you run into issues with earthquakes changing the length of the day etc. The basic assumptions they where working from don’t actually hold up.
Also, I’m reasonably sure you couldn’t actually write out an infinite decimal representation of the irrational number
e using a finite number of epicycles. Not something I’ve really considered deeply, but it seems like an obvious counter example.
The first is overlooking the issue of overfitting using hand calculation and imperfect observations. The calculated “best fit” for the data available did involved adding a bunch of epicycles and there was no theoretical reason to avoid doing so.
The second is playing fast and loose with a fat line drawn over a squiggly line based on a better model. It’s being mathematically rigorous but intentionally deceptive. You can fairly trivially construct a set of epicycles to fit some desired shape, but working backwards from observation there’s nothing guiding you to the most elegant possible solution for a given situation.
Today with vastly more data and more accurate measurements you’d need effectively infinite terms, which makes it more obvious but you don’t need that level of absurdity to render judgment.