We’ve remarked elsewhere that Ptolemy’s Almagest is not a work of cosmology; it’s a manual for ephemeris-makers. Ptolemy’s goal was not to describe the mechanism of planetary orbits, but to predict where the sun, moon and planets would be found in the future.
Viewed in this light, we can better make sense of Ptolemy’s use of nested circular orbits – epicycles.
At the time that Ptolemy worked, the most advanced mathematical tools available were geometry and trigonometry, both planar and spherical. The ancient Greeks made important contributions to these fields, along with early forays into number theory, some of which bordered on mysticism. But these were essentially static techniques, useful in describing the location and relationships among the planets at a fixed time, but clumsy for prediction. Algebra, calculus, and later methods provided tools which could be used to generate values for planetary locations at a specified future time. The geometric models of Ptolemy lent themselves to a clumsy, iterative series of ratcheting calculations.
Using the techniques available in Ptolemy’s mathematical toolbox, a savant might use 3, or 6, or 12 months of observations as a baseline for predicting planetary locations 6, 15 or 36 months into the future. The results of this first round of calculations would be added to the baseline measurements, and used to calculate a second set of predictions, perhaps 12, 48, even 60 months. And on for further iterations, reaching perhaps a decade into the future. [Numbers cited are for illustration only. We don’t pretend to know how far into the future astronomers could map the movements of the planets using geometric constructs.]
The problem is that the further one proceeds from a fixed observation, the greater the errors in predicted values. There was always a call for newer and more accurate ephemerides.
For more than a thousand years, Ptolemy’s text and methods remained a standard among savants who sought to calculate an ephemeris.
In 1543 CE, De Revolutionibus was published as Copernicus lay on his deathbed (it’s a complicated story). Modern historians credit Copernicus with revolutionizing astronomy, but that gives him too much credit. Copernicus modeled his text and his geometry after Almagest, using sets of circular epicycles to describe the motions of sun, moon and planets. The only difference was the placement of the sun, instead of the earth, at the center of all.
Copernicus didn’t replace Ptolemy’s model of the universe. He simply rearranged the furniture.
Inspired by Newton, it was Kepler who knocked down Ptolemy’s house, and laid a new foundation for planetary science.
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While epicycles are typically viewed today as quaintly archaic constructs, they can be mathematically interesting. In theory, by placing epicycles on epicyles on epicycles, it’s possible to model any orbit, or any well-behaved closed curve to whatever level of precision is called for. A modern counterpart to such a mathematical technique is the Fourier transform, whereby any well-behaved waveform can be modeled by adding together a series of simpler waveforms.
(A secret to success in mathematics is setting forth parameters
and boundary conditions to define what’s ‘well-behaved’.
Mathematical tools impose limits on possible inputs and outputs.)
For an animated demonstration of the use of the epicyclic method in tracing an extremely complicated (but well-behaved) orbit, pay a visit to https://www.youtube.com/watch?v=QVuU2YCwHjw ; for a more extended discussion, see https://www.youtube.com/watch?v=qS4H6PEcCCA