# Equivalence of minor epicycle and eccentric

In epicycle-deferent astronomy, adding a second minor epicycle to account for observational discrepancies is mathematically equivalent to shifting the deferent into a so-called eccentric, or a circle with a center not at the Earth (see e.g. DeWitt (2010), Worldviews, p. 120), as was pointed out already by Hipparchus. Ptolemy famously chose the eccentric approach for the Almagest. I’m trying to wrap my head around how to geometrically prove the equivalence. Any ideas?

• Is it in connection with the double generation of the epicycloid ? See animation here. Feb 15 '21 at 17:37
• Fascinating! Though the idea I was after has to do with adding yet another (minor) epicycle, and that this addition is supposed to have the same effect upon the movement as projected unto the outer circle of fixed stars by a line of sight from Earth as just shifting the original epicycloid by a distance equal to the diameter of the minor epicycle. I think it’s the earliest proof of an ”empirical equivalence” of two hypotheses. My question was obviously way too imprecise, I will update it. Feb 15 '21 at 19:31