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?