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More details can be found in Neugebauer (1955, 1959).

It is not known whether Apollonius used his theorem to determine the numerical values of the parameters necessary to reproduce the observed phenomena, but Ptolemy later used it in Book XII of the Almagest to construct a table of stationary points even though it is not strictly valid for Ptolemy's more complicated scheme.

model retrograde motion using epicycles, still made extensive use of them to account for other irregularities.33

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