See Ptolemy's _Almagest_ (X 7) for his fixed-point iteration for
determining the eccentricities of the deferents of the superior planets
(O. Neugebauer's _History of Ancient Mathematical Astronomy_, I pp. 174--178,
gives a helpful exposition). The method probably considerably predates
Ptolemy, but it may be difficult to find an explicit discussion of it in
the classic Hellenic mathematical works, since it falls far short of
Euclidean standards of rigorous demonstration. It appears in Indian
texts from the first few centuries of this era, but not identifiably
earlier than the Almagest, as far as I know; I don't know what evidence
there may be for such a method in Egyptian or Chinese math. Good luck,
Kim Plofker
Dept. of History of Mathematics
Brown University