Your love of Greek mathematics should be extended to
the methods that Greeks themselves loved, Egyptian
mathematics. For example, Greeks learned from Egyptians
how to take infinite series of several forms, particularly
geometric progessions, and exactly convert them concise
finite series.
As you may have seen before, Archimedes general rule for:
4/3 = A + B + C + D + E + E/3
where B + A/4, C = B/4, D = C/4 and E = D/4
was first stated in a geometric context. However, the
historical method used followed Hultsch's 1895 view of
exact Egyptian fraction methods, as I'll happily detail,
if you wish.
Regards,
Milo Gardner
Sacramento, Calif.