It is written in *Riemann Surfaces* (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that:

"[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus"

Can someone please provide an articulated commentary on this statement.

Specifically, the statement suggests, [or seems to suggest], that Riemann surfaces were the logical / mathematical outcome of many years of careful development and refinement of traditional calculus. But: (i) what was / were the major milestones(s) in this road? and (ii) when the author uses the word 'culmination' what specifically is it the culmination of, and what problems / issues did the introduction of Riemann surfaces help to solve / clarify / etc.?

(This question was originally posted on Math SE, but I'm also posting it here because I'm seeking an expert's [in Riemann surface theory] feedback if possible.)

againcrossposting; three hours is a ridiculously short time to wait. – quid Sep 15 '11 at 16:01