Most of people believed the earth was flat throughout most of human history

Probably 90% of the populace never gave the subject any thought at all. I don't think they can be said to have "believed" anything about it.

The educated, on the other hand--people who knew how very much larger the world was than just their own village and the village next door--have known that the world was round for millennia, though as far as I know its radius wasn't accurately calculated till Eratosthenes accomplished it in the third century B.C.

http://en.wikipedia.org/wiki/Eratosthenes

Before that, they had "the mountains in the distance coming up over the horizon as we travel toward them" and a host of other observed phenomena to demonstrate that the earth was round.

I wish people would get this through their heads--that people didn't believe the earth was flat in the past. Columbus didn't sail west to prove the earth was round. He sailed west precisely because everyone knew the earth was round. Fortunately Columbus was a better sailor than he was a navigator and he had miscalculated the width of the ocean west of Europe. He was the only person on the planet who didn't know the trip to the Indies would take six months on the caravels available in his day. He really thought, using "proof" from the Bible and ignoring the seventeen centuries old calculations of that nasty ol' pagan Eratosthenes, that the earth was much smaller than it really was and thought he could cross the ocean westward in just three months. He didn't know there was a continent in the way that would rescue him just as his sailors' willingness to proceed farther and farther from land was giving out.