Which of these is correct, and why?

They're both correct, but the second one - "... knew the world was round" - is the more idiomatic and more frequently used. I doubt there's an identifiable reason; it's just idiom, the way it's conventionally expressed, regardless of logic.

We can get a rough idea of usage from published books. See Google Books:

"knew the world is round" 32 hits http://books.google.co.uk/books?oe=utf-8&client=fi...

"knew the world was round" 575 hits http://books.google.co.uk/books?oe=utf-8&client=fi...

For centuries before Columbus, educated Europeans knew the world is round

because the world is ALWAYS round(the shapes of the world never change),if,"The world WAS round"so maybe it was flat before Round

For centuries before and some time after Galileo, Newton and Darwin, it was a death sentence to disagree the sun revolved around the earth. The death penalty for heresy was only removed in the mid 1800's, not that long ago. Centuries before Columbus, nobody knew any different seeing as they were nearly all illiterate.

Practicing Shaman... quantum physics rocks

was.

Because youre talking in past tense... sure the world is still round, but the most correct way to use "is" would have to be: "Europeans know the world is round"

was.

end of.