I think there's an interesting fillip of information added when you use "know" in some sentences. The distinction "know" vs. "believe" seems to me to have within it an extra bit of a claim. When we say "the ancient Mayans knew that the year is a bit under 365 and a quarter days," we are really saying two things: (1) they believed that was so, and (2) we (the speaker of the sentence) also believe it. That is, "know" equals "believe correctly" (at least, correctly according to the speaker's standards). We can tell this is happening because we don't use this same phrasing for claims we believe (know?) not to be true. We say "Medieval Europeans believed that tomatoes were poisonous." Their belief was presumably every bit as strong as that of the Mayans, but they get stuck with "believed" instead of "knew," only because we, the speakers of the sentence, disagree with them (or should I say, because we "know better").
True, you sometimes will see something like, "People knew that the Earth was the center of the universe, and couldn't countenance any dissenting opinion," but I believe that that usage is more ironic than serious. The speaker here deliberately uses the "wrong" verb (and noticed in such sentences it's nearly always given extra stress) just to emphasize its wrongness. They believed it, so strongly... it was as if it were even right! But alas, I Know Better.
Just some thoughts; they were expressed a while ago by some folks pondering this notion on the Lojban mailing list, regarding the exact meanings of the Lojban predicates for "know" and "believe" and "opine" and such.