Among the many bullshit facts kids are (were?) fed about Christopher Columbus is that his voyage to the New World disproved the prevailing notion that the Earth was flat. If I remember correctly, the idea impressed upon us cica elementary school was that until Columbus's most noble voyage in 1492 it was believed that a foolish sailor was liable to just fall off the edge of the planet if they ventured too far from land, and that they might even encounter some kind of edge monster on the way down. Columbus believed otherwise, however, and would prevail against the ignorance of the fearful and would reach his eastern destination by traveling west. What a guy.
I suspect, on observing the throngs of supporters of a certain politician, that this particular Columbus myth and many others still run rampant among minimally educated adults and probably well beyond. Washington Irving's mostly fictional The Life and Voyages of Christopher Columbus is often credited as one of the myth's key proponents. Jean Antoine Letronne's On the Cosmographical Ideas of the Church Fathers likewise pushed the idea, this time in an effort to paint the early-days Christian Church Fathers as anti-science hicks. Which is fair enough in many respects, but most evidence points to their being fully aware that the Earth was round and not flat.
When Columbus lived, people thought that the earth was flat. They believed the Atlantic Ocean to be filled with monsters large enough to devour their ships, and with fearful waterfalls over which their frail vessels would plunge to destruction. Columbus had to fight these foolish beliefs in order to get men to sail with him. He felt sure the earth was round.
Where did this knowledge come from? That would be Eratosthenes, the Greek mathematician and astronomer. Right around this time of year in 240 BC—today is often given as the anniversary, in fact—Eratosthenes was pursuing a papyrus text preserved at the library at Alexandria, which he was the director of, and noted a peculiar observation. As reported in the city of Syene in southern Egypt, the text's author noted that, "the shadow of someone looking down a deep well would block the reflection of the Sun at noon."
The Sun then was directly overhead. A perfectly vertical object would at that time cast no shadows, at least in Syene. In Alexandria, this was not the case, Eratosthenes noted. The Sun here was not directly overhead and so it did indeed cast shadows. To get to the bottom of things, he did an experiment. By measuring the length of the shadow cast by a stick at noon on the solstice, he could calculate the angle between the Sun and the vertical stick: 7.2 degrees.
And yet, in Syene, the angle was 0 degrees. What gives? Eratosthenes realized that the difference must be the result of a curvature on the Earth's surface. With respect to a vertical stick in Syene, a vertical stick in Alexandria would be tilted away by this amount. He then realized that if he knew the distance between the cities, he could calculate the circumference of the planet itself. And so he did, arriving at a value either 2 percent less (39,375 kilometers) than the actual circumference or 16 percent more (46,620 kilometers), depending on Eratosthenes' exact metric (e.g. what a "stadia" translates to in modern terms). Which is not bad.
Even if civilization had its collective brain wiped between Eratosthenes day and Columbus's, it seems highly unlikely that someone, anyone else would not have noted something similar. In any case, Ptolemy's Geography, written in 150 AD, remained a classic reference by the time of Columbus's voyage and it is quite clear about the spherical shape of the planet.