Educated Europeans knew the Earth was round the whole time

The points below about the medieval flat Earth are ordered by how much difference they make, not by how often they get repeated.

What medieval scholars actually thought

The spherical shape of the Earth was established in ancient Greek astronomy and was never seriously lost to European learning afterwards. Standard teaching texts used in medieval universities describe the planet as a sphere and explain the astronomical evidence for it. Popular symbols of the period show it plainly, including the orb held by rulers in portraits, which is a globe surmounted by a cross.

Debate certainly existed about the size of the sphere, about what lay in the unknown hemisphere, and about whether it could be inhabited. None of that debate involved anybody arguing that the world was a flat plate with an edge to fall off.

Where the story was written

A widely read biography of Columbus published in the late 1820s dramatised his negotiations as a struggle against a superstitious council. Its author was primarily a writer of fiction and openly blended narrative invention with the documentary record. The book was hugely popular in both Europe and America and was treated by many readers as straightforward history.

Later writers arguing that religion had systematically obstructed science found the scene extremely useful and repeated it. Within decades it had entered school textbooks, where it stayed for a very long time indeed.

What the real dispute was about

The council examining his proposal did not doubt that sailing west would eventually reach the east, since that follows from a sphere. Their objection was that the distance was far too great for any ship of the period to cover without resupply.

Ancient estimates of the Earth's circumference were reasonably close to the true value, and the experts were working from those. Columbus had assembled a much smaller figure by combining the most favourable estimates available, which suited his case perfectly. He was wrong and they were right, and he survived only because an unexpected continent happened to be in the way.

The evidence anyone could see

A ship sailing away disappears hull first and mast last, which is difficult to explain on a flat surface and obvious on a curved one. The changing height of stars as one travels north or south had been used to estimate the planet's size in antiquity.

The curved shadow the Earth casts on the Moon during an eclipse is always circular, whatever the orientation. Sailors in particular had daily practical experience of these effects, and navigation methods depended on them. Any belief in a flat Earth would have had to survive contradiction by the working knowledge of every pilot in Europe.

Why the myth was attractive

It flatters the present by making the past not merely less informed but actively stupid, which is a comfortable position to occupy. It also provided a clean narrative for a nineteenth-century argument about the relationship between religion and scientific enquiry.

Somewhere in the retelling, neat stories with a lone visionary and an obstructive establishment travel far better than accounts of committees disagreeing about arithmetic. The myth was, in short, doing political work, and factual accuracy was not what it was being used for. Recognising that is more useful than simply noting the error, because the same narrative shape is still in heavy use.

Printing a correction has a poor record of taking a myth out of circulation.

What actually happened to flat-Earth belief

Organised flat-Earth advocacy is a modern phenomenon, emerging in the nineteenth century as a deliberately contrarian movement. That is to say the belief became prominent after the myth about the middle ages had been invented, not before it.

The sequence is almost exactly backwards from what most people assume, which makes it one of the more enjoyable corrections available. Medieval maps that look flat are usually schematic diagrams of relationships and pilgrimage routes rather than attempts at physical geography. Reading a symbolic map as a literal claim about shape is roughly like reading an underground railway diagram as a survey.

Everything above, in order of what to do first

1. What medieval scholars actually thought. The spherical shape of the Earth was established in ancient Greek astronomy and was never seriously lost to European learning afterwards.
2. Where the story was written. A widely read biography of Columbus published in the late 1820s dramatised his negotiations as a struggle against a superstitious council.
3. What the real dispute was about. The council examining his proposal did not doubt that sailing west would eventually reach the east, since that follows from a sphere.
4. The evidence anyone could see. A ship sailing away disappears hull first and mast last, which is difficult to explain on a flat surface and obvious on a curved one.
5. Why the myth was attractive. It flatters the present by making the past not merely less informed but actively stupid, which is a comfortable position to occupy.
6. What actually happened to flat-Earth belief. Organised flat-Earth advocacy is a modern phenomenon, emerging in the nineteenth century as a deliberately contrarian movement.

The takeaway

The council was doing better arithmetic than the explorer, which is not the story anybody wanted to tell.

Ask who gained from the claim being repeated. That usually explains the rest.