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Columbus did not prove the Earth was round
Educated people already knew
Educated Europeans had known since ancient times that the Earth is a sphere. The dispute with Columbus's critics was over the size of the Earth, and they were right that Asia was too far to reach.
Greek scholars were arguing that the Earth is a sphere by the fourth century BCE, and Eratosthenes of Cyrene, working in the third century BCE, used shadow angles at two Egyptian cities to estimate its circumference. His result was surprisingly close to the modern value, though how close depends on which length of the ancient stadion unit one assumes, so claims of exact accuracy should be treated with care.
By the late fifteenth century, university-trained Europeans took a round Earth for granted. The real disagreement before 1492 was about distance. Columbus argued for a much smaller Earth and a much wider Asia than most scholars accepted, which made the western route look short. His critics at the Spanish court used larger and more accurate figures for the distance, and by their numbers the trip across open ocean was too long to survive.
Columbus was lucky rather than right: the Americas lay in the way, and the crew reached land before supplies ran out. The flat-Earth version of the story became widespread in the nineteenth century, helped by popular writing about Columbus, and it has been repeated in school lessons ever since. The tale persists because it makes a neat contrast between a bold explorer and timid, ignorant opponents.