There's a commonly held belief that Columbus thought the world was round when everyone else thought it was flat, and it was brave but sensible of him to sail West to reach the Spice Islands, which were in East Asia.
Everyone knew that the world was round, except people who really didn't care (which is most landlubbers). When you see a ship at sea coming towards you, you see the mast and sails before you see the hull. If you hold a straight-edge against the sea horizon, you can actually see the curvature. If you see the shadow of the earth on the moon during a lunar eclipse, you can see it's curved.
The ancient Greeks knew that the world was round. 2200 years ago, Eratosthenes not only knew that, he actually measured the circumference. He got between 44,100 and 46,100 kilometers. Archimedes estimated 48,300 km. The actual value is 40,000 km.
So what happened?
Columbus got it wrong. He got confused between Roman miles and Arabic miles, and thought that the circumference was 30,000 km. If Columbus had been right, then you could load up a ship with tons of food and water, and sail west, and you'll get to the Indies before you run out.
So Columbus thought that the distance from the Canary Islands to Japan was 3,700 km. Actually, it's 20,000 km. He set off on September 6. On October 10, his crew nearly mutinied. This was not, as the myth goes, because they thought they were about to sail off the edge of the world. It was because they'd almost reached the point of no return. If they turned back now, they'd get back to the Canaries, and not starve at sea. Columbus persuaded them to go on two more days; they sighted land the next day. It was not, of course, the East Indies or Japan.
If it hadn't been for the previously unknown existence of the West Indies (so called because Columbus thought he'd reached what we now call the East Indies), he, and all his crew, would have died of thirst or starvation.