Why did Columbus believe that he could reach the east sailing west?

Answer 1

Because... the map in his time was wrong (but only in scale):

China was thought to be farther east than it was,

the Americas were not on the maps

the combined Atlantic and Pacific was thought to be much narrower than they are.

Answer 2

Because, contrary to common believe, medieval scholars were well aware of the fact that the earth is a sphere.

Columbus disagreed with the scientific community concerning the size of earth, that's why he thought it possible to actually reach India by sailing West using contemporary ships.

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Answer 3

Because the knowledge of the world did not include the vast expanse of the Pacific Ocean.

Answer 4

more than likely by noticing that the moon, sun and other stars were round. or how the horizon changed the higher you are elevated.