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Geographical considerations

Despite a popular misconception to the contrary, nearly all educated Westerners of Columbus's time knew that the Earth is spherical, a concept that had been understood since antiquity. The techniques of celestial navigation, which uses the position of the Sun and the stars in the sky, had long been in use by astronomers and were beginning to be implemented by mariners.

However Columbus made several errors in calculating the size of the Earth, the distance the continent extended to the east, and therefore the distance to the west to reach his goal.

First, as far back as the 3rd century BC, Eratosthenes had correctly computed the circumference of the Earth by using simple geometry and studying the shadows cast by objects at two remote locations. In the 1st century BC, Posidonius confirmed Eratosthenes's results by comparing stellar observations at two separate locations. These measurements were widely known among scholars, but Ptolemy's use of the smaller, old-fashioned units of distance led Columbus to underestimate the size of the Earth by about a third.

Second, three cosmographical parameters determined the bounds of Columbus's enterprise: the distance across the ocean between Europe and Asia, which depended on the extent of the oikumene, i.e., the Eurasian land-mass stretching east–west between Spain and China; the circumference of the Earth; and the number of miles or leagues in a degree of longitude, which was possible to deduce from the theory of the relationship between the size of the surfaces of water and the land as held by the followers of Aristotle in medieval times.

From Pierre d'Ailly 's Imago Mundi (1410), Columbus learned of Alfraganus 's estimate that a degree of latitude (equal to approximately a degree of longitude along the equator) spanned 56.67 Arabic miles (equivalent to 66.2 nautical miles, 122.6 kilometers or 76.2 mi), but he did not realize that this was expressed in the Arabic mile (about 1,830 meters or 1.14 mi) rather than the shorter Roman mile (about 1,480 m) with which he was familiar. Columbus therefore estimated the size of the Earth to be about 75% of Eratosthenes's calculation.

Third, most scholars of the time accepted Ptolemy's estimate that Eurasia spanned 180° longitude, rather than the actual 130° (to the Chinese mainland) or 150° (to Japan at the latitude of Spain). Columbus believed an even higher estimate, leaving a smaller percentage for water. In d'Ailly's Imago Mundi, Columbus read Marinus of Tyre 's estimate that the longitudinal span of Eurasia was 225° at the latitude of Rhodes. Some historians, such as Samuel Eliot Morison, have suggested that he followed the statement in the apocryphal book 2 Esdras (6:42) that "six parts are habitable and the seventh is covered with water." He was also aware of Marco Polo's claim that Japan (which he called "Cipangu") was some 2,414 km (1,500 mi) to the east of China ("Cathay"), and closer to the equator than it is. He was influenced by Toscanelli's idea that there were inhabited islands even farther to the east than Japan, including the mythical Antillia, which he thought might lie not much farther to the west than the Azores, and the distance westward from the Canary Islands to the Indies as only 68 degrees, equivalent to 3,080 nmi (5,700 km; 3,540 mi) (a 58% error).

Based on his sources, Columbus estimated a distance of 2,400 nmi (4,400 km; 2,800 mi) from the Canary Islands west to Japan; the actual distance is 10,600 nmi (19,600 km; 12,200 mi). No ship in the 15th century could have carried enough food and fresh water for such a long voyage, and the dangers involved in navigating through the uncharted ocean would have been formidable. Most European navigators reasonably concluded that a westward voyage from Europe to Asia was unfeasible. The Catholic Monarchs, however, having completed the Reconquista, an expensive war against the Moors in the Iberian Peninsula, were eager to obtain a competitive edge over other European countries in the quest for trade with the Indies. Columbus's project, though far-fetched, held the promise of such an advantage.