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When Columbus "discovered" the New World, the Old World had little to no knowledge of it. What little was known was perhaps Greenland, and - based on accurate knowledge of the size of the world - a vast expanse of unknown between Europe and East Asia, which was assumed to be empty ocean until Columbus proved otherwise, setting off the Age of Discovery and the associated mania with finding unknown lands, some of which proved true (Australia, Antarctica).

At this time, what did the New World know of, or speculate about the Old World? They were reasonably advanced, for example the Mayans were very good at Astronomy, and so may have known the size of the world. If so, they would have realised they occupied a narrow strip of land surrounded by a vast, unknown expanse. Did they know this, and if so what did they speculate would be there?

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    Probably nothing. Only central and southern American cultures might have had some knowledge about the size of the earth, but even if they did, they never ventured out and could not tell what was across (if anything) the ocean, nor did they have any incentive to find out. Probably we don't even have any sources about what they thought on the matter, because there is so little written evidence. But i don't really know, don't take my word for it, and i'd love to be proven wrong.
    – Matthaeus
    Jan 21, 2015 at 13:55

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How would you know this? For most tribes, being illiterate and having no writings, the only way would be to ask them, and of course by that time they would know about you, so you have spoiled the experiment, so to speak.

That said, the Mayans did have codices that recorded their myths and fables, notably the Popul Vuh, sometimes called "The Dawn of Life", in which their civilization was created by gods that came from the east over the sea. So, the Mayans at least definitely believed in a world beyond the sea, but this was a mythological belief, not firm knowledge.

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    I have edited the question to clarify that I am also interested in what they speculated was there, such as the Popul Vuh.
    – congusbongus
    Jan 21, 2015 at 23:50
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    @congusbongus The ancient world certainly had legends -- Atlantis, for example -- of large land masses out in the Atlantic. Yet it seems more likely this was making the random-ass-guess that that big a body of water might have something in it. If that counts as speculating about the New World, then who cares?
    – Mark Olson
    Aug 19, 2022 at 20:17
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They (i.e. those who understood Astronomy) certainly knew that something existed elsewhere, but it is doubtfull that they knew what.

The same is true for Europe. By the time of Columbus, they had a good idea how far away the Asian coastline was, but knew nothing about what was in between.

The combined effect of these mistakes was that Columbus estimated the distance to Japan to be only about 5,000 km (or only to the eastern edge of the Caribbean) while the true figure is about 20,000 km. The Spanish scholars may not have known the exact distance to the east coast of Asia, but they believed that it was significantly further than Columbus's projection; and this was the basis of the criticism in Spain and Portugal, whether academic or among mariners, of the proposed voyage.
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The ships barely reached the eastern Caribbean islands. Already the crews were mutinous, not because of some fear of "sailing off the edge", but because they were running out of food and water with no chance of any new supplies within sailing distance.

Columbus problem was that he used an Italian/Roman Mile (1481.76 meters) instead of the Arabic mile (1,972 meters) for the 56⅔ miles per degree calculation.

Another estimate given by his [Caliph Al-Ma'mun] astronomers was 56⅔ Arabic miles (111.8 km per degree), which corresponds to a circumference of 40,248 km, very close to the current values of 111.3 km per degree and 40,068 km circumference, respectively.

Al-Farghani and the “Short Degree"

It is assumed that the legal cubit (49.8 centimeters) was used since it was most commonly used unit during al-Farghani's lifetime.

  • Arab mile = 4000 cubits
    • black cubit: 54.04 cm
      • 2161.60 m
        • 122.4906666667 Kilometer = 56⅔ = 1 Degree
          • 44096.64 km = 360 Degrees
    • surveying cubit: 48.25
      • 1930.00 m
        • 109.3666666667 Kilometer = 56⅔ = 1 Degree
          • 39372 km = 360 Degrees
    • legal cubit: 49.80
      • 1992.00 m
        • 112.88 Kilometer = 56⅔ = 1 Degree
          • 40636.8 km = 360 Degrees
  • circumference: 40075.017 km
    • 1 Degree: 111.3194916667

Al-Farghani and the “Short Degree”
But we know from other sources that the black cubit had not yet been introduced during the reign of al-Ma'mun, when the length of a degree was measured on the plain of Sinjar. So in spite of the terminology al-Farghani uses, his "black cubit" must in fact refer to either the "surveying cubit" of 48.25 centimeters, or to the legal cubit of 49.8 centimeters. The latter is the more likely, since we know that it was the most commonly used unit during al-Farghani's lifetime.

type meters Arabic miles Roman miles
Arabic 1992.00 4000 legal cubits
(each 0.498 km)		 1.3443472627
Roman 1481.76 0.7438554217 Nippur cupit=518.616 mm
Foot=4/7 Nippur cupit
(296.352 mm)
Roman mile=5000 Feet
Roman cubit=6/7 Nippur cupit
(444.528 mm)

Sources:

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