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.
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
- legal cubit: 49.80
- 1992.00 m
- 112.88 Kilometer = 56⅔ = 1 Degree
- circumference: 40075.017 km
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.
||4000 legal cubits
(each 0.498 km)
||Nippur cupit=518.616 mm
Foot=4/7 Nippur cupit
Roman mile=5000 Feet
Roman cubit=6/7 Nippur cupit