326 11 POST-EUCLIDEAN GEOMETRY
FIGURE 4. Ptolemy's first method of mapping.
the Equator. The extreme southern limit of his map was the circle 16° 25' south of
the equator, which he called "anti-Meroe," since Meroe was 16° 25' north.
Since he knew only the geography of what is now Europe, Africa, and Asia, he
did not need 360° of longitude. He took his westernmost point to be the Blessed
Islands (possibly the Canary Islands, at 17° west). That was his prime meridian,
and he measured longitude out to 180° eastward from there, to the Seres^5 and the
Chinese (Sinai) and "Kattigara." According to Dilke (1985, p. 81), "Kattigara"
may refer to Hanoi. Actually, the east-west span from the Canary Islands to
Shanghai (about 123° east) is only 140° of longitude. Ptolemy's inaccuracy is due
partly to unreliable reports of distances over trade routes and partly to his decision
to accept 500 stades as the length of a degree of latitude when the true distance is
about 600 stades.^6 We are not concerned with geography, however, only with its
mathematical aspects.
The problem Ptolemy faced was to draw a flat map of the Earth's surface
spanning 180° of longitude and about 80° degrees of latitude, from 16° 25' south
to 63° north. Ptolemy described three methods of doing this, the first of which we
shall now discuss. The latitude and longitude coordinates of the inhabited world
(oikumene) known to Ptolemy represent a rectangle whose width is | of its length.
But Ptolemy did not like to represent parallels of latitude as straight lines; he
preferred to draw them as arcs of concentric circles while keeping the meridians
(^5) The Seres were a Hindu people known to the Greeks from the silk trade.
(^6) It has become a commonplace that Christopher Columbus, relying on Ptolemy's geography,
expected to reach the Orient at a distance that would have placed him in the middle of the Pacific
Ocean had North America not been in the way. If he believed Ptolemy, he would have thought
it about 180° of longitude, which at a latitude of 40° would have been about 138 great-circle
degrees. But he thought a degree was 500 stades (92 km), and hence that the distance to Japan
was about 12,700 km. Since a degree is actually 600 stades (110 km), the journey would have
been more than 15,000 km. But the latitude of Japan is slightly south of the latitude of Spain.