Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. Details were given in his treatise On the measurement of the Earth which is now lost. However, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo. Eratosthenes compared the noon shadow at midsummer between Syene (now Aswan on the Nile in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumfer! ence of the Earth as 250,000 stadia.
Of course how accurate this value is depends on the length of the stadium and scholars have argued over this for a long time. The article [11] discusses the various values scholars have given for the stadium. It is certainly true that Eratosthenes obtained a good result, even a remarkable result if one takes 157.2 metres for the stadium as some have deduced from values given by Pliny. It is less good if 166.7 metres was the value used by Eratosthenes as Gulbekian suggests in [11].
Several of the papers referenced, for example [10], [15] and [16], discuss the accuracy of Eratosthenes' result. The paper [15] is particularly interesting. In it Rawlins argues convincingly that the only measurement which Eratosthenes made himself in his calculations was the zenith distance on the summer solstice at Alexandria, and that he obtained the value of 712'. Rawlins argues that this is in error by 16' while other data which Eratosthenes used, from unknown sources, was considerably more accurate.
Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured the tilt of the Earth's axis with great accuracy obtaining the value of 11/83 of 180, namely 23
51' 15".
The value 11/83 has fascinated historians of mathematics, for example the papers [9] and [17] are written just to examine the source of this value. Perhaps the most commonly held view is that the value 11/83 is due to Ptolemy and not to Eratosthenes. Heath [4] argues that Eratosthenes used 24 and that 11/83 of 180
was a refinement due to Ptolemy. Taisbak [17] agrees with attributing 11/83 to Ptolemy although he believes that Eratosthenes used the value 2/15 of 180
. However Rawlins [15] believes that a continued fraction method was used to calculate the value 11/83 while Fowler [9] proposes that the anthyphairesis (or Euclidean algorithm) method was used (see also [3])."
Of course the fact that Ptolemy quotes E. on this file seems very significant. Ptolemy is the archetyp of tradtional astrology and it s not sure IMHO that he favoured the flat earth theory. Such things are not that simple and Copernicus obviously had predecessors.
Regards,
Jean
Mark Hagerman <mhagerman@xxxxxxxxxxxxxxxx> wrote: