Eratosthenes' TooBig Earth & TooTiny Universe
Abstract
Sostratos' legendary lighthouse on Pharos Island in Alexandria's harbor is proposed as what enabled Eratosthenes to displace the prior 1significantdigit crudity of Aristotle & Dikaearchos with 3significantdigit precision in Earthcircumference, 256000 stades. Bishop Eusebius' testimony that Eratosthenes' distance to the Sun was 4080000 stades shows, in the Greek tradition of setting solar distance as a power of 10 in Earthradii, that Eratosthenes' Earthradius was 40800 stades, which multiplied by 2π matches the circumference cited. Reverse calculation fails, so the primary empirical datum was radius, which is what the Pharos provided by the visibility of its flame down the Mediterranean coast from Alexandria over the intervening sea, since Earthradius is found by dividing the visibilitydistance's square by twice the Pharos' height. Ancient testimony indicates the height was c.300', but if Sostratos' tallestever lighthouse was made exactly 300' or 1/2 a stade, then the equation's denominator is unity so squaring the visibilitydistance in stades equals Earthradius in stades: squaring visibilitydistance 202 stades yields 40800, conventionally rounded. It is emphasized that 40800 stades is 19% or about 6/5 higher than Earth's actual radius, and PoseidoniosPtolemy's 180000stade circumference is 5/6 too small. Atmospheric refraction affects the Pharosflame method by 6/5, while shrinking by 5/6 circumference found by the doublesunset method of gauging the Earth'sgirth (AmJPhys 47:126128). So the 2 standard ancient values cited differ by 40% but are explained by airbent light to within 1% each, using the standard 185meter stade, while the everwheelspinning ad hoc search for oddball stades to explain the ancient errors don't match either value so exactly.
 Publication:

DIO
 Pub Date:
 March 2008
 Bibcode:
 2008DIO....14....3R
 Keywords:

 Sostratos;
 Eratosthenes;
 Eusebius;
 Earth radius;
 Earth circumference;
 atmospheric refraction;
 doublesunset;
 Pharos;
 Alexandria lighthouse