# Eratosthenes and the Mystery of the Stades - Eratosthenes' Correction

Author(s):
Newlyn Walkup

Most of what is known about Eratosthenes’ geometric argument comes from the writings of Cleomedes, in the first century BCE.  Eratosthenes’ argument, as described by Cleomedes, gives the circumference of the Earth as approximately 250,000 stades.  However, most other ancient authors give 252,000 stades as Eratosthenes’ value for the circumference of the Earth.  Strabo’s Geography, written in the late first century BCE, cites Hipparchus as the source of this figure.  Another reference to this length appears in a letter written by Heron of Alexandria (ca. 75 CE) in the second century CE [4, pp.60-63 ].

The perimeter [circumference] of the Earth is 252,000 stades, as Eratosthenes, who investigated this question more accurately than others, has shown in the book he wrote “On the Measurement of the Earth” [4, p.63 ].

In light of the many textual references stating 252,000 stades as the circumference given by Eratosthenes, many scholars believe that the addition of 2000 stades was a correction made by Eratosthenes shortly after his original calculation.

What could be the reason for Eratosthenes’ correction?  There are many theories as to why this correction may have been made.  Let use examine three prevailing theories.

Adding 40 stades to the original 5000 stades between Alexandria and Syene produces a final result of 252,000 stades, but it is unlikely that this was the correction made by Eratosthenes.  As was mentioned earlier, the stretch of land between Alexandria and Syene was measured every year, decade after decade.  It is doubtful that one year the measurement would be increased by a full 40 stades (over 7 kilometers).

Similarly, changing the angle formed by the shadow in Alexandria from the original 7 1/5° to 7 1/7°, a decrease of 2/35°, gives a final result of 252,000 stades.  This is also unlikely. The best piece of astronomical equipment available at the time was the scaphe, which is basically a sundial [5, p.153].  It is doubtful that even the most precise scaphe was precise enough distinguish between 7 1/5° and 7 1/7° [6, p.412 ].

It may be that the correction was not due to an improved measurement, but instead to simplify future calculations involving the result.  Some scholars believe that Eratosthenes added 2000 stades simply to make the final figure divisible by 60.  Recalling that Eratosthenes divided the circle into 60 parts called hexacontades; dividing 250,000 stades by 60 results in approximately 4166.7 stades per hexacontade, whereas dividing 252,000 by 60 results in a round 4200 stades per hexacontade [5, p.154].  This reason seems far more likely.  Today, altering measurements in order to obtain a simple result is considered highly unscientific, but in the ancient world practicality often took priority over accuracy [2, p.45 ].

Having established that Eratosthenes probably gave 252,000 stades as his best approximation of the Earth’s circumference, and given four stade lengths which might represent reasonable approximations of the stade used by Eratosthenes, it is now possible to obtain some modern equivalents to Eratosthenes’ approximation.  Below is a table listing four approximations of the Earth’s circumference by Eratosthenes’ method, using each of the four previously mentioned types of stade.

 Type of Stade Equivalent Modern Length Stade x 252,000 Percent Difference from Modern Circumference Olympic 176.4 meters 44,450 kilometers +10.9% Italian 184.8 meters 46,560 kilometers +16.2% (Rawlins’ 17%) Babylonian-Persian 196.1 meters 49,410 kilometers +23.3% Phoenician-Egyptian 209.2 meters 52,710 kilometers +31.5%