A history of Greek mathematics - Wilbourhall.org

MEASUREMENT OF THE EARTH 107
a, or l/50th of four right angles. Now the distance from S
to A was known by measurement to be 5,000 stades ; it
followed that the circumference of the earth was 250,000
stades. This is the figure given by Cleomedes, but Theon of
Smyrna and Strabo both give it as 252,000 stades. The
reason of the discrepancy is not known ; it is possible that
Eratosthenes corrected 250,000 to 252,000 for some reason,
perhaps in order to get a figure divisible by 60 and, incidentally,
a round number (700) of stades for one degree. If
Pliny is right in saying that Eratosthenes made 40 stades
equal to the Egyptian a\o1vos, then, taking the o-yolvos at
12,000 Royal cubits of 0-525 metres, we get 300 such cubits,
or 157-5 metres, i.e. 516-73 feet, as the length of the stade.
On this basis 252,000 stades works out to 24,662 miles, and
the diameter of
the earth to about 7,850 miles, only 50 miles
shorter than the true polar diameter, a surprisingly close
approximation, however much it owes to happy accidents
in the calculation.
We learn from Heron's Dioptra that the measurement of
the earth by Eratosthenes was given in a separate work On
the Measurement of the Earth. According to Galen 1 this work
dealt generally with astronomical or mathematical geography,
treating of ' the size of the equator, the distance of the tropic
and polar circles, the extent of the polar zone, the size and
distance of the sun and moon, total and partial eclipses of
these heavenly bodies, changes in the length of the day
according to the different latitudes and seasons'. Several
details are preserved elsewhere of results obtained by
Eratosthenes, which were doubtless contained in this work.
He is supposed to have estimated the distance between the
tropic circles or twice the obliquity of the ecliptic at 1 l/83rds
of a complete circle or 47° 42' 39"; but from Ptolemy's
language on this subject it is not clear that this estimate was
not Ptolemy's own. What Ptolemy says is that he himself
found the distance between the tropic circles to lie always
between 47° 40' and 47° 45', 'from which we obtain about
(ayeSov) the same ratio as that of Eratosthenes, which
Hipparchus also used. For the distance between the tropics
becomes (or is found to be, yiverai) very nearly 1 1 parts
Galen, Instit. Logica, 12 (p. 26 Kalbfleisch).