A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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).
108 ERATOSTHENES
out of 83 contained in the whole meridian circle'. 1 The
mean of Ptolemy's estimates, 4 7° 42' 30", is of course nearly
ll/83rds of 360°. It is consistent with Ptolemy's language
to suppose that Eratosthenes adhered to the value of the
obliquity of the ecliptic discovered before Euclid's time,
namely 24°, and Hipparchus does, in his extant Commentary
on the Phaenomena of Aratus and Eudoxus, say that the
summer tropic is very nearly 24° north of the equator'.
'
The Doxographi state that Eratosthenes estimated the
distance of the moon from the earth at 780,000 stades and
the distance of the sun from the earth at 804,000,000 stades
(the versions of Stobaeus and Joannes Lydus admit 4,080,000
as an alternative for the latter figure, but this obviously
cannot be right). Macrobius 2 says that Eratosthenes made
the 'measure' of the sun to be 27 times that of the earth.
It is not certain whether measure means ' solid content ' or
'
diameter ' in this case ; the other figures on record make the
former more probable, in which case the diameter of the sun
would be three times that of the earth. Macrobius also tells
us that Eratosthenes's estimates of the distances of the sun
and moon were obtained by means of lunar eclipses.
Another observation by Eratosthenes, namely that at Syene
(which is under the summer tropic) and throughout a circle
round it with a radius of 300 stades the upright gnomon
throws no shadow at noon, was afterwards made use of by
Posidonius in his calculation of the size of the sun. Assuming
that the circle in which the sun apparently moves round the
earth is 10,000 times the size of a circular section of the earth
through its centre, and combining with this hypothesis the
datum just mentioned, Posidonius arrived at 3,000,000 stades
as the diameter of the sun.
Eratosthenes wrote a poem called Hermes containing a good
deal of descriptive astronomy ; only fragments of this have
survived. The work Catasterismi (literally placings among
'
the stars ') which is extant can hardly be genuine in the form
in which it has reached us ; it goes back, however, to a genuine
work by Eratosthenes which apparently bore the same name
alternatively it is alluded to as KardXoyoi or by the general
1
2
Ptolemy, Syntaxis, i. 12, pp. 67. 22-68. 6.
Macrobius, In Somn. Scip. i. 20. 9.