106 
ERATOSTHENES 
geometric mean between CP, CP', the loci of A, G respectively 
are conics. Zenthen therefore suggests that these loci and 
the corresponding loci of the points on CPP' at a distance 
from G equal to the subcontraries of the geometric and 
harmonic means between CP and CP' are the ‘loci with 
reference to means ’ of Eratosthenes; the latter two loci are 
‘linear’, i.e. higher curves than conics. Needless to say, we 
have no confirmation of this conjecture. 
Eratosthenes’s measurement of the Earth. 
But the most famous scientific achievement of Eratosthenes 
was his measurement of the earth, Archimedes mentions, as 
we have seen, that some had tried to prove that the circum 
ference of the earth is about 300,000 stades. This was 
evidently the measurement based on observations made at 
Lysimachia (on the Hellespont) and Syene. It was observed 
that, while both these places were on one meridian, the head 
of Draco was in the zenith at Lysimachia, and Cancer in the 
zenith at Syene; the arc of the meridian separating the two 
in the heavens was taken to be 1 /15th of the complete circle. 
^ . The distance between the two towns 
was estimated at 20,000 stades, and 
accordingly the whole circumference of 
the earth was reckoned at 300,000 
stades. Eratosthenes improved on this. 
He observed (1) that at Syene, at 
noon, at the summer solstice, the 
sun cast no shadow from an upright 
gnomon (this was confirmed by the 
observation that a well dug at the 
same place was entirely lighted up at 
the same time), while (2) at the same moment the gnomon fixed 
upright at Alexandria (taken to be on the same meridian with 
Syene) cast a shadow corresponding to an angle between the 
gnomon and the sun’s rays of 1 /50th of a complete circle or 
four right angles. The sun’s rays are of course assumed to be 
parallel at the two places represented by S and A in the 
annexed figure. If a be the angle made at A by the sun’s rays 
with the gnomon (OA produced), the angle SO A is also equal to 
a, ci 
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