A history of Greek mathematics - Wilbourhall.org

106 ERATOSTHENES
geometric mean between CP, CP' ', the loci of A, G respectively
are conies. Zeuthen therefore suggests that these loci and
the corresponding loci of the points on CPP' at a distance
from C 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 conies. 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 circumference
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/I5th 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
Eratosthenes improved on this.
stades.
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 l/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 oc be the angle made at A by the sun's rays
with the gnomon (DA produced), the angle SOA is also equal to