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

PLATONICUS AND ON MEANS 105
there, which was cubical in form, should be doubled in size.
The book evidently contained a disquisition on 'proportion
(dvaXoyia); a quotation by Theon on this
subject shows that
Eratosthenes incidentally dealt with the fundamental definitions
of geometry and arithmetic. The principles of music
were discussed in the same work.
We have already described Eratosthenes' s solution of the
problem of Delos, and his contribution to the theory of arithmetic
by means of his sieve (kovkivov) for finding successive
prime numbers.
He wrote also an independent work On means. This was in
two Books, and was important enough to be mentioned by
Pappus along with works by Euclid, Aristaeus and Apollonius
as forming part of the Treasury of Analysis 1 ;
this
proves that it was a systematic geometrical treatise. Another
passage of Pappus speaks of certain loci which Eratosthenes
called 'loci with reference to means' (tottol irpbs fieo-oTrjTas) 2 ;
these were presumably discussed in the treatise in question.
What kind of loci these were is quite uncertain ; Pappus (if it
is not an interpolator who speaks) merely says that these loci
'
belong to the aforesaid classes of loci ', but as the classes are
numerous (including ' plane ', ' solid ', ' linear ', ' loci on surfaces ',
&c), we are none the wiser. Tannery conjectured that they
were loci of points such that their distances from three fixed
straight lines furnished a ' mediete^', i.e. loci (straight lines
and conies) which we should represent in trilinear coordinates
by such equations as 2y = x + z,
y 2 = xz, y(x + z) = 2xz,
x(x — y) — z(y — z), x(x — y) = y(y — z), the first three equations
representing the arithmetic, geometric and harmonic means,
while the last two represent the ' subcontraries ' to the
harmonic and geometric means respectively. Zeuthen has
a different conjecture. 3 He points out that, if QQ' be the
polar of a given point C with reference to a conic, and GPOP'
be drawn through meeting QQ f in and the conic in P, P f ,
then GO is the harmonic mean to GP, GP' ;
the locus of for
all transversals GPP' is then the straight line QQ\ If A, G
are points on PP f
such that GA is the arithmetic, and GG the
1
2
Pappus, vii, p. 636. 24.
lb., p. 662. 15 sq.
3 Zeuthen, Die Lehre von den Kegelschnitten im Altertum, 1886, pp.
320, 321.
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
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 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