A history of Greek mathematics - Wilbourhall.org

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.