PLATONICUS AND ON MEANS 
105 
there, which was cubical in form, should be doubled in size. 
The book evidently contained a disquisition on ‘proportion 
{draXoyLa); a quotation by Theon on this subject shows that 
Eratosthenes incidentally dealt with the fundamental defini 
tions 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 arith 
metic by means of his sieve (koxklvov) 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 Apol 
lonius as forming part of the Treasury of Analysis'; 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 upas p.e<r6TrjTas) 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 ‘medidtd’, i.e. loci (straight lines 
and conics) which we should represent in trilinear coordinates 
by such equations as 2y = x + z, y 2 =xz, y(x + z) = 2xz, 
x i x ~y) = z{y — z), x{x — y) = yfy — 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 CPOP' 
be drawn through C meeting QQ' in 0 and the conic in P, P', 
then CO is the harmonic mean to CP, CP'; the locus of 0 for 
all transversals CPP' is then the straight line QQ'. If A, G 
are points on PP' such that CA is the arithmetic, and GG the 
’ Pappus, vii, p. 636. 24. 2 lb., p. 662. 15 sq. 
Zeuthen, Die Lehre von den Kegelschnitten im Altertum, 1886, pp. 
320, 321.