the “Treasury of Analysis” (topos analuomenos), a corpus of resources for the solution of
geometrical problems by analysis, but he does not describe the work’s contents. It was likely
in On Means that Eratosthene ̄s discussed “loci on means,” which according to an obscure
statement of Pappos’ (7.22) seem to have comprised straight lines and circles, conic sec-
tions, and other curved lines. We do not know the context in which Eratosthene ̄s presented
the so-called “sieve,” an algorithm for finding prime numbers (N, Introductio
Arithmetica 1.13).
E (In Arch. Sph. Cyl. pp. 88–96 Heiberg) quotes what purports to be a letter
addressed by Eratosthene ̄s to “King Ptolemy,” describing a geometrical and instrumental
solution of the problem of finding two mean proportionals between two given rectilinear
magnitudes; that is, given linear magnitudes A and D, to find magnitudes B and C such that
A : B = B : C = C : D. The letter originates the problem in the story of how the Delians
consulted the “geometers around P” on how to obey an oracle commanding the
doubling of a cubical altar. Eratosthene ̄s’ geometrical solution is to erect A and D as
perpendiculars to a base line, and to construct three similar right triangles adjacent to one
another on this base such that the first has the end point of A as its vertex and the other
two have their vertices collinear with the end points of A and D (Fig.); the heights of these
latter triangles are the mean proportionals. The solution is to be implemented mechanically
by an arrangement of rigid triangles sliding along grooves. According to the letter, Eratos-
thene ̄s made a votive dedication of a bronze specimen of this mesolabon (“mean-obtainer”)
accompanied by a proud epigram in elegiacs, reproduced at the end of the letter, asserting
the superiority of Eratosthene ̄s’ solution to those of A, E, and
M. (N would in turn castigate Eratosthene ̄s’ approach as both
unmechanical and ungeometrical; cf. Eutokios p. 98 Heiberg.) Modern scholarship has, for
the most part, followed Wilamowitz in considering the letter spurious but the epigram
authentic, though Knorr has argued that the whole is genuine. It seems plausible in any
case that Eratosthene ̄s did commemorate his discovery through a votive object and
inscription.
T S (p. 2, Hiller) reports that Eratosthene ̄s gave a similar account of the
Delians’ efforts to double their altar in Plato ̄nikos, “the Platonist.” This seems to have been a
discursive book on the philosophy of mathematical objects and relations, and Theo ̄n’s
several citations of Eratosthene ̄s on the topic of ratios probably come from it. Theo ̄n
(p. 142) also ascribes to Eratosthene ̄s a discussion of the harmonies of the celestial spheres
that was partly in verse and contained an etiological myth for the origins of the celestial
tuning.
Eratosthene ̄s’ mechanical method of finding two mean proportionals between
magnitudes A and D. The left and right triangles are slid along grooves until by trial and error
the four vertices are collinear © Jones
ERATOSTHENE ̄S OF KURE ̄NE ̄