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THE DUPLICATION OF THE CUBE
theorem that, if the distance of a point from a fixed point is
in a given ratio to its distance from a fixed line, the locus of
the point is a conic section which is an ellipse, a parabola,
or a hyperbola according as the given ratio is less than, equal
to, or greater than, unity. The importance of these passages
lies in the fact that the Lemma was required for the
understanding of Euclid’s treatise. We can hardly avoid
the conclusion that the property was used by Euclid in his
Surface-Loci, but was assumed as well known. It was, there
fore, probably taken from some treatise current in Euclid’s
time, perhaps from Aristaeus’s work on Solid Loci.
The Duplication of the Cube, or the problem
of the two mean proportionals.
(a) History of the problem.
In his commentary on Archimedes, On the Sphere and
Cylinder, II. 1, Eutocius has preserved for us a precious
collection of solutions of this famous problem. 1 One of the
Solutions is that of Eratosthenes, a younger contemporary of
Archimedes, and it is introduced by what purports to be
a letter from Eratosthenes to Ptolemy. This was Ptolemy
Euergetes, who at the beginning of his reign (245 b.c.) per
suaded Eratosthenes to come from Athens to Alexandria to be
tutor to his son (Philopator). The supposed letter gives the
tradition regarding the origin of the problem and the history of
its solution up to the time of Eratosthenes. Then, after some
remarks on its usefulness for practical purposes, the author
describes the construction by which Eratosthenes himself solved
it, giving the proof of it at some length and adding directions
for making the instrument by which the construction could
be effected in practice. Next he says that the mechanical
contrivance represented by Eratosthenes was, c in the votive
monument ’, actually of bronze, and was fastened on with lead
close under the arefdvri of the pillar. There was, further,
on the pillar the proof in a condensed form, with one figure,
and, at the end, an epigram. The supposed letter of Eratos
thenes is a forgery, but the author rendered a real service
1 Archimedes, ed. Heib., vol. iii, pp. 54. 26-106. 24.
