276 10. EUCLIDEAN GEOMETRYgave it a more exotic origin. In The Utility of Mathematics, the commentator
Theon of Smyrna, who lived around the year 100 CE, discusses a work called
Platonicus that he ascribes to Eratosthenes, to the effect that the citizens of Delos
(the island that was the headquarters of the Athenian Empire) consulted an oracle
in order to be relieved of a plague, and the oracle told them to double the size
of an altar (probably to Apollo). Plagues were common in ancient Greece; one
is described in Sophocles' Oedipus the King, and another decimated Athens early
in the Peloponnesian War, claiming Pericles as one of its victims. According to
Theon, Eratosthenes depicted the Delians as having turned for technical advice to
Plato, who told them that the altar was not the point: The gods really wanted the
Delians to learn geometry better. In his commentary on Archimedes' work on the
sphere and cylinder, Eutocius gives another story, also citing Eratosthenes, but he
says that Eratosthenes told King Ptolemy in a letter that the problem arose on the
island of Crete when King Minos ordered that a tomb built for his son be doubled
in size.
Whatever the origin of the problem, both Proclus and Eutocius agree that Hip-
pocrates was the first to reduce it to the problem of two mean proportionals. The
Pythagoreans knew that the mean proportional between any two square integers is
an integer, for example, · 49 = 28 and that between any two cubes such as 8
and 216 there are two mean proportionals (Euclid, Book 8, Propositions 11 and 12);
for example, 8 : 24 :: 24 : 72 :: 72 : 216. If two mean proportionals could be found
between two cubes—as seems possible, since every volume can be regarded as the
cube on some line—the problem would be solved. It would therefore be natural
for Hippocrates to think along these lines when comparing two cubes. Eutocius,
however, was somewhat scornful of this reduction, saying that the new problem
was just as difficult as the original one. That claim, however, is not true: One can
easily draw a figure containing two lines and their mean proportional (Fig. 1): the
two parts of the diameter on opposite sides of the endpoint of the half-chord of a
circle and the half-chord itself. The only problem is to get two such figures with the
half-chord and one part of the diameter reversing roles between the two figures and
the other parts of the diameters equal to the two given lines, as shown in Fig. 4. It
is natural to think of using two semicircles for this purpose and moving the chords
to meet these conditions.
In his commentary on the treatise of Archimedes on the sphere and cylinder,
Eutocius gives a number of solutions to this problem, ascribed to various authors,
including Plato. The earliest one that he reports is due to Archytas (ca. 428-
350 BCE). This solution requires intersecting a cylinder with a torus and a cone.