248
THE DUPLICATION OF THE CUBE
when the god proclaimed to the Delians by the oracle that, if
they would get rid of a plague, they should construct an altar
double of the existing one, their craftsmen fell into great
perplexity in their efforts to discover how a solid could be made
double of a (similar) solid; they therefore went to ask Plato
about it, and he replied that the oracle meant, not that the god
wanted an altar of double the size, but that he wished, in
setting them the task, to shame the Greeks for their neglect
of mathematics and their contempt for geometry.’ 1
Eratosthenes’s version may well be true; and there is no
doubt that the question was studied in the Academy, solutions
being attributed to Eudoxus, Menaechmus, and even (though
erroneously) to Plato himself. The description by the pseudo-
Eratosthenes of the three solutions by Archytas, Eudoxus and
Menaechmus is little more than a paraphrase of the lines about
them in the genuine epigram of Eratosthenes,
£ Do not seek to do the difficult business of the cylinders of
Archytas, or to cut the cones in the triads of Menaechmus, or
to draw such a curved form of lines as is described by the
god-fearing Eudoxus.’
The different versions are reflected in Plutarch, who in one
place gives Plato’s answer to the Delians in almost the same
words as Eratosthenes, 2 and in another place tells us that
Plato referred the Delians to Eudoxus and Helicon of Cyzicus
for a solution of the problem. 3
After Hippocrates had discovered that the duplication of
the cube was equivalent to finding two mean proportionals in
continued proportion between two given straight lines, the
problem seems to have been attacked in the latter form
exclusively. The various solutions will now be reproduced
in chronological order.
(/3) Archytas.
The solution of Archytas is the most remarkable of all,
especially when his date is considered (first half of fourth cen
tury b. c.), because it is not a construction in a plane but a bold
1 Theon of Smyrna, p. 2. 3-12.
2 Plutarch, Be E apud Delphos, c. 6, 886 E.
3 Be genio Socratis, c. 7, 579 c, n.
