plague lasted for almost four years, and killed a third of the Athenian
population. Only recently have we discovered through DNA analysis that
the disease was actually typhoid fever.^2
One can only imagine the desperation of the people, who were almost
certainly willing to attempt anything that had even the remotest chance
of alleviating the devastation. The oracle at Delos was consulted, and the
recommended remedy was to double the size of the existing altar, which
was in the shape of a cube.
It was easy to double the edge of a cube, but this would have created an
altar with a volume eight times the size of the initial one. The Greeks
were highly skilled in geometry, and realized that in order to construct a
cube whose volume was double the size of the initial one, the edge of the
doubled cube would have to exceed the length of the original cube by
a factor of the cube root of 2. None of the sages could use these instru-
ments to construct an edge of the desired length using only the compass
and unmarked straightedge. Eratosthenes relates that when the crafts-
men, who were to construct the altar, went to Plato to ask how to resolve
the problem, Plato replied that the oracle didn’t really want an altar of that
size, but by so stating the oracle intended to shame the Greeks for their
neglect of mathematics and geometry.^3 In the midst of a plague, receiving
a lecture on the mathematical deficiencies of Greek education was prob-
ably not what the craftsmen or the Athenian populace wanted to hear.
It took four years for the plague to burn out, but the problem of con-
structing a line segment of the desired length endured—either because
the Greeks relished the intellectual challenge of the problem, or as a pos-
sible defense against a recurrence of the plague. At any rate, the problem
of constructing a line segment of the desired length was solved by several
different mathematicians using a variety of approaches.
Probably the most elegant of the solutions was that proposed by Ar-
chytas, who constructed a solution based on the intersection of three
surfaces: a cylinder, a cone, and a torus (a torus looks like the inner tube
of a tire). This solution demonstrated a considerable amount of sophisti-
cation—solid geometry is considerably more complex than plane geome-
try (I took a course in solid geometry in high school and received a B
minus; to this day, it remains one of the toughest math courses I’ve ever
taken). Two simpler solutions were found by Menaechmus using plane
curves: the intersection of two parabolas, and the intersection of a hyper-
bola and a parabola.^4
Archytas’s and Menaechmus’s solutions are representative of a theme
we shall see throughout this book—the quest for solutions to a problem,
even an impossible one, often leads to fruitful areas where no man, or
Impossible Constructions 69