mathematician, has gone before. Menaechmus is credited with the dis-
covery of the hyperbola and parabola,^5 which are two of the four conic
sections, the others being the circle and the ellipse. Each of the curves is
the intersection of a plane and a cone, and each of the curves not only re-
curs constantly in nature, but has also been incorporated in many of the
devices that characterize our technological age: the parabola in the para-
bolic ref lectors of satellite dishes; the ellipse in lithotripsy machines,
which break up kidney stones using sound waves rather than invasive
surgery; and the hyperbola in navigational systems such as loran.
The Greeks did more than simply discover solutions to mathematical
problems; they also used them. Plato invented a device known as Plato’s
machine, which utilized geometry, to construct a line segment whose
length was the cube root of the length of a given line segment. Plato, how-
ever, was not the only savant to tackle the physical construction of dou-
bling the cube. Another person to undertake this task was Eratosthenes.
His construction, involving simple rotations of lines and attached trian-
gles, was capable of being adapted to construct not only cube roots, but
any integer root. Eratosthenes supplied color commentary on his con-
struction, complete with a denigration of rival techniques.
“If, good friend, thou mindest to obtain from any small cube a cube the
double of it, and duly to change any solid f igure into another, this is in thy
power; thou canst find the measure of a fold, a pit, or the broad basin of a
hollow well, by this method, that is, if thou thus catch between two rulers
two means with their extreme ends converging. Do not thou seek to do
the difficult business of Archytas’s cylinders, or to cut the cone in the
triads of Menaechmus, or to compass such a curved form of lines as is
described by the god-fearing Eudoxus.”^6
Eratosthenes’s self-serving remarks may have been prompted by his be-
ing given the nickname Beta (the second letter of the Greek alphabet) by
his contemporaries, who felt that his not-inconsiderable achievements
(among which were the first accurate measurement of the circumference
of Earth, the compilation of a star catalog, and numerous contributions to
mathematics, astronomy, and geography) never merited the supreme ac-
colades given to the best of the best.
“[Eratosthenes] was, indeed, recognised by his contemporaries as a man
of great distinction in all branches of knowledge, though in each subject
he just fell short of the highest place. On the latter ground he was called
Beta, and another nickname applied to him, Pentathlos, has the same
implication, representing as it does an all-round athlete who was not the
first runner or wrestler but took the second prize in these contests as well
70 How Math Explains the World