said that a Roman soldier came across Archimedes con-

centrating on geometric figures he had drawn in the

sand. Not knowing who the scholar was, or what he

was doing, the soldier simply killed him.

Archytas of Tarentum (ca. 428–350

B

.

C

.

E

.) Greek

Geometry, Philosophy The Greek scholar Archytas of

Tarentum was the first to provide a solution to the clas-

sic

DUPLICATING THE CUBE

problem of antiquity. By

reducing the challenge to one of constructing certain

ratios and proportions, Archytas developed a geometric

construct that involved rotating semicircles through

certain angles in three-dimensional space to produce a

length essentially equivalent to the construct of the

cube root of two. (Creating a segment of this length is

the chief stumbling block to solving the problem.)

Although his innovative solution is certainly correct, it

uses tools beyond what is permissible with straightedge

and compass alone. In the development of his solution,

Archytas identified a new mean between numbers,

which he called the

HARMONIC MEAN

.

Archytas of Tarentum 21

A 19th-century engraving of Archimedes’ water screw, a device for pumping water (Photo courtesy of AAAC/Topham/The Image Works)