From the Beginning to Plato

Figure 8.14
Hippias of Elis and the teaching of the mathematical sciences
There is one other fifth-century figure to be mentioned in connection with
quadrature, the sophist Hippias of Elis, who described a curve known as the
quadratix (hē tetragōnizousa grammē) because of its use in squaring the circle. It
seems probable that Hippias used it only for the trisection of angles. 55 The
quadratix is defined as follows (see Figure 8.14). Let ABCD be a square, ABED
the quadrant of a circle with radius AB. Let AB make a uniform sweeping
motion through the quadrant in the same time in which BC falls uniformly to
coincide with AD. The quadratix is the curve BFGH described by the
intersection of the two moving straight lines.
The quadratix enables one to divide an angle in any given ratio, since it is easy
to prove that if the given ratio is IJ:FJ and GK is made equal and parallel to IJ, then
angle GAD:angle FAD::IJ:FJ. The quadrature of the circle is more difficult. One
first establishes the length of circumference of the circle by showing that:
arc BED:AB :: AB:AH 56
GREEK ARITHMETIC, GEOMETRY AND HARMONICS 285
and then uses a result, associated with Archimedes 57 and proved by the method of
exhaustion:
A circle has the same area as a right triangle with one leg equal to its
radius and the other equal to its circumference.
Hippias of Elis is known to us from Plato’s dialogues, where he is represented as
an intellectual jack-of-all trades, who has a prodigious memory, performs
numerical calculations at a speed which amazes his audience, and teaches the
‘quadrivium’, calculation (logismos), astronomy, geometry and music
(Protagoras 318e). In the Theaetetus (145a) Plato represents Theatetus’
instructor Theodoras, the remaining fifth-century mathematician mentioned in
Proclus’ history, as knowledgeable in the subjects of the quadrivium.