- THE EARLIEST GREEK GEOMETRY 271
plan, since one could not get into the Pyramid to measure the distance from the
center to the tip of the shadow directly. One might use the Pythagorean theorem,
which Thales could well have known, to measure the distance from the center of
the pyramid to the point where its outer wall intersects the vertical plane through
the top of the pyramid and the tip of its shadow. A simpler way of computing the
distance, however, is to reflect a triangle about one of its vertices. This technique
is known to have been used by Roman surveyors to measure the distance across a
river without leaving shore.
According to Diogenes Laertius, a Roman historian named Pamphila, who lived
in the time of Nero, credits Thales with being the first to inscribe a right triangle in
a circle. To achieve this construction, one would have to know that the hypotenuse
of the inscribed triangle is a diameter. Diogenes Laertius goes on to say that others
attribute this construction to Pythagoras.
1.2. Pythagoras and the Pythagoreans. Half a century later than Thales the
philosopher Pythagoras was born on the island of Samos, another of the Greek
colonies in Ionia. No books of Pythagoras survive, but many later writers mention
him, including Aristotle. Diogenes Laertius devotes a full chapter to the life of
Pythagoras. He acquired even more legends than Thales. According to Diogenes
Laertius, who cites the logicist Apollodorus, Pythagoras sacrificed 100 oxen when
he discovered the theorem that now bears his name. If the stories about Pythagoras
can be believed, he, like Thales, traveled widely, to Egypt and Mesopotamia. He
gathered about him a large school of followers, who observed a mystical discipline
and devoted themselves to contemplation. They lived in at least two places in
Italy, first at Croton, then, after being driven out,^4 at Metapontion, where he died
sometime around 500 BCE.
According to Book I, Chapter 9 of Attic Nights, by the Roman writer Aulus
Gellius (ca. 130 180), the Pythagoreans first looked over potential recruits for phys-
ical signs of being educable. Those they accepted were first classified as akoustikoi
(auditors) and were compelled to listen without speaking. After making sufficient
progress, they were promoted to mathematikoi (learners).^5 Finally, after pass-
ing through that state they became physikoi (natural philosophers). In his Life
of Pythagoras Iamblichus uses these terms to denote the successors of Pythago-
ras, who split into two groups, the akoustikoi and the mathematikoi. According to
Iamblichus, the mathematikoi recognized the akoustikoi as genuine Pythagoreans,
but the sentiment was not reciprocated. The akoustikoi kept the pure Pythagorean
doctrine and regarded the mathematikoi as followers of the disgraced Hippasus
mentioned in Chapter 8.
Diogenes Laertius quotes the philosopher Alexander Polyhistor (ca. 105-35
BCE) as saying that the Pythagoreans generated the world from monads (units).
By adding a single monad to itself, they generated the natural numbers. By al-
lowing the monad to move, they generated a line, then by further motion the line
generated plane figures (polygons), the plane figures then moved to generate solids
(polyhedra). From the regular polyhedra they generated the four elements of earth,
air, fire, and water.
(^4) Like modern cults, the Pythagoreans attracted young people, to the despair of their parents.
Accepting new members from among the local youth probably aroused the wrath of the citizenry.
(^5) Gellius remarks at this point that the word mathematikoi was being inappropriately used in
popular speech to denote a "Chaldean" (astrologer).