270 10. EUCLIDEAN GEOMETRYuse of the sexagesimal system of measuring angles and in Ptolemy's explicit use of
Mesopotamian astronomical observations. It may also appear in Book 2 of Euclid's
Elements, which contains geometric constructions equivalent to certain algebraic
relations that are frequently encountered in the cuneiform tablets. This relation,
however, is controversial. Leaving aside the question of Mesopotamian influence,
we do see a recognition of their debt to Egypt, which the Greeks never concealed.
And how could they? Euclid actually lived in Egypt, and the other two of the "big
three" Greek geometers, Archimedes and Apollonius, both studied there, in the
Hellenistic city of Alexandria at the mouth of the Nile.
1.1. Thales. The philosopher Thales, who lived in the early sixth century BCE,
was a citizen of Miletus, a Greek colony on the coast of Asia Minor. The ruins
of Miletus are now administered by Turkey. Herodotus mentions Thales in several
places. Discussing the war between the Medes and the Lydian king Croesus, which
had taken place in the previous century, he says that an eclipse of the Sun frightened
the combatants into making peace. Thales, according to Herodotus, had predicted
that an eclipse would occur no later than the year in which it actually occurred.
Herodotus goes on to say that Thales had helped Croesus to divert the river Halys
so that his army could cross it.
These anecdotes show that Thales had both scientific and practical interests.
His prediction of a solar eclipse, which, according to the astronomers, occurred
in 585 BCE, seems quite remarkable, even if, as Herodotus says, he gave only a
period of several years in which the eclipse was to occur. Although solar eclipses
occur regularly, they are visible only over small portions of the Earth, so that their
regularity is difficult to discover and verify. Lunar eclipses exhibit the same period
as solar eclipses and are easier to observe. Eclipses recur in cycles of about 19 solar
years, a period that seems to have been known to many ancient peoples. Among
the cuneiform tablets from Mesopotamia there are many that discuss astronomy,
and Ptolemy uses Mesopotamian observations in his system of astronomy. Thales
could have acquired this knowledge, along with certain simple facts about geometry,
such as the fact that the base angles of an isosceles triangle are equal. Bychkov
(2001) argues that the recognition that the base angles of an isosceles triangle are
equal probably did come from Egypt. In construction, for example, putting a roof
on a house, it is not crucial that the cross section be exactly an isosceles triangle,
since it is the ridge of the roof that must fit precisely, not the edges. However,
when building a symmetric square pyramid, errors in the base angles of the faces
would make it impossible for the faces to fit together tightly. Therefore, he believes,
Thales must have derived this theorem from his travels in Egypt.
In his Discourses on the Seven Wise Men, Plutarch reports that Thales traveled
to Egypt and was able to calculate the height of the Great Pyramid by driving a pole
into the ground and observing that the ratio of the height of the pyramid to that of
the pole was the same as the ratio of their shadow lengths. In his Lives of Eminent
Philosophers, Diogenes Laertius cites the historian Hieronymus (fourth or third
century BCE) in saying that Thales calculated the height of the pyramid by waiting
until his shadow was exactly as long as he was tall, then measuring the length of the
shadow of the Great Pyramid.^3 There are practical difficulties in executing this
(^3) A very interesting mystery/historical novel by Denis Guedj, called Le thioreme du perroquet,
uses this history to connect its story line. An English translation of this novel now exists, The
Parrot's Theorem, St. Martin's Press, New York, 2002.