The Greeks actually constructed both numerical and geometrical proofs
of this fact—the numerical proof was based on the concept of odd and
even numbers. If the square root of 2 could be expressed as the ratio p/q
of whole numbers, those numbers could be chosen to have no common
factor (we learned in elementary school to cancel common factors to re-
duce fractions). If p/q�2, then p^2 /q^2 2, and so p^2 2 q^2. Since p^2 is a
multiple of 2, p must be an even number, as odd numbers have odd
squares. Since p and q have no common factor, q must be odd. Letting
p 2 n, we see that (2n)^2 2 q^2 , and so q^2 2 n^2 ; the same reasoning as we
used to show that p is even shows that q is even—and we have thus con-
cluded that q is both even and odd.
The discovery of the incommensurability of the square root of 2 affected
the development of Greek mathematics as profoundly as the discovery of
the moons of Jupiter affected the development of astronomy. The Greeks
turned from the philosophy of arithmos (the belief in number that is obvi-
ously the root of our word arithmetic) to the logical deductions of geome-
try, whose validity were assured.
The geometry of the Greeks—later to be formalized by Euclid—was ini-
tially based on the line and the circle. The tools for exploring geometry
were the straightedge, for drawing lines and line segments, and the com-
pass, for creating circles. There does not seem to be a record as to why the
Greeks required that the straightedge be unmarked, so that no distances
were inscribed upon it. Possibly the earliest Greek geometers had access
only to the simplest tools, and the use of compass and unmarked straight-
edge simply became the traditional way to do geometry. However, it wasn’t
until the Greeks began the exploration of figures other than those con-
structed with lines and circles that the utility of the marked straightedge
began to reveal itself; it brought a certain ungainly aspect into geometri-
cal constructions, but greatly increased their scope. That exploration did
not begin until four hundred years before the birth of Christ, and after
another profound event was to shake the foundations of ancient Greece.
The First Pandemic
In 430 BC, the Athenians were engaged in the Peloponnesian War when
a plague overtook the city. The historian Thucydides was taken ill but
survived, and described the horrifying course of the disease.^1 The eyes,
throat, and tongue became red and bloody, followed by sneezing, cough-
ing, diarrhea, and vomiting. The skin was covered in ulcerated sores and
pustules, accompanied by a burning, unquenchable thirst. The disease
started in Ethiopia, and spread to Egypt, Libya, and then to Greece. The
68 How Math Explains the World