use of letters to list items in order (the Iliad was divided into
24 books, which were given consecutive letters of the alpha-
bet), but this is not a true counting system.
Neither of the number systems was very easy to use for
calculation, but they were designed for recording numbers,
not manipulating them. Calculation was done with devices
such as the abacus: Th e word comes from the Greek word
abax, meaning a “board” or “slab,” which would be divided
into columns (sometimes covered with sand); numbers were
marked in writing or represented by pebbles.
Although the Greeks acquired mathematical knowledge
from the Babylonians (and to a lesser extent the Egyptians),
their original contributions to the fi eld were immense. Th e
work of the mathematical pioneers Pythagoras and Th ales
(both from the sixth century b.c.e.) is shrouded in legend:
the fi rst mathematician whose work is directly known is
Hippocrates of Chios, active at Athens in the late fi ft h cen-
tur y. He is said to have compiled an “Elements of Geometr y,”
which anticipated the work of Euclid by more than 200 years,
and to have worked on the problem of squaring the circle
(that is, the construction of a square with area equal to that
of a given circle). Th e greatest signifi cance of his work is that
it exhibits the concept of formal proof, which remains the
single greatest contribution of the Greeks to the develop-
ment of mathematics.
Th e existence of irrational numbers was a problem for
early geometers. (Colorful stories are told about Pythagoras’s
reaction to the discovery of the square root of 2, geometri-
cally the most obvious irrational number.) Infi nitesimals
(infi nitely small numbers) also caused concern and may be
the source of Zeno’s famous paradoxes about the impossibil-
ity of motion (Zeno “proved” that an arrow could not travel
through the air and that the famously swift -footed Achilles
could not catch up to a tortoise in a footrace). Logic and ge-
ometry developed greatly in the fourth century, as seen in
the works of Eudoxus and Aristotle. A culmination of these
trends is the work of Euclid, whose Elements provides a sum-
mary of the geometric knowledge of his day and is perhaps
the single most infl uential textbook of all time, especially for
its use of the axiomatic method, in which all conclusions are
derived from a small set of simple, apparently self-evident
statements (“axioms”). Euclid exhibits knowledge of alge-
braic geometry (showing further Babylonian infl uence) and
devotes part of the Elements to number theory, including a
proof that the number of primes is infi nite. His other sur-
viving works include the Phaenomena (an application of the
geometry of spheres to astronomy) and the Optics, a treatise
on perspective; that is, Euclid worked in what is known as
both applied and pure mathematics, as did most Greek math-
ematicians.
An even more noteworthy example is provided by the
career of Archimedes, whose work included systematic treat-
ments of statics and hydrostatics (branches of mathematics
dealing with the equilibrium of weights, in and out of wa-
ter) but who also put this knowledge of mechanics to use
as a designer of siege engines. (He was killed when Roman
troops sacked his native city of Syracuse in 212 b.c.e.) His
work exhibits an originality far beyond that of any other an-
cient mathematician. In describing mechanical theorems, he
uses a procedure of dividing fi gures into infi nitely thin strips,
which is quite similar to the procedure that led to the devel-
opment of the calculus in the 17th century c.e. In his book
Th e Sand Reckoner he describe a numerical system capable
of dealing with extremely large numbers far superior to any
system used in antiquity. Paradoxically, his infl uence on sub-
sequent mathematics was relatively small: few ancient math-
ematicians were capable of understanding the implications of
his work.
While the greatest achievements of the Greeks were in
geometry, there was also important work in arithmetic and
something approaching algebra. Th e work of Heron and Di-
ophantus (in the fi rst and second centuries c.e.) shows the
Greek attitude toward these branches of math: their works
present specifi c number problems (for example, “fi nd three
numbers such for which the product of any two of them plusPainted Greek stele with metrical inscription (Alison Frantz
Photographic Collection, American School of Classical Studies at Athens)804 numbers and counting: Greece