We should mention that when speaking of “Greek
mathematics,” historians include any mathematician
who wrote in the Greek language and followed the
Greek tradition of mathematical thought. Greek was
the common language of the Mediterranean world dur-
ing ancient times, and many intellectuals from different
parts of that region are today considered Greek schol-
ars. For instance, the great Archimedes was from Syra-
cuse, now a part of Italy, and E
UCLID
(ca. 300–260
B
.
C
.
E
.) is believed to have lived in Alexandria, Egypt.
There are very few original records of Greek work.
Initially, knowledge was transmitted only orally from
teacher to student. Around 450
B
.
C
.
E
. the Greeks
adopted the ancient Egyptian practice of writing on
papyrus scrolls. Unfortunately, papyrus—a grasslike
plant grown in the Nile Delta region—decays rapidly
away from the exceptionally dry climate of Egypt. The
Greeks combated this problem by repeatedly making
copies of their works but, because of the effort
involved, copied only those pieces they deemed of
utmost importance. The first mathematical work pre-
served and honored this way was Euclid’s masterpiece
T
HE
E
LEMENTS
of ca. 300
B
.
C
.
E
. Historians have had
to rely on commentary made by later scholars to
deduce what was accomplished mathematically before
the time of Euclid.
Greek scholars approached all of mathematics
through the study of
GEOMETRY
. Even their work on
the properties of whole numbers, ratios, and propor-
tions, as well as mechanics and astronomy was done in
a geometric style. A “number,” for instance, was liter-
ally a line segment, and a “ratio” was understood in
terms of
COMMENSURABLE
segments. It is interesting to
note that Greek scholars took careful steps to avoid
speaking directly of the infinite. (The fifth-century
B
.
C
.
E
. paradoxes on the nature of motion and the
infinitely small developed by Z
ENO OF
E
LEA
deeply
affected Greek thinking.) For instance, Euclid stated
that any line segment could be extended to any arbi-
trary length, but never spoke of lines that were
infinitely long. In E
UCLID
’
S PROOF OF THE INFINITUDE
OF PRIMES
, Euclid stated that from any finite list of
PRIME
numbers one can always construct one more, but
never spoke of the set of primes as infinite.
Many historians regard T
HALES OF
M
ILETUS
(ca.
625–547
B
.
C
.
E
.) as the first Greek mathematician of
note. Commentaries suggest that Thales identified, and
proved, seven key geometric propositions, including
that the base angles of an
ISOSCELES TRIANGLE
are
always equal and that the inscribed angle from the
diameter of a
CIRCLE
is always a right angle, for
instance. The great scholar and mystic P
YTHAGORAS
lived a century later, and he and his followers are cred-
ited with the discovery of the famous result about right
triangles (today called P
YTHAGORAS
’
S THEOREM
) and
the discovery of
IRRATIONAL NUMBERS
. A great deal of
mystery surrounds the life and legend of Pythagoras.
He founded a semireligious sect called the Pythagorean
Brotherhood (women were equal members) based on
certain mystic significances ascribed to whole numbers
and their ratios.
The great philosopher P
LATO
(428–348
B
.
C
.
E
.)
wrote a great deal about mathematics in his famous dia-
logues, demonstrating a deep personal respect for the
subject. The five regular polyhedra—the P
LATONIC
SOLID
s—are named in his honor. In his philosophical
treatises, Plato used the example of mathematics as
something that cannot be discovered by the senses, but
can nonetheless be discovered by the power of logical
reasoning. He also believed mathematics to be an essen-
tial part of a cultured person’s education. Philosopher
A
RISTOTLE
(384–322
B
.
C
.
E
.) adopted the same view and
used mathematics as examples in his development of
FORMAL LOGIC
and his analysis of
ARGUMENT
s.
Today, the Greek scholar Euclid is considered to be
the most influential mathematics scholar of all time. In
his famous work The Elements, Euclid collated all
mathematical knowledge known at his time into a sin-
gle tome. Although an impressive feat, it was the orga-
nization of the text that had the greatest impact.
Beginning with a small collection of “self-evident
truths,” Euclid showed that all mathematical knowl-
edge of his time could be deduced by pure logical rea-
soning alone. This work demonstrated the power of the
mind and set the model for all mathematical research in
the future. Mathematicians today still work to the stan-
dards of rigor as set by Euclid. Next to the Bible,
Euclid’s The Elements is the most widely published
book of all time.
After producing The Elements, Euclid continued
work on the
CONIC SECTIONS
, on optics, and on general
problems in geometry. He continued interest in
CON
-
STRUCTIBLE
numbers and no doubt contemplated the
classic Greek problem of
SQUARING THE CIRCLE
. (In
The Elements Euclid had demonstrated general proce-
dures for squaring arbitrary polygonal figures.) This
238 Greek mathematics