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 during ancient times, and many intellectuals from different
parts of that region are today considered Greek scholars. For instance, the great Archimedes was from Syracuse, now a part of Italy, and EUCLID (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 preserved and honored this way was Euclid’s masterpiece
THE ELEMENTS 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 proportions, as well as mechanics and astronomy was done in
a geometric style. A “number,” for instance, was literally 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 ZENO OF ELEA deeply
affected Greek thinking.) For instance, Euclid stated
that any line segment could be extended to any arbitrary length, but never spoke of lines that were
infinitely long. In EUCLID’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 THALES OF MILETUS (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 PYTHAGORAS
lived a century later, and he and his followers are credited with the discovery of the famous result about right
triangles (today called PYTHAGORAS’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 PLATO (428–348 B.C.E.)
wrote a great deal about mathematics in his famous dialogues, demonstrating a deep personal respect for the
subject. The five regular polyhedra—the PLATONIC
SOLIDs—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 essential part of a cultured person’s education. Philosopher
ARISTOTLE (384–322 B.C.E.) adopted the same view and
used mathematics as examples in his development of
FORMAL LOGIC and his analysis of ARGUMENTs.
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 single tome. Although an impressive feat, it was the organization of the text that had the greatest impact.
Beginning with a small collection of “self-evident
truths,” Euclid showed that all mathematical knowledge of his time could be deduced by pure logical reasoning 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 standards 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 CONSTRUCTIBLE numbers and no doubt contemplated the
classic Greek problem of SQUARING THE CIRCLE. (In
The Elements Euclid had demonstrated general procedures for squaring arbitrary polygonal figures.) This
238 Greek mathematics
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 during ancient times, and many intellectuals from different
parts of that region are today considered Greek scholars. For instance, the great Archimedes was from Syracuse, now a part of Italy, and EUCLID (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 preserved and honored this way was Euclid’s masterpiece
THE ELEMENTS 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 proportions, as well as mechanics and astronomy was done in
a geometric style. A “number,” for instance, was literally 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 ZENO OF ELEA deeply
affected Greek thinking.) For instance, Euclid stated
that any line segment could be extended to any arbitrary length, but never spoke of lines that were
infinitely long. In EUCLID’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 THALES OF MILETUS (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 PYTHAGORAS
lived a century later, and he and his followers are credited with the discovery of the famous result about right
triangles (today called PYTHAGORAS’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 PLATO (428–348 B.C.E.)
wrote a great deal about mathematics in his famous dialogues, demonstrating a deep personal respect for the
subject. The five regular polyhedra—the PLATONIC
SOLIDs—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 essential part of a cultured person’s education. Philosopher
ARISTOTLE (384–322 B.C.E.) adopted the same view and
used mathematics as examples in his development of
FORMAL LOGIC and his analysis of ARGUMENTs.
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 single tome. Although an impressive feat, it was the organization of the text that had the greatest impact.
Beginning with a small collection of “self-evident
truths,” Euclid showed that all mathematical knowledge of his time could be deduced by pure logical reasoning 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 standards 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 CONSTRUCTIBLE numbers and no doubt contemplated the
classic Greek problem of SQUARING THE CIRCLE. (In
The Elements Euclid had demonstrated general procedures for squaring arbitrary polygonal figures.) This
238 Greek mathematics