- GREEK AND ROMAN MATHEMATICS 43
Theon of Smyrna (ca. 100 CE) was the author of an introduction to mathemat-
ics written as background for reading Plato, a copy of which still exists. It contains
many quotations from earlier authors.
Diogenes Laertius (third century CE) wrote a comprehensive history of phi-
losophy, Lives of Eminent Philosophers, which contains summaries of many earlier
works and gives details of the lives and work of many of the pre-Socratic philoso-
phers. He appears to be the source of the misnomer "Pythagorean theorem" that
has come down to us (see Zhmud, 1989, p. 257).
Iamblichus (285-330 CE) was the author of many treatises, including 10 books
on the Pythagoreans, five of which have been preserved.
Pappus (ca. 300 CE) wrote many books on geometry, including a comprehensive
treatise of eight mathematical books. He is immortalized in calculus books for his
theorem on the volume of a solid of revolution. Besides being a first-rate geometer
in his own right, he wrote commentaries on the Almagest of Ptolemy and the tenth
book of Euclid's Elements.
Proclus (412-485 CE) is the author of a commentary on the first book of
Euclid, in which he quoted a long passage from a history of mathematics, now lost,
by Eudemus, a pupil of Aristotle.
Simplicius (500-549 CE) was a commentator on philosophy. His works contain
many quotations from the pre-Socratic philosophers.
Eutocius (ca. 700 CE) was a mathematician who lived in the port city of Askelon
in Palestine and wrote an extensive commentary on the works of Archimedes.
Most of these commentators wrote in Greek. Knowledge of Greek sank to a
very low level in western Europe as a result of the upheavals of the fifth century.
Although learning was preserved by the Church and all of the New Testament
was written in Greek, a Latin translation (the Vulgate) was made by Jerome in
the fifth century. From that time on, Greek documents were preserved mostly in
the Eastern (Byzantine) Empire. After the Muslim conquest of North Africa and
Spain in the eighth century, some Greek documents were translated into Arabic and
circulated in Spain and the Middle East. From the eleventh century on, as secular
learning began to revive in the West, scholars from northern Europe made journeys
to these centers and to Constantinople, copied out manuscripts, translated them
from Arabic and Greek into Latin, and tried to piece together some long-forgotten
parts of ancient learning.
1.2. General features of Greek mathematics. Greek mathematics—that is,
mathematics written in ancient Greek—is exceedingly rich in authors and works.
Its most unusual feature, compared with what went before, is its formalism. Math-
ematics is developed systematically from definitions and axioms, general theorems
are stated, and proofs are given. This formalism is the outcome of the entanglement
of mathematics with Greek philosophy. It became a model to be imitated in many
later scientific treatises, such as Newton's Philosophise naturalis principia mathe-
matica. Of course, Greek mathematics did not arise in the finished form found in
the treatises. Tradition credits Thales only with knowing four geometric proposi-
tions. By the time of Pythagoras, much more was known. The crucial formative
period was the first half of the fourth century BCE, when Plato's Academy flour-
ished. Plato himself was interested in mathematics because he hoped for a sort of
"theory of everything," based on fundamental concepts perceived by the mind.