42 3. MATHEMATICAL CULTURES IIobservations." Among the techniques transmitted to the Greeks and ultimately
to the modern world was the convention of dividing a circle into 360 equal parts
(degrees). Greek astronomers divided the radius into 60 equal parts so that the
units of length on the radius and on the circle were very nearly equal.
The amount that the Greeks learned from Egypt is the subject of controversy.
Many scholars who have read the surviving mathematical texts from papyri have
concluded that Egyptian methods of computing were too cumbersome for applica-
tion to the complicated measurements of astronomers. Yet both Plato and Aristotle
speak approvingly of Egyptian computational methods and the ways in which they
were taught. As for geometry, it is generally acknowledged that the Egyptian in-
sight was extraordinary; the Egyptians knew how to find the volume of a pyramid,
for example. They even found the area of a hemisphere, the only case known before
Archimedes in which the area of a curved surface is found.^1 The case for advanced
Egyptian mathematics is argued in some detail by Bernal (1992), who asserts that
Ptolemy himself was an Egyptian. The argument is difficult to settle, since little is
known of Ptolemy personally; for us, he is simply the author of certain works on
physics and astronomy.
Because of their extensive commerce, with its need for counting, measuring,
navigation, and an accurate calendar, the Ionian Greek colonies such as Miletus
on the coast of Asia Minor and Samos in the Aegean Sea provided a very favor-
able environment for the development of mathematics, and it was there, with the
philosophers Thales of Miletus (ca. 624 547 BCE) and Pythagoras of Samos (ca.
570-475 BCE), that Greek mathematics began.1.1. Sources. Since the material on which the Greeks wrote was not durable, all
the original manuscripts have been lost except for a few ostraca (shells) found in
Egypt. We are dependent on copyists for preserving the information in early Greek
works, since few manuscripts that still exist were written more than 1000 years ago.
We are further indebted to the many commentators who wrote summary histories
of philosophy, including mathematics, for the little that we know about the works
that have not been preserved and their authors. The most prominent among these
commentators are listed below. They will be mentioned many times in the chapters
that follow.
Marcus Vitruvius (first century BCE) was a Roman architect who wrote an
extremely influential treatise on architecture in 10 books. He is regarded as a
rather unreliable source for information about mathematics, however.
Plutarch (45-120 CE) was a pagan author, apparently one of the best educated
people of his time, who wrote on many subjects. He is best remembered as the
author of the Parallel Lives of the Greeks and Romans, in which he compares
famous Greeks with eminent Romans who engaged in the same occupation, such
as the orators Demosthenes and Cicero.^2 Plutarch is important to the history of
mathematics for what he reports on natural philosophers such as Thales.
(^1) Some authors claim that the surface in question was actually half of the lateral surface of a
cylinder, but the words used seem more consistent with a hemisphere. In either case it was a
curved surface.
(^2) Shakespeare relied on Plutarch's account of the life of Julius Caesar, even describing the mirac-
ulous omens that Plutarch reported as having occurred just before Caesar's death.