From the Beginning to Plato

CHAPTER 8
Greek arithmetic, geometry and harmonics:
Thales to Plato
lan Mueller
INTRODUCTION: PROCLUS’ HISTORY OF GEOMETRY
In a famous passage in Book VII of the Republic starting at Socrates proposes to
inquire about the studies (mathēmata) needed to train the young people who will
become leaders of the ideal polis he is describing, that is, the subjects that will
draw their souls away from the sensible world of becoming to the intelligible
world of being and the dialectical study of the Forms. Socrates goes on to discuss
five such studies: arithmetic, plane geometry, solid geometry, astronomy and
harmonics. The purpose of this chapter is to discuss some important aspects of
the development of these mathematical sciences other than astronomy in the
Greek world down to the later fourth century.
As will become clear, discussion of this topic involves a wide range of
interrelated historical, philosophical, and philological questions on many of
which opinion still remains sharply divided. My goal here is more to explain
what the questions are than to offer answers to them. As a framework for my
discussion I shall use a passage from Proclus’ commentary on Book I of Euclid’s
Elements ([8.74], 64.7ff.), which is sometimes referred to as the Eudemian
summary, on the assumption that its ultimate source is Eudemus’ history of
geometry written c.300. 1 I shall simply call it Proclus’ history. Proclus begins by
mentioning the origin of geometry in Egyptian land-surveying and the origin of
arithmetic in Phoenician commercial activity. He then turns to the
accomplishments of the Greeks, beginning with the proverbial Thales, standardly
supposed to have flourished c.585, of whom he says:
Thales, who had traveled to Egypt, was the first to introduce geometry into
Greece. He made many discoveries himself and taught his successors the
principles for many other discoveries, treating some things in a more
universal way, others more in terms of perception.
([8.74], 65.7–11)