Chapter 10. Euclidean Geometry
We shall divide the history of Greek mathematics into four periods. The first period,
from about 600 to 400 BCE, was the time when the Greeks acquired geometry from
Egypt and Mesopotamia and turned it in the direction of logical argument. The
second period came in the fourth century, when the logical aspects of the subject
were debated in Plato's Academy^1 and Aristotle's Lyceum,^2 proofs were improved,
and basic principles and assumptions were isolated. The third period began in the
third century, when the mature subject was expounded in Euclid's Elements, and
further research continued on more complicated curves and surfaces. The fourth
and final period was a long decline in originality, in which no revolutionary changes
occurred and commentaries were the main literary form.
1. The earliest Greek geometry
The history of Greek mathematics up to the time of Euclid (300 BCE) was written
by Eudemus, a pupil of Aristotle. This history was lost, but it is believed to be
the basis of the first paragraph of a survey given by Proclus in the fifth century
CE in the course of his commentary on the first book of Euclid. In this passage
Proclus mentions 25 men who were considered to have made significant contri-
butions to mathematics. Of these 25, five are well known as philosophers (Thales,
Pythagoras, Anaxagoras, Plato, and Aristotle); three are famous primarily as math-
ematicians and astronomers (Euclid, Eratosthenes, and Archimedes). The other 17
have enjoyed much less posthumous fame. Some of them are so obscure that no
mention of them can be found anywhere except in Proclus' summary. Some others
(Theodorus, Archytas, Menaechmus, Theaetetus, and Eudoxus) are mentioned by
other commentators and by Plato. The 13 just named are the main figures we shall
use to sketch the history of Greek geometry. It is clear from what Proclus writes
that something important happened to mathematics during the century of Plato
and Aristotle, and the result was a unique book, Euclid's Elements.
Missing from the survey of Proclus is any reference to Mesopotamian influence
on Greek geometry. This influence is shown clearly in Greek astronomy, in the(^1) This word has become so common in English that its original, legendary meaning is mostly
forgotten. In his biography of the Athenian king Theseus, who had slain the Minotaur on Crete
as a youth, Plutarch says that at the age of 50 the widowed king abducted the beautiful 12-year-
old Helen of Sparta and hid her away. (This was before she married Menelaus and ran off with
Paris, becoming the cause of the Trojan War.) Her twin brothers Castor and Polydeukes (Pollux)
threatened to destroy Athens in revenge. Akademos, however, averted the calamity by telling
them where she was hidden. For this deed he was venerated as the savior of the city, and a grove
of trees on its northwest side, supposedly his burial place, was dedicated to his memory. Plato
gave his lectures in that grove, and hence arose the phrase "the groves of Academe."
(^2) Here is another word whose origins are lost in common usage. The Lyceum was so named
because it was near the temple to Apollo Lykeios ("Apollo of the Wolves").
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