3.2 Pre-Euclidean Greek Mathematics
Euclid’s Elements (c. 300 BC) forms a natural break point in Greek mathematics, since much of what
came before was subsumed by it. In this section, we consider the contributions of several pre-
Euclidean mathematicians. There are very few sources for Greek mathematics & philosophy before
400 BC, so almost everything is inferred from the writings and commentaries of others.
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Thales of Miletus (c. 624–546 BC) One of the first people known to state abstract general principles,
Thales was a trader based in Miletus, a city-state in Anatolia. He travelled widely and was likely
exposed to mathematical ideas from all round the Mediterranean. Here are some statements at least
partly attributed to Thales:
• The angles at the base of an isosceles triangle are equal.
• Any circle is bisected by its diameter.
• A triangle inscribed in a semi-circle is right-angled (still known as Thales’ Theorem).
Thales’ major development is generality: his propositions concern all triangles,
circles, etc. The Babylonians/Egyptians observed such results in examples, but
we have little indication that they believed these could be discerned by pure
reason. Thales’ reasoning was almost certainly visual. As an example of typical
geometric reasoning of the period, by 425 BC Socrates could describe how to
halve/double the area of a square by joining the midpoints of edges.
Pythagoras of Samos (c. 572–497 BC) Like Thales, Pythagoras travelled widely, eventually settling
in Croton (southeast Italy) where he founded a school lasting 100 years after his death. It is believed
that Plato learned much of his mathematics from a Pythagorean named Archytas.
The Pythagoreans practiced a mini-religion with ideas out of the mainstream of Greek society.
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One
of their mottos, “All is number,” emphasised their belief in the centrality of pattern and proportion.
The following quote
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gives some flavor of the Pythagorean way of life.
After a testing period and after rigorous selection, the initiates of this order were allowed
to hear the voice of the Master [Pythagoras] behind a curtain; but only after some years,
when their souls had been further purified by music and by living in purity in accordance
with the regulations, were they allowed to see him. This purification and the initiation
into the mysteries of harmony and of numbers would enable the soul to approach [be-
come] the Divine and thus escape the circular chain of re-births.
The Pythagoreans were particularly interested in musical harmony and the relationship of such to
number. For instance, they related intervals in music to the ratios of lengths of vibrating strings:
• Identical strings whose lengths are in the ratio 2:1 vibrate an octave apart.
• A perfect fifth corresponds to the ratio 3:2.
• A perfect fourth corresponds to the ratio 4:3.
The use of these intervals to tune musical instruments is still known as Pythagorean tuning.
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For instance, most of our knowledge of Socrates comes from the voluminous writings of Plato and Aristotle. The
earliest known Greek textbook/compilation (Elements of Geometry) was written around 430 BC by Hippocrates of Chios; no
copy survives, though most of its material probably made it into Book I of Euclid.
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They were vegetarians, believed in the transmigration of souls, and accepted women as students; controversial indeed!
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Van der Waerden, Science Awakening pp 92–93
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