Notes on Pythagoras (
For the test: review the following facts/legends about Pythagoras.
(Comments in parenthesis are for your amusement, or for reference, but not for the test).
One difficulty in assessing the contribution of Pythagoras to mathematics or
astronomy is separating what he did from what his students and followers did.
Another difficulty is that Pythogoras did not leave any writings, and histories of
his life were written much later. Thus much of what is presented here are legends and
not provable fact. (To add to the confusion there were forgeries claiming to be written
(His name written in Greek is
Pythagoras lived, only upper-case letters were used.
So he would have written
his name more like
Pythagoras grew up on Samos, an island off the coast of Ionia during the 6th century BC.
His father was involved in trade, and took Pythagoras as a boy on trading trips
to far away places such as Italy. His father also provided him with excellent teachers
so that he could get the best possible education.
When he was still young, he is said to have met several wise men
including Thales of Miletus who was old and famous at the time.
Thales is said to have advised him to travel to Egypt, since that was how
Thales himself got his reputation for wisdom. (In Miletus he also studied
astronomy and geometry under the famous Anaximander.
Anaximander had been himself a student of Thales.)
(Pythagoras was also interested in religious mysteries, and wanted to be initiated in all
rites of Greece, and other lands as well. Much of his learning is said to have come
from a Priestess to Apollo at Delphi).
Pythagoras took the advice of Thales to heart and traveled to Egypt.
He left as a young man, and spent a long time in Egypt (perhaps decades)
learning the mathematics and ancient religious mysteries of the Egyptians.
Later he was in Phoenicia (Syria) and Babylon where he also
learned much about mathematics and religion. (Babylon was ruled by the Persians
at the time, and Pythagoras was especially interested in the teachings of
the Magi, the name for Persian priests in the Zoroastrian religion.
Some sources also mention that he learned from the Hebrews when he
was in Phoenicia.)
When he returned from his travels (around age 40) he set up a school at Samos.
He also set up a cave where he lived, and did his philosophical work.
It wasn't long, however, before he decided to leave again.
(Several sources say he wanted to leave due to the violence and oppression of
the tyrant Polycrates, ruler of Samos.)
Pythagoras abandoned Samos and moved to Croton (
in southern Italy. (At that time, southern Italy was settled by Greeks.
Here is a map
of the region at the time.)
Pythagoras was a gifted speaker, and was charismatic and attractive.
His message appealed to the leading citizens of Croton who sent their
sons and daughters to hear Pythagoras speak. So soon he had gathered a large
following (not just in Croton, but among the leading citizens in other Greek cities
in southern Italy. In fact, non Greek people living in Italy came to hear him,
including some Romans).
Part of his message was the importance of just laws. Since many of his followers
were influential, he was able to inspire several cities in southern Italy to adopt better laws,
and these cities were well-governed, at least partially, by his followers.
This state of affairs lasted quite a while. Pythagoras lived to a ripe old age.
At some point a large number of non-Pythagoreans became angry with the
Pythagoreans. (We do not know the reason why, but it could have had something
to do with the political situation in the region, or might have been part of a larger
war or conflict.)
Because of this many Pythagoreans were put to death in an anti-Pythagorean uprising.
Pythagoras himself died at this time (there are several contradictory accounts
of exactly how).
In mathematics he is famous for his view that number is the source
of everything. He viewed the universe as in some sense being made up of numbers.
(Some give him credit for the view that the world is a sphere. However, others give
that credit to Parmenides, who was likely influenced by Pythagoras, and lived a bit later.)
Pythagoras was actually more of a religious leader than a mathematician or astronomer
However he felt that mathematics, and the search or love of wisdom (which he called philosophy)
played an important role in spiritual growth and personal enlightenment.
His teaching inspired generations of Greek mathematicians and philosophers to work
not just for material gain or applications, but for the love of knowledge itself.
(His religous teaching seems to center around Apollo and the Muses.
His followers thought of Pythagoras a practically a god. In fact, some said
that his real father was actually Apollo. He is credited with many miracles and
predicitions. For example, he was once in two places at the same time:
talking to friends in Italy, and at the same time talking to friends in Sicily.)
Some of his teachings were secret. Pythagoreans did not think that everyone
needed to be told everything. Consequently we do not know all of his teachings.
Unlike many schools and societies in ancient Greece, women played an important
role in the Pythagorean society. Some Pythagorean women became famous.
There is evidence that his followers shared their possessions.
Pythagoras taught that the soul was immortal, and that everything that happens
will happen again someday, and has happened in the past. He believed in
reincarnation. (In fact, he claimed to have remembered some of his past lives.)
He taught to avoid excess, and to do things in just proportion. This is related to
his teachings about harmony (see below). (So of course, he believed in not overeating,
not having too much sex, not getting drunk. He also told his followers to honor their
parents. He told his followers not to pray for things for themselves,
a person does not know what is good for them. Do not punish someone when
you are angry.)
(Pythagoras advocated a vegetarian diet. He thought that we are related to animals,
and to the gods, so that we should avoid killing them. He taught that animals
have souls. Also, he felt that meat was hard on the digestion, and made people overweight.
He also taught against eating beans since they were hard on the digestion, caused flatulence
and unrestful sleep. There might have been more reasons for not eating beans: it might
have had something to do with his religious teachings.)
(Some of his teachings sound funny. For example, do not poke a fire
with a sword. Do not travel on a public road.
However, these were probably meant symbolically not literally. For example,
don't poke a fire with a sword means do not provoke an angry person.
Do not travel on a public road means to avoid following the opinions of the crowd.)
was an important symbol in Pythagoreanism.
Britannica for a picture.)
It consists of the number ten arranged as a triangular number in four rows.
To us, looks like a bowling pin configuration.
Contains some of the important ratios in music: 1:2 (octave), 2:3 (fifth), and 3:4 (fourth).
Also illustrates the sequence of dimensions: point, line, plane, and space.
Proportion and Harmony
Pythagoras noticed that harmonious notes in music are the result of
simple proportions in string length.
(His followers studied by playing with a monochord).
This idea of harmony and proportion
carried over in the teaching of the music of the cosmos (or spheres).
The cosmos is ordered according to number and proportion: so it has a harmony of its own.
Pythagoras was a skilled musician, and believed in using music to affect the mood.
He is even said to have been able to heal people with his music.
In general, even outside of music and mathematics,
Pythagoras advocated that everything should be done in harmony and proportion,
and not to excess. (An illustration: the best time of the year is when the contrasting qualities,
light and darkness, hot and cold, dry and moist, are equally balanced.)
He invented the words philosophy (
and/or philosopher (
The word philosophy is the combination of
φιλέω) meaning “to love”
σοφία) meaning “wisdom, learning”
Before Pythagoras, the term cosmos
meant “order; decoration”.
With Pythagoras it acquired the meaning “the (beautiful) order of the universe”.
(In English cosmos means “the universe as a well-ordered whole”,
or a famous PBS
by Carl Sagen).
Before Pythagoras mathematics (
meant “things that are learned”.
(Coming from the Greek verb
μανθάνω meaning “to learn”).
Among the Pythagoreans it meant specificially arithmetic, geometery, music (theory), and
Among the Pythagoreans there were two groups of followers:
(i) the listeners (
ἀκουσματικοί ), and
(ii) the mathematicians (
The listeners were the casual students who just wanted to get direction
on how to live their lives without detailed explanations.
The mathematicians were the advanced students who wanted to
learn and study deeply.
Pentagon, Pentagram, and Dodecahedron
The Pythagoreans used the pentagram,
as a password, (and called it
ὑγίεια meaning “health”.)
The pentagon with pentagram contains the golden ration, incommensurable segments,
and other nice properties that Pythagoras, or the later Pythagoreans noticed.
The Pythagoreans knew about the “sphere of the twelve pentagons”.
This was later called the dodecahedron.
Know what commensurable and incommensurable mean.
The Pythagoreans discovered that there exists incommensurable lengths.
In modern terms: the ratio of the distances give irrational numbers.
So the Pythagoreans essentially discovered irrational numbers.
Many sources guess that the diagonal of the square (compared to the
side of the same square) was the first example.
However, we know that the Pythagoreans were interested
in the pentagon (and the related solid: the dodecahedron)
and the inscribed
After all, the pentagram was the sign of health among
the followers of Pythagoras.
So some scholars now believe that the first known incommensurable segments were
the length of a side of the star compared with the length of a side of the pentagon.
(Also, we know that Euclid, who drew from the Pythagoreans, used the golden section to
construct pentagons, so it is quiet likely that the Pythagoreans
discovered the golden section in the pentagram.)
Go over the class notes on the
Pythagoras sacrificed an ox
for his theorem about the right triangle, but one source says it was
actually made of flour.
(There are other topics: Pythagorean theorem, perfect numbers, Pythagorean triples,
triangular numbers, gnomens, perfect numbers, other properties of integers,
golden section, et cetera.)