Pythagoras was born
on the island of Samos in 568 BC to a Phoenician merchant from Tyre
called Mnesarchus^{12,13}. His mother, Pythais^{8} was
a native of Samos. He is often described as the first pure mathematician.
He is an extremely important figure in the development of mathematics
yet we know relatively little about his mathematical achievements. Unlike
many later mathematicians, where at least we have some of the books
which they wrote, we have nothing of Pythagoras's writings. The society
which he led, half religious and half scientific, followed a code of
secrecy which certainly means that today Pythagoras is a mysterious
figure.
We do have details
of Pythagoras's life from early biographies which use important original
sources yet are written by authors who attribute divine powers to him,
and whose aim was to present him as a god-like figure. What we present
below is an attempt to collect together the most reliable sources to
reconstruct an account of Pythagoras's life. There is fairly good agreement
on the main events of his life but most of the dates are disputed with
different scholars giving dates which differ by 20 years. Some historians
treat all this information as merely legends but, even if the reader
treats it in this way, being such an early record it is of historical
importance.

**Pythagoras's
father, Mnesarchus was a merchant who came from Tyre,** and there
is a story^{12,13} that he brought corn to Samos at a time of
famine and was granted citizenship of Samos as a mark of gratitude.
As a child Pythagoras spent his early years in Samos but travelled widely
with his father. There are accounts of Mnesarchus returning to Tyre
with Pythagoras and that he was taught there by the Chaldaeans and the **learned men of Phoenicia and was initiated into the 'Ancient Mysteries'
of the Phoenicians c. 548 B.C. and studied for about 3 years in the
temples of Tyre, Sidon, and Byblos**. It seems that he also visited
Italy with his father.

Little is known
of Pythagoras's childhood. All accounts of his physical appearance are
likely to be fictitious except the description of a striking birthmark
which Pythagoras had on his thigh. It is probable that he had two brothers
although some sources say that he had three. Certainly he was well educated,
learning to play the lyre, learning poetry and to recite Homer.
There were, among his teachers, three philosophers who were to influence
Pythagoras while he was a young man. One of the most important was Pherekydes
who many describe as the teacher of Pythagoras.

The other two philosophers
who were to influence Pythagoras, and to introduce him to mathematical
ideas, were Thales and his pupil Anaximander
who both lived on Miletus. It is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old^{8}.
By this time Thales was an old man and, although
he created a strong impression on Pythagoras, he probably did not teach
him a great deal. However he did contribute to Pythagoras's interest
in mathematics and astronomy, and advised him to travel to Egypt to
learn more of these subjects. Thales's pupil,
Anaximander, lectured on Miletus and Pythagoras attended these lectures.
Anaximander certainly was interested in geometry and cosmology
and many of his ideas would influence Pythagoras's own views.

In about 535 BC
Pythagoras went to Egypt. This happened a few years after the tyrant
Polycrates seized control of the city of Samos. There is some evidence
to suggest that Pythagoras and Polycrates were friendly at first and
it is claimed^{5} that Pythagoras went to Egypt with a letter
of introduction written by Polycrates. In fact Polycrates had an alliance
with Egypt and there were therefore strong links between Samos and Egypt
at this time. The accounts of Pythagoras's time in Egypt suggest that
he visited many of the temples and took part in many discussions with
the priests. According to Porphyry^{12,13}. Pythagoras was refused admission
to all the temples except the one at Diospolis where he was accepted
into the priesthood after completing the rites necessary for admission.

It is not difficult
to relate many of Pythagoras's beliefs, ones he would later impose on
the society that he set up in Italy, to the customs that he came across
in Egypt. For example the secrecy of the Egyptian priests, their refusal
to eat beans, their refusal to wear even cloths made from animal skins,
and their striving for purity were all customs that Pythagoras would
later adopt. Porphyry says that Pythagoras learnt geometry from the
Egyptians but it is likely that he was already acquainted with geometry,^{12,13}
certainly after teachings from Thales and
Anaximander.

In 525 BC Cambyses
II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance
with Egypt and sent 40 ships to join the Persian fleet against the Egyptians.
After Cambyses had won the Battle of Pelusium in the Nile Delta and
had captured Heliopolis and Memphis, Egyptian resistance collapsed.
Pythagoras was taken prisoner and taken to Babylon. Iamblichus
writes that Pythagoras^{8}:

*... was transported
by the followers of Cambyses as a prisoner of war. Whilst he was there
he gladly associated with the Magoi *...* and was instructed
in their sacred rites and learnt about a very mystical worship of
the gods. He also reached the acme of perfection in arithmetic and
music and the other mathematical sciences taught by the Babylonians...*

In about 520 BC
Pythagoras left Babylon and returned to Samos. Polycrates had been killed
in about 522 BC and Cambyses died in the summer of 522 BC, either by
committing suicide or as the result of an accident. The deaths of these
rulers may have been a factor in Pythagoras's return to Samos but it
is nowhere explained how Pythagoras obtained his freedom. Darius of
Persia had taken control of Samos after Polycrates' death and he would
have controlled the island on Pythagoras's return. This conflicts with
the accounts of Porphyry and Diogenes Laertius who state that Polycrates
was still in control of Samos when Pythagoras returned there.

Pythagoras made
a journey to Crete shortly after his return to Samos to study the system
of laws there. Back in Samos he founded a school which was called the
semicircle. Iamblichus^{8} writes in the third century AD that:

*... he formed
a school in the city *[*of Samos*]*, the 'semicircle' of
Pythagoras, which is known by that name even today, in which the Samians
hold political meetings. They do this because they think one should
discuss questions about goodness, justice and expediency in this place
which was founded by the man who made all these subjects his business.
Outside the city he made a cave the private site of his own philosophical
teaching, spending most of the night and daytime there and doing research
into the uses of mathematics...*

Pythagoras left
Samos and went to southern Italy in about 518 BC (some say much earlier).
Iamblichus^{8} gives some reasons for him leaving. First he
comments on the Samian response to his teaching methods:

*... he tried
to use his symbolic method of teaching which was similar in all respects
to the lessons he had learnt in Egypt. The Samians were not very keen
on this method and treated him in a rude and improper manner.*

This was, according
to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:

*... Pythagoras
was dragged into all sorts of diplomatic missions by his fellow citizens
and forced to participate in public affairs. *...* He knew that
all the philosophers before him had ended their days on foreign soil
so he decided to escape all political responsibility, alleging as
his excuse, according to some sources, the contempt the Samians had
for his teaching method.*

Pythagoras founded
a philosophical and religious school in Croton (now Crotone, on the
east of the heel of southern Italy) that had many followers. Pythagoras
was the head of the society with an inner circle of followers known
as mathematikoi. The mathematikoi lived permanently with the Society,
had no personal possessions and were vegetarians. They were taught by
Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras
held were:^{2}

(1) *that at
its deepest level, reality is mathematical in nature,*

(2) *that philosophy can be used for spiritual purification,*

(3) *that the soul can rise to union with the divine,*

(4) *that certain symbols have a mystical significance, and*

(5) *that all brothers of the order should observe strict loyalty
and secrecy.*

Both men and women
were permitted to become members of the Society, in fact several later
women Pythagoreans became famous philosophers. The outer circle of the
Society were known as the akousmatics and they lived in their own houses,
only coming to the Society during the day. They were allowed their own
possessions and were not required to be vegetarians.

Of Pythagoras's
actual work nothing is known. His school practised secrecy and communalism
making it hard to distinguish between the work of Pythagoras and that
of his followers. Certainly his school made outstanding contributions
to mathematics, and it is possible to be fairly certain about some of
Pythagoras's mathematical contributions. First we should be clear in
what sense Pythagoras and the mathematikoi were studying mathematics.
They were not acting as a mathematics research group does in a modern
university or other institution. There were no 'open problems' for them
to solve, and they were not in any sense interested in trying to formulate
or solve mathematical problems.

Rather Pythagoras
was interested in the principles of mathematics, the concept of number,
the concept of a triangle or other mathematical figure and the abstract
idea of a proof. As Brumbaugh writes:

*It is hard
for us today, familiar as we are with pure mathematical abstraction
and with the mental act of generalisation, to appreciate the originality
of this Pythagorean contribution.*^{3}

In fact today we
have become so mathematically sophisticated that we fail even to recognise
2 as an abstract quantity. There is a remarkable step from 2 ships +
2 ships = 4 ships, to the abstract result 2 + 2 = 4, which applies not
only to ships but to pens, people, houses etc. There is another step
to see that the abstract notion of 2 is itself a thing, in some sense
every bit as real as a ship or a house.

Pythagoras believed
that all relations could be reduced to number relations. As Aristotle
wrote:

*The Pythagorean
*...* having been brought up in the study of mathematics, thought
that things are numbers *...* and that the whole cosmos is a
scale and a number.*

This generalisation
stemmed from Pythagoras's observations in music, mathematics and astronomy.
Pythagoras noticed that vibrating strings produce harmonious tones when
the ratios of the lengths of the strings are whole numbers, and that
these ratios could be extended to other instruments. In fact Pythagoras
made remarkable contributions to the mathematical theory of music. He
was a fine musician, playing the lyre, and he used music as a means
to help those who were ill.

Pythagoras studied
properties of numbers which would be familiar to mathematicians today,
such as even and odd numbers, triangular
numbers, perfect numbers
etc. However to Pythagoras numbers had personalities which we hardly
recognise as mathematics today:^{3}

*Each number
had its own personality - masculine or feminine, perfect or incomplete,
beautiful or ugly. This feeling modern mathematics has deliberately
eliminated, but we still find overtones of it in fiction and poetry.
Ten was the very best number: it contained in itself the first four
integers - one, two, three, and four *[1 + 2 + 3 + 4 = 10]* -
and these written in dot notation formed a perfect triangle.*

Of course today
we particularly remember Pythagoras for his
famous geometry theorem. Although the theorem, now known as Pythagoras's
theorem, was known to the Babylonians 1000 years earlier he may have
been the first to prove it. Proclus, the last major Greek philosopher,
who lived around 450 AD wrote:^{7}

*After *[Thales*,
etc.*]* Pythagoras transformed the study of geometry into a liberal
education, examining the principles of the science from the beginning
and probing the theorems in an immaterial and intellectual manner:
he it was who discovered the theory of irrational
and the construction of the cosmic figures.*

Again Proclus, writing
of geometry, said:

*I emulate the
Pythagoreans who even had a conventional phrase to express what I
mean "a figure and a platform, not a figure and a sixpence", by which
they implied that the geometry which is deserving of study is that
which, at each new theorem, sets up a platform to ascend by, and lifts
the soul on high instead of allowing it to go down among the sensible
objects and so become subservient to the common needs of this mortal
life.*

Heath^{7}
gives a list of theorems attributed to Pythagoras, or rather more generally
to the Pythagoreans.

(i) The sum of the
angles of a triangle is equal to two right angles. Also the Pythagoreans
knew the generalisation which states that a polygon with *n* sides
has sum of interior angles 2*n* - 4 right angles and sum of exterior
angles equal to four right angles.

(ii) The theorem
of Pythagoras - for a right angled triangle the square on the hypotenuse
is equal to the sum of the squares on the other two sides. We should
note here that to Pythagoras the square on the hypotenuse would certainly
not be thought of as a number multiplied by itself, but rather as a
geometrical square constructed on the side. To say that the sum of two
squares is equal to a third square meant that the two squares could
be cut up and reassembled to form a square identical to the third square.

(iii) Constructing
figures of a given area and geometrical algebra. For example they solved
equations such as *a* (*a* - *x*) = *x*^{2}
by geometrical means.

(iv) The discovery
of irrationals. This is certainly attributed to the Pythagoreans but
it does seem unlikely to have been due to Pythagoras himself. This went
against Pythagoras's philosophy the all things are numbers, since by
a number he meant the ratio of two whole numbers. However, because of
his belief that all things are numbers it would be a natural task to
try to prove that the hypotenuse of an isosceles right angled triangle
had a length corresponding to a number.

(v) The five regular
solids. It is thought that Pythagoras himself knew how to construct
the first three but it is unlikely that he would have known how to construct
the other two.

(vi) In astronomy
Pythagoras taught that the Earth was a sphere at the centre of the Universe.
He also recognised that the orbit of the Moon was inclined to the equator
of the Earth and he was one of the first to realise that Venus as an
evening star was the same planet as Venus as a morning star.

Primarily, however,
Pythagoras was a philosopher. In addition to his beliefs about numbers,
geometry and astronomy described above, he held^{2}:

*... the following
philosophical and ethical teachings: ... the dependence of the dynamics
of world structure on the interaction of contraries, or pairs of opposites;
the viewing of the soul as a self-moving number experiencing a form
of metempsychosis, or successive reincarnation in different species
until its eventual purification *(*particularly through the intellectual
life of the ethically rigorous Pythagoreans*)*; and the understanding
...that all existing objects were fundamentally composed of form and
not of material substance. Further Pythagorean doctrine *...*
identified the brain as the locus
of the soul; and prescribed certain secret cultic practices.*

Their practical
ethics are also described:

*In their ethical
practices, the Pythagorean were famous for their mutual friendship,
unselfishness, and honesty.*^{3}

Pythagoras's Society
at Croton was not unaffected by political events despite his desire
to stay out of politics. Pythagoras went to Delos in 513 BC to nurse
his old teacher Pherekydes who was dying. He remained there for a few
months until the death of his friend and teacher and then returned to
Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris
and there is certainly some suggestions that Pythagoras became involved
in the dispute. Then in around 508 BC the Pythagorean Society at Croton
was attacked by Cylon, a noble from Croton itself. Pythagoras escaped
to Metapontium and the most authors say he died there, some claiming
that he committed suicide because of the attack on his Society. Iamblichus
quotes one version of events:

*Cylon, a Crotoniate
and leading citizen by birth, fame and riches, but otherwise a difficult,
violent, disturbing and tyrannically disposed man, eagerly desired
to participate in the Pythagorean way of life. He approached Pythagoras,
then an old man, but was rejected because of the character defects
just described. When this happened Cylon and his friends vowed to
make a strong attack on Pythagoras and his followers. Thus a powerfully
aggressive zeal activated Cylon and his followers to persecute the
Pythagoreans to the very last man. Because of this Pythagoras left
for Metapontium and there is said to have ended his days.*^{8}

This seems accepted
by most but Iamblichus himself does not accept this version and argues
that the attack by Cylon was a minor affair and that Pythagoras returned
to Croton. Certainly the Pythagorean Society thrived for many years
after this and spread from Croton to many other Italian cities. Gorman^{6}
argues that this is a strong reason to believe that Pythagoras returned
to Croton and quotes other evidence such as the widely reported age
of Pythagoras as around 100 at the time of his death and the fact that
many sources say that Pythagoras taught Empedokles to claim that he
must have lived well after 480 BC.

The evidence is
unclear as to when and where the death of Pythagoras occurred. Certainly
the Pythagorean Society expanded rapidly after 500 BC, became political
in nature and also spilt into a number of factions. In 460 BC the Society:^{2}

*... was violently
suppressed. Its meeting houses were everywhere sacked and burned;
mention is made in particular of "the house of Milo" in Croton, where
*50* or *60* Pythagoreans were surprised and slain. Those
who survived took refuge at Thebes and other places.*

Primary
source: *J
J O'Connor*
and *E F Robertson*, School
of Mathematics and Statistics, University of St Andrews, Scotland