PYTHAGORAS (569 BC-475 BC)

Pythagoras was a Greek mathematician born in

569 BC in Samos, Ionia.

His father was Mnesarchus, a merchant from

Tyre and his mother Pythais a native of Samos.

He is often described as the first pure

mathematician who has contributed immensely

towards the development of mathematics.

Little is known of Pythagoras’s childhood.

Pythagoras, spent his early years in Samos but

travelled widely with his father.

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.

Thales contributed to Pythagoras’s interest in

mathematics and astronomy, and advised him

to travel to Egypt to learn more of these

subjects

In 525 BC when Cambyses invaded Egypt,

Pythagoras was taken to Babylon as prisoner.

There he became perfect in arithmetic and

music and other mathematical sciences taught

by Babylonians

In 520 BC Pythagoras left Babylon and

returned to Samos . He founded a school

called ‘Semicircle’.

Pythagoras founded a philosophical and

religious school in Croton 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, obeyed strict rules, and were taught by

Pythagoras himself. Bothe 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, coming to the society during day only.

Of Pythagoras’s actual work nothing is known.

Certainly his school made outstanding

contributions to mathematics, and it is possible

to be fairly certain about some of Pythagoras’s

mathematical contributions.

For Pythagoras and the mathematikoi 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.

•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 nibblers 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 e extended to other instruments. In fact

Pythagoras made remarkable contributions to the

mathematical theory of music.

Contributions

•Pythagoras studied properties of numbers which would

be familiar to mathematicians today, such as even and

odd numbers, triangular numbers, perfect numbers etc

•He discovered that any odd number (say 2n+1) can be

expressed as the difference of two squares: 2n+1 =

(n+1)2 –n2.

•The list of theorems attributed to Pythagoras, or rather

more generally to the Pythagoreans are:

•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 2n-4 right angles and sum of exterior

angles equal to four right angles.

•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.

•Constructing figures of a given area and geometrical

algebra:-

For example they solved equations such as a(a-x) = x2 by

geometrical means

•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.

•He constructed a polygon equivalent to one given

polygon and similar to another and could construct the

five regular polyhedron

•Many mathematical terms like parabola, lying side by

side, ellipse etc. Can be attributed to Pythagoras.

•Pythagoras studied the properties of areas and

volumes and he was the first to prove that the circle

contains a greater area than any plane figure with the

same perimeter while the sphere contains a greater

volume than any other shape bounded by the same

surface

•In solid geometry, Pythagoras called sphere the most

perfect of all solids. He knew that there were five

regular solids which lie exactly in a sphere namely

tetrahedron, hexahedron, octahedron, dodecahedron

and icosahedrons

•Pythagoras studied the properties of areas and volumes

and he was the first to prove that the circle contains a

greater area than any plane figure with the same

perimeter while the sphere contains a greater volume

than any other shape bounded by the same surface

•In solid geometry, Pythagoras called sphere the most

perfect of all solids. He knew that there were five regular

solids which lie exactly in a sphere namely tetrahedron,

hexahedron, octahedron, dodecahedron and

icosahedrons

•Many mathematical terms like parabola, lying side by

side, ellipse etc. Can be attributed to Pythagoras.