## Pythagoras of Samos (570BC - 475BC)

Pythagoras was born in Greece in Samos. Little is known of his childhood, but he may have travelled to Egypt, Babylon and perhaps even India before he went to Croton in Southern Italy around 520 BC.

Here he established a school and sect welcoming women as well as men. The sect was secretive and it is likely that knowledge was held in common rather than seen as the exclusive prize of one man, we should therefore talk of ‘Pythagoreans’ rather than just one man Pythagoras.

The sect was certainly the first reported one to think of mathematics as an intellectual pursuit rather than a practical one to support agriculture and finance. The group believed passionately that all relations could be reduced to number relations. Their life was intimately connected to numbers, with some, such as the number 10, ascribed mystical properties. Ten, of course, is made up from the first four whole numbers, 10 = 1 + 2 + 3 + 4.

They held a view of the world which fused mathematics, music and astronomy. Pythagoras was a keen lyre player and noticed that two strings plucked simultaneously produced harmonies when the ratios of the length of strings are whole numbers - this is why the harmonic sequence is so-named. They saw a great mechanical harmony in the universe, believing the planets and sun to revolve around the earth producing a harmony known as the music of the spheres.

This idea - of the earth being the centre of the universe - was to hold until Copernicus first realised that the earth rotated about the sun; followed by Kepler and Newton who converted this observation into powerful scientific theory.

## Pythagoras’ mathematics

The Pythagoreans were entranced by the properties of certain numbers, especially properties that appeared to be connected with simplicity or with geometric arrangements. For example, the triangular numbers, made up of the sum of successive whole numbers:

 1 1+2=3 1+2+3=6 1+2+3+4=10 1+2+3+4+5=15

This leads, in turn, to the Tetrahedral numbers - successive sums of Triangular numbers:

Each number had its own unique personality - masculine or feminine, perfect or incomplete, beautiful or ugly depending on its geometric configuration.

Of course today we particularly remember the Pythagoreans for the famous geometry theorem now known as Pythagoras’s theorem, This was probably known to the Babylonians 1000 years earlier but the Pythagoreans may have been the first to prove it. The theorem concerns the sides of a right-angled triangle:

 c^2 = a^2 + b^2 The side opposite the right angle is called the hypotenuse.

The simplest such triangle is the so-called ‘3,4,5 triangle’ because,

5^2 = 3^2 + 4^2

another is, the ‘5,12,13 triangle’. In fact, any triangle of the form,

c = k(m^2 + n^2), a = 2kmn and b = k(m^2 - n^2)

is a right-angled triangle. Here, k can be any whole number greater than 1, and m, n are whole numbers, with m>n. For example:

• the ‘3,4,5 triangle’ has k=1, m=2 and n=1;
• the ‘5,12,13 triangle’ has k=1, m=3 and n=2;

Pythagoras’s Theorem also has a converse - which we used above - if the sum of the squares of the two shorter sides is the square of the hypotenuse, then the triangle is right-angled.

This theorem has many different proofs - here are two.

First proof - it has three parts:

We start with two squares whose sides are of length a and b, placed side by side. The total area of the two squares is just a^2+b^2.
First we draw two right angled triangles each with sides a and b and hypotenuse c. (The segment common to the two squares has been removed.) Now we have two triangles and a strange looking shape.
For the last step, we rotate the triangles through 90° about their top vertices. The triangle on the right one is rotated clockwise; the one on the left is rotated anti-clockwise.The final result is a square of side c and area c^2.

Second proof - this is a proof without words:

The Pythagoreans also showed that the sum of the angles of a triangle is equal to two right angles, and in spite of it being contrary to his theory of number relations, he may have shown the existence of irrational. It is unclear when Pythagoras died, but his sect thrived after his death and spread to a number of other Italian cities, while the work ascribed to him with its revolutionary focus on numbers and their relations has influenced human thinking ever since.