Pythagoras (c. 580–500 BC)

This states that the area of a square drawn on the longest side of a right-angled triangle (the hypotenuse), will be equal to the sum of the areas of the squares drawn on the other two sides. The theorem is likely to have been known long before the time of Pythagoras. It was probably used by the ancient Egyptians to lay out the pyramids.

Greek mathematician and philosopher who formulated Pythagoras' theorem.

Much of Pythagoras' work concerned numbers, to which he assigned mystical properties. For example, he classified numbers into triangular ones (1, 3, 6, 10, ...), which can be represented as a triangular array, and square ones (1, 4, 9, 16, ...), which form squares. He also observed that any two adjacent triangular numbers added together form a square number (for example, 1 + 3 = 4; 3 + 6 = 9; 6 + 10 = 16).

Pythagoras was born on the island of Samos and may have been obliged to flee the despotism of its ruler. He went on to found a school and religious brotherhood in Croton, southern Italy. Its tenets included the immortality and transmigration of souls. As a mathematical and philosophical community the Pythagorean brotherhood extended science; politically its influence reached the western Greek colonies. This caused conflict which saw Pythagoras exiled to Metapontum, on the Gulf of Taranto, until he died. The school continued for 50 or 60 years before being totally suppressed.

Using geometrical principles, the Pythagoreans were able to prove that the sum of the angles of any regular-sided triangle is equal to that of two right angles (using the theory of parallels), and to solve any algebraic quadratic equations having real roots. They formulated the theory of proportion (ratio), which enhanced their knowledge of fractions, and used it in their study of harmonics upon their stringed instruments: the harmonic of the octave was made by touching the string at 1/2 its length, of a fifth at 2/3 its length, and so on. Pythagoras himself is said to have made this the basis of a complete system of musical scales and chords.