Pythagoras, a native of the island of Samos, was one of the important men that ever lived. He is the true father of demonstrative deductive argument and in a sense; he is a first pure mathematician. He is an extremely important figure in the development of mathematics despite of the fact we have nothing of Pythagoras’s writings. We know very little about Pythagoras as a person. We have some details of Pythagoras’s life from early biographies, which are written by authors who are more interested in portraying Pythagoras a divine figure rather than a truth.

    Pythagoras thrived around 532 B.C. At that time, Samos was a commercial rival of Miletus. There are two theories about his family tree. Firstly, Pythagoras’s father Mnesarchus was from Tyre and his mother Pythais was a native of Samos. The father of Pythagoras’s was a merchant, brought corn to Samos at a time of famine, and as a mark of gratitude, was granted citizenship of Samos as a mark of gratitude. Secondly, he was the son of the god Apollo. As we have said, little is known of Pythagoras’s childhood. All accounts of his physical appearance are likely to be fictitious except the description of a birthmark on his thigh.

    Certainly, Pythagoras was well educated, learning to play the lyre, learning poetry and to recite Homer. Among his teachers, three philosophers (Pherekydes, Thales, and Anaximander) were to influence Pythagoras while he was a young. The most important was Pherekydes who many describe the teacher of Pythagoras. Thales, who lived in Miletus, introduced mathematical ideas to Pythagoras. Thales also introduced him to mathematics and astronomy and advised him to travel to Egypt to learn more of these subjects. Anaximander was interested in geometry and cosmology and many of his ideas would influence Pythagoras’s own views.

    After the tyrant Polycrates seized control of the city of Samos, Pythagoras went to Egypt in about 545 BC with a letter of introduction written by Polycrates. In 525 BC Cambyses II, the king of Persia, invaded Egypt. Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to Babylon. In about 520 BC Pythagoras left Babylon and returned to Samos (nowhere explained how Pythagoras obtained his freedom). Pythagoras disliked the government of Samos and left to Crotone in southern Italy in about 518 B.C. At Croton Pythagoras founded a society of disciples. It is kind of a philosophical and religious school. The society had many followers known as mathematikoi. The mathematikoi lived permanently with the Society. They had no personal possessions and were vegetarians. Pythagoras himself taught them and they obeyed strict rules. In the beginning, the society was quite influential in the city but eventually people of Croton turned against him. Pythagoras, then moved to another city in Southern Italy named, Metapontion, where he died.

    After his dead two opposing tradition disputed his memory. On one hand, he became the founder of a great scientific and philosophical movement that spread all over Greece and on the other hand, he became a mythical figure, credited with miracles and magic powers.

    First, we outline the various religious doctrines that the Pythagoras developed and then look into the mathematical contributions.

Religion

    The main idea of religion that Pythagoras was founded was the transmigration of souls. The secondary purpose, like any other religion, was try to acquire control of the state and established a rule of the saints. The Pythagoreans believed that the soul was a divine and immortal being, imprisoned in the material body. The body contaminated the soul so through the process of purification the soul must be freed. Purification depended largely upon the individual’s conduct and especially upon his observing a code of prescribed rules. If the soul was not purified, it passed, on the death of the body, into another body, possibly of some other species of animal, but if purification could be achieved, the soul was released altogether from its connection with the body and returned to the abode of the gods.

Some of the rules of the Pythagorean religion were [Russell]:

1. To abstain from beans.
2. Not to pick up what has fallen.
3. Not to break bread.
4. Not to step over a crossbar.
5. Not to eat from whole loaf.
6. Not to walk on highways.
7. Do not look in a mirror beside a light.

    It is important to note that the beliefs of Pythagorean religion have much in common with the orphic religion of Greece. I think we can safely say that this religion is a cult or sect of Orphic religion. Pythagorean religion performed elaborate purification rites involving the use of music to induce an emotional state of ecstasy through which worshipper’s attained purification. It seems like to Pythagorean for intellectual activity it is necessary to first pure the soul.

Mathematics

    Pythagoras said that “all things are numbers” and the numerical aspect of things was the main interest to the Pythagoreans. Pythagoras discovered the relationship between music and numbers that survive in the mathematical terms “harmonic mean” and “harmonic progression”. Since at that early data there was no systematic numerical notation, be thought of numbers as shapes i.e., numbers were represented by dots or pebbles in the sand. We still speak of squares and cubes of numbers, which are terms that we owe to Pythagoras.

    The greatest discovery of Pythagoras was the proposition about right-angled triangles, that the sum of squares of the adjoining sides of the right angle is equal to the square of the remaining side (hypotenuse). The Egyptians had known that triangle whose sides are 3, 4, and 5 has a right angle, but the Greeks were the first to observe that 3² + 4² = 5² and acting on this observation discovered a proof of general proposition [Russell].

    Pythagoras is great not only in his wisdom but also in his idiocy. He is no doubt one of the most interesting and puzzling men in history.