Pythagoras, the Pythagorean Theorem and

¶ … Pythagoras, the Pythagorean theorem and its relationship to the area of a circle. Biography of Pythagoras: Pythagoras was a Greek sage of the 6th century B.C. He was born on the Greek island of Samos, off the coast of Asia Minor. Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander, according Iamblichus, the Syrian historian. He traveled to Egypt, around 535 B.C., to continue his studies, but was captured by Cambyses II of Persia, in 535 B.C., and was taken to Babylon ("Pythagorean," 2007).

Eventually, Pythagoras emigrated to the Greek colonial city-state of Croton, in Southern Italy (Mourelatos, 2007; "Pythagoras," 2007). Pythagoras was a "teacher and leader of extraordinary charisma. Pythagoras founded in Croton a society or brotherhood of religious-ethical orientation. The society fostered strong bonds of friendship and a sense of elitism among its initiates through ritual, esoteric symbolism and a code of rigorous self-control, including lists of taboos" (Mourelatos, 2007). This was known as Pythagoreanism. Pythagoreanism became politically influential in Pythagoras' home town of Croton, and eventually spread to other cities in the region ("Pythagoras," 2007).

Pythagoras' teachings were basically ethical, mystical, and religious. He believed in the transmigration of souls from one body to another, known as metempsychosis, either human or animal. It's unclear whether Pythagoras believed that this led to the immortality of the soul; however, it did lay the foundations for some of the practices of the Pythagorean society he founded. These included vegetarianism and the rituals of purification, in an effort to promote the chances of superior reincarnation (Mourelatos, 2007). A legend grew around Pythagoras, according to Mourelatos (2007), involving superhuman abilities and feats.

However, he believes that this legend was based on the historical reality that Pythagoras was a Greek shaman. Some modern scholars theorize that the religious movement of Orphism, as well as Indian and Persian religious beliefs, influenced Pythagoras. Although Pythagoras' contemporaries honored him as a polymath, modern scholars question this. Today, many "discount the tradition that he was the founder of Greek mathematics, or even that he proved the geometric theorem named for him" (Mourelatos, 2007). Pythagoras died in Metapontum, near modern-day Metaponto, in approximately 500 B.C. ("Pythagorean," 2007).

History of the Pythagorean Theorem: The Pythagorean theorem holds that "the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides" (Meserve, 2007). During Pythagoras' lifetime, the square of a number was represented by the area of a square with the side of a length of that number.

With this representation, the Pythagorean theorem can then be stated as "the area of the square on the hypotenuse of a right triangle is equal to the sum of the reas of the squares on the legs" (Meserve). It was then notated that if triangle ABC is a right triangle, with a right angle at C, then c2 = a2 + b2. Earlier, the converse of this theorem appears to have been used. This became proposition number 47 from Book I of Euclid's Elements ("Pythagorean," 2007).

Although this theorem is traditionally associated with Pythagoras, it is actually much older. More than a millennium before the birth of Pythagoras, four Babylonian tablets were created demonstrating some knowledge of this theorem, circa 1900-1600 B.C. At the very least, these works represent the knowledge of at least special integers known as Pythagorean triples that satisfy it. In addition, the Rhind papyrus, created around 1650 B.C., shows that Egyptians had knowledge of the theorem as well.

However, the first proof of this theorem is still credited to Pythagoras, despite the fact that some scholars believe it was independently discovered in several different cultures ("Pythagorean," 2007). In Euclid's Book I of the Elements, the work ends with the famous 'windmill' proof of the Pythagorean theorem. In Book VI of the Elements, Euclid later gives an even easier demonstration of the theorem, "using the proposition that the areas of similar triangles are proportionate to the squares of their corresponding sides.

Apparently, Euclid invented the windmill proof so that he could place the Pythagorean theorem as the capstone to Book I" ("Pythagorean," 2007). The Pythagorean Theorem's Relation to the Area of Circles: The Pythagorean theorem can be.