Pythagoras, the Pythagorean Theorem, and Circle Area

Introduction

This paper examines the life of Pythagoras, the history of the Pythagorean theorem, and the theorem's 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. According to Iamblichus, the Syrian historian, Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander. He traveled to Egypt around 535 B.C. to continue his studies, but was captured by Cambyses II of Persia and 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." He 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 movement became known as Pythagoreanism. It grew politically influential in Croton and eventually spread to other cities in the region ("Pythagoras," 2007).

Pythagoras' teachings were essentially ethical, mystical, and religious. He believed in the transmigration of souls from one body to another — known as metempsychosis — into either human or animal form. It is unclear whether Pythagoras believed this process led to the immortality of the soul; however, it did lay the foundations for several practices of the Pythagorean society he founded. These included vegetarianism and rituals of purification, undertaken in an effort to promote the chances of superior reincarnation (Mourelatos, 2007).

A legend grew around Pythagoras, according to Mourelatos (2007), involving claims of superhuman abilities and feats. Mourelatos believes, however, 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 him.

Although Pythagoras' contemporaries honored him as a polymath, modern scholars are more skeptical. 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 whose side had a length equal to that number. With this representation, the theorem can be stated as: the area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the legs (Meserve, 2007). This is notated such that if triangle ABC is a right triangle with a right angle at C, then c² = a² + b². The converse of this theorem appears to have been in use even earlier. It became Proposition 47 of 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 his birth, four Babylonian tablets were created demonstrating knowledge of the theorem, dating to approximately 1900–1600 B.C. At the very least, these works show familiarity with special integers known as Pythagorean triples that satisfy the theorem. In addition, the Rhind Papyrus, created around 1650 B.C., shows that Egyptians also possessed knowledge of the relationship. Despite this, the first formal proof of the theorem is still credited to Pythagoras, even though some scholars believe it was independently discovered in several different cultures ("Pythagorean," 2007).

In Book I of Euclid's Elements, the work ends with the famous "windmill" proof of the Pythagorean theorem. In Book VI, Euclid later provides an even more straightforward demonstration, "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).

Conclusion

Pythagoras remains one of the most influential figures in the early history of mathematics and philosophy, even though modern scholars debate the extent of his personal contributions. The theorem that bears his name has roots stretching back to ancient Babylon and Egypt, was formally demonstrated by Euclid, and continues to serve as a foundational tool in geometry — including applications such as deriving the area of a circle from an inscribed square.

References

Meserve, B. E. (2007). Pythagoras, theorem of. Grolier Multimedia Encyclopedia. Retrieved December 6, 2007, from Grolier Online.

Mourelatos, A. P. D. (2007). Pythagoras and Pythagoreanism. Encyclopedia Americana. Retrieved December 6, 2007, from Grolier Online.

Pythagoras. (2007). Encyclopedia Britannica. Retrieved December 5, 2007, from Encyclopedia Britannica Online.

Pythagorean theorem. (2007). Encyclopedia Britannica. Retrieved December 5, 2007, from Encyclopedia Britannica Online.