Bryson of Heraclea (ca. 450 BCE - ca. 390 BCE) was an ancient Greek mathematician and sophist who contributed to solving the problem of squaring the circle and calculating pi. Sophism was originally a term for the techniques taught by a highly respected group of philosophy and rhetoric teachers in ancient Greece. ... Squaring the circle: the areas of this square and this circle are equal. ... When a circles diameter is 1, its circumference is π. Pi or π is the ratio of a circles circumference to its diameter in Euclidean geometry. ...

Life and work

Although little is known about the life of Bryson, it is believed that he was probably a pupil of Socrates. He is said to have upset Aristotle by asserting that obscene language does not exist. This page is about the ancient Greek philosopher. ... Profanity is a word choice or usage which many consider to be offensive. ...

Pi and circling the square

Bryson, along with his companion Antiphon, was the first to inscribe a polygon inside of a circle, find the polygon's area, double the number of sides of the polygon, and repeat the process several more times, resulting in a lower bound approximation of the area of a circle. "Sooner or later (they figured), ...[there would be] so many sides that the polygon ...[would] be a circle" (Blatner, 16). Bryson later followed the same procedure for polygons circumscribing a circle, resulting in an upper bound approximation of the area of a circle. With these calculations they were able to approximate pi and further place lower and upper bounds on pi's true value. But due to the complexity of their work they only calculated pi to a few digits. They further applied this method to the problem of squaring the circle. Aristotle criticized this method, but Archimedes would later use a method similar to that of Bryson and Antiphon to calculate pi, only Archimedes calculated the perimeter of a polygon instead of the area. To meet Wikipedias quality standards, this article or section may require cleanup. ... In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... The following is a proof for the area of a circle and an ellipse using calculus : An ellipse The area of a circle A circle is a form of ellipse, denoted by the equation . Solving the equation of the ellipse for , the following is derived: Because the ellipse is symmetric... A black circle circumscribed by a red square In geometry, a circumscribed planar shape or solid is one that encloses and fits snugly around another geometric shape or solid. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... Squaring the circle: the areas of this square and this circle are equal. ... Aristotle (Greek: AristotélÄ“s) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Archimedes of Syracuse (Greek: c. ...

References

• Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997.
• Philosophy Dictionary definition of Bryson of Heraclea. The Oxford Dictionary of Philosophy. Copyright © 1994, 1996, 2005 by Oxford University Press.

External links

• The history of pi
• The mac-tutor biography of Bryson