Bryson of Heraclea
Encyclopedia
Bryson of Heraclea was an ancient Greek
mathematician
and sophist who contributed to solving the problem of squaring the circle
and calculating pi
.
, and he may have been a pupil of Socrates
. He is mentioned in the 13th Platonic Epistle, and Theopompus
even claimed in his Attack upon Plato that Plato
stole many ideas for his dialogues from Bryson of Heraclea. He is known principally from Aristotle
who criticizes his method of squaring the circle. He also upset Aristotle by asserting that obscene language does not exist. Diogenes Laërtius
and the Suda
refer several time to a Bryson as a teacher of various philosophers, but since some of the philosophers mentioned lived in the late 4th-century BCE, it is possible that Bryson became confused with a Bryson of Achaea
who may have lived around that time.
, was the first to inscribe
a polygon inside a circle, find the polygon
's area, double the number of sides of the polygon, and repeat the process, 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." 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 Bryson was able to approximate π and further place lower and upper bounds on π's true value. But due to the complexity of the method, he only calculated π to a few digits. Aristotle
criticized this method, but Archimedes
would later use a method
similar to that of Bryson and Antiphon to calculate π; however, Archimedes calculated the perimeter
of a polygon instead of the area.
described Bryson's attempt of proving the quadrature of the circle as a sophistical
syllogism---one which "deceives in virtue of the fact that it promises to yield a conclusion producing knowledge on the basis of specific considerations and concludes on the basis of common considerations that can produce only belief." His account of the syllogism is as follows:
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and sophist who contributed to solving the problem of squaring the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
and calculating pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
.
Life and work
Little is known about the life of Bryson; he came from Heraclea PonticaHeraclea Pontica
Heraclea Pontica , an ancient city on the coast of Bithynia in Asia Minor, at the mouth of the river Lycus. It was founded by the Greek city-state of Megara c.560-558 and was named after Heracles who the Greeks believed entered the underworld at a cave on the adjoining Archerusian promontory .The...
, and he may have been a pupil of Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
. He is mentioned in the 13th Platonic Epistle, and Theopompus
Theopompus
Theopompus was a Greek historian and rhetorician- Biography :Theopompus was born on Chios. In early youth he seems to have spent some time at Athens, along with his father, who had been exiled on account of his Laconian sympathies...
even claimed in his Attack upon Plato that Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
stole many ideas for his dialogues from Bryson of Heraclea. He is known principally from Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
who criticizes his method of squaring the circle. He also upset Aristotle by asserting that obscene language does not exist. Diogenes Laërtius
Diogenes Laertius
Diogenes Laertius was a biographer of the Greek philosophers. Nothing is known about his life, but his surviving Lives and Opinions of Eminent Philosophers is one of the principal surviving sources for the history of Greek philosophy.-Life:Nothing is definitively known about his life...
and the Suda
Suda
The Suda or Souda is a massive 10th century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Suidas. It is an encyclopedic lexicon, written in Greek, with 30,000 entries, many drawing from ancient sources that have since been lost, and often...
refer several time to a Bryson as a teacher of various philosophers, but since some of the philosophers mentioned lived in the late 4th-century BCE, it is possible that Bryson became confused with a Bryson of Achaea
Bryson of Achaea
Bryson of Achaea was an ancient Greek philosopher.Very little information is known about him. He was said to have been a pupil of Stilpo and Clinomachus, which would mean that he was a philosopher of the Megarian school. He was said to have taught Crates the Cynic, Pyrrho the Skeptic, and...
who may have lived around that time.
Pi and squaring the circle
Bryson, along with his contemporary AntiphonAntiphon (person)
Antiphon the Sophist lived in Athens probably in the last two decades of the 5th century BC. There is an ongoing controversy over whether he is one and the same with Antiphon of the Athenian deme Rhamnus in Attica , the earliest of the ten Attic orators...
, was the first to inscribe
Inscribe
right|thumb|An inscribed triangle of a circleIn geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "Figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about...
a polygon inside a circle, find the polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
's area, double the number of sides of the polygon, and repeat the process, resulting in a lower bound approximation of the area of a circle
Area of a circle
The area of a circle is πr2 when the circle has radius r. Here the symbol π denotes, as usual, the constant ratio of the circumference of a circle to its diameter...
. "Sooner or later (they figured), ...[there would be] so many sides that the polygon ...[would] be a circle." Bryson later followed the same procedure for polygons circumscribing a circle, resulting in an upper bound
Upper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
approximation of the area of a circle. With these calculations Bryson was able to approximate π and further place lower and upper bounds on π's true value. But due to the complexity of the method, he only calculated π to a few digits. Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
criticized this method, but Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
would later use a method
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...
similar to that of Bryson and Antiphon to calculate π; however, Archimedes calculated the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
of a polygon instead of the area.
Robert Kilwardby on Bryson's Syllogism
The 13th-century English philosopher Robert KilwardbyRobert Kilwardby
Robert Kilwardby was an Archbishop of Canterbury in England and as well as a cardinal.-Life:Kilwardby studied at the University of Paris, then was a teacher of grammar and logic there. He then joined the Dominican Order and studied theology, and became regent at Oxford University before 1261,...
described Bryson's attempt of proving the quadrature of the circle as a sophistical
Sophism
Sophism in the modern definition is a specious argument used for deceiving someone. In ancient Greece, sophists were a category of teachers who specialized in using the tools of philosophy and rhetoric for the purpose of teaching aretê — excellence, or virtue — predominantly to young statesmen and...
syllogism---one which "deceives in virtue of the fact that it promises to yield a conclusion producing knowledge on the basis of specific considerations and concludes on the basis of common considerations that can produce only belief." His account of the syllogism is as follows: