**Bryson of Heraclea** (Greek: Βρύσων Ἡρακλεώτης, *gen*.: Βρύσωνος; fl. late 5th-century BCE) was an ancient Greek mathematician and sophist who studied the solving the problems of squaring the circle and calculating pi.

Little is known about the life of Bryson; he came from Heraclea Pontica, and he may have been a pupil of Socrates. He is mentioned in the *13th Platonic Epistle*,^{[1]} and Theopompus even claimed in his *Attack upon Plato* that Plato stole many ideas for his dialogues from Bryson of Heraclea.^{[2]} He is known principally from Aristotle, who criticizes his method of squaring the circle.^{[3]} He also upset Aristotle by asserting that obscene language does not exist.^{[4]} Diogenes Laërtius^{[5]} and the Suda^{[6]} refer several times 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 Bryson of Achaea, who may have lived around that time.^{[7]}

Bryson, along with his contemporary, Antiphon, 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."^{[8]} 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 appears to have made little progress.^{[citation needed]} Aristotle criticized this method,^{[9]} 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.

The 13th-century English philosopher Robert Kilwardby 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."^{[10]} His account of the syllogism is as follows:

Bryson's syllogism on the squaring of the circle was of this sort, it is said:

In any genus in which one can find a greater and a lesser than something, one can find what is equal; but in the genus of squares one can find a greater and a lesser than a circle; therefore, one can also find a square equal to a circle.This syllogism is sophistical not because the consequence is false, and not because it produces a syllogism on the basis of apparently readily believable things-for it concludes necessarily and on the basis of what is readily believable. Instead, it is called sophistical and contentious [litigiosus] because it is based on common considerations and is dialectical when it should be based on specific considerations and be demonstrative.^{[11]}