Bryson of Heraclea

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. 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]

• Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997.
• Diels, H. and Kranz, W. Die Fragmente der Vorsokratiker, 2 vv. Zürich and Berlin 1964. (DK)
• Kilwardby, Robert. De ortu scientiarum. Auctores Britannici Medii Aevi IV ed. A.G. Judy. Toronto: PIMS, 1976. Published for the British Academy by the Oxford University Press. (The translation of this quote is found in: N. Kretzmann & E. Stump (eds. & trns.), The Cambridge Translations of Medieval Philosophical Texts: Volume 1, Logic and the Philosophy of Language. Cambridge: Cambridge UP, 1989.)
• Philosophy Dictionary definition of Bryson of Heraclea. The Oxford Dictionary of Philosophy. Copyright © 1994, 1996, 2005 by Oxford University Press.
• Heath, Thomas (1981). A History of Greek Mathematics, Volume I: From Thales to Euclid. Dover Publications, Inc. ISBN 0-486-24073-8.

• The History of Pi
• O'Connor, John J.; Robertson, Edmund F., "Bryson of Heraclea", MacTutor History of Mathematics Archive, University of St Andrews