# Biography:Bryson of Heraclea

Short description: Greek mathematician, 5th–4th century BC

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.

## Life and work

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]

### Pi and squaring the circle

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.

### Robert Kilwardby on Bryson's syllogism

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:

## Notes

1. Platonic Epistles, xiii. 360c
2. Athenaeus, xi. ch. 118, 508c-d
3. Aristotle, Posterior Analytics, 75b4; Sophistical Refutations, 171b16, 172a3
4. Aristotle, Rhetoric, 3.2, 1405b6-16
5. Diogenes Laërtius, i. 16, vi. 85, ix. 61
6. Suda, Pyrrhon, Krates, Theodoros
7. Robert Drew Hicks, Diogenes Laertius: Lives of Eminent Philosophers, page 88. Loeb Classical Library
8. Blatner, page 16
9. Aristotle, Posterior Analytics, 75b37-76a3.
10. Robert Kilwardby, De ortu scientiarum, LIII, §512, pp. 272f.
11. Robert Kilwardby, De ortu scientiarum, LIII, §512, pp. 273.

## References

• Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997.
• 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.