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{{Short description|5th-century BC Greek mathematician}}
'''Bryson of Heraclea''' ({{langx|el|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 ''[[Thirteenth Letter (Plato)|13th Platonic Epistle]]'',<ref>Platonic Epistles, xiii. 360c</ref> and [[Theopompus]] even claimed in his ''Attack upon Plato'' that [[Plato]] stole many ideas for his dialogues from Bryson of Heraclea.<ref>Athenaeus, xi. ch. 118, 508c-d</ref> He is known principally from [[Aristotle]], who criticizes his method of squaring the circle.<ref>Aristotle, ''Posterior Analytics'', 75b4; ''Sophistical Refutations'', 171b16, 172a3</ref> He also upset Aristotle by asserting that [[obscene language]] does not exist.<ref>Aristotle, ''Rhetoric'', 3.2, 1405b6-16</ref> [[Diogenes Laërtius]]<ref>Diogenes Laërtius, i. 16, vi. 85, ix. 61</ref> and the [[Suda]]<ref>Suda, ''Pyrrhon'', ''Krates'', ''Theodoros''</ref> 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.<ref>Robert Drew Hicks, ''Diogenes Laertius: Lives of Eminent Philosophers'', page 88. Loeb Classical Library</ref>

===Pi and squaring the circle===
Bryson, along with his contemporary, [[Antiphon (person)|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."<ref>Blatner, page 16</ref> Bryson later followed the same procedure for polygons [[circumscribe|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,<ref>Aristotle, ''Posterior Analytics'', 75b37-76a3.</ref> but [[Archimedes]] would later use a [[method of exhaustion|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 [[Sophism|sophistical]] [[Prior Analytics#The syllogism|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."<ref>Robert Kilwardby, ''De ortu scientiarum'', LIII, §512, pp. 272f.</ref> His account of the syllogism is as follows:

{{cquote|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.<ref>Robert Kilwardby, ''De ortu scientiarum'', LIII, §512, pp. 273.</ref>}}

==Notes==
{{reflist}}

==References==
*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.
*{{cite book
| first=Thomas
| last=Heath
| authorlink=Thomas Little Heath
| title=A History of Greek Mathematics, Volume I: From Thales to Euclid
| publisher=Dover Publications, Inc
| year=1981
| isbn=0-486-24073-8
}}

== External links ==
*[http://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html The History of Pi]
*{{MacTutor Biography|id = Bryson}}

{{Ancient Greek mathematics}}

{{Authority control}}
{{DEFAULTSORT:Bryson of Heraclea}}
[[Category:Ancient Greek geometers]]
[[Category:4th-century BC Greek people]]
[[Category:5th-century BC births]]
[[Category:4th-century BC deaths]]
[[Category:Sophists]]
[[Category:5th-century BC Greek mathematicians]]

Bryson of Heraclea - Revision history

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