Bryson of Heraclea (Greek:Βρύσων Ἡρακλεώτης,gen.: Βρύσωνος; fl. late 5th-century BCE) was anancient Greekmathematician andsophist who studied the solving the problems ofsquaring the circle and calculatingpi.
Little is known about the life of Bryson; he came fromHeraclea Pontica, and he may have been a pupil ofSocrates. He is mentioned in the13th Platonic Epistle,[1] andTheopompus even claimed in hisAttack upon Plato thatPlato stole many ideas for his dialogues from Bryson of Heraclea.[2] He is known principally fromAristotle, who criticizes his method of squaring the circle.[3] He also upset Aristotle by asserting thatobscene language does not exist.[4]Diogenes Laërtius[5] and theSuda[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 withBryson of Achaea, who may have lived around that time.[7]
Bryson, along with his contemporary,Antiphon, was the first toinscribe a polygon inside a circle, find thepolygon's area, double the number of sides of the polygon, and repeat the process, resulting in alower bound approximation of thearea 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 polygonscircumscribing a circle, resulting in anupper 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] butArchimedes would later use amethod similar to that of Bryson and Antiphon to calculate π; however, Archimedes calculated theperimeter of a polygon instead of the area.
The 13th-century English philosopherRobert Kilwardby described Bryson's attempt of proving the quadrature of the circle as asophisticalsyllogism—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]
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