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 ,and Theopompus even claimed in his Attack upon Plato that Plato stole many ideas for his dialogues from Bryson of Heraclea. He is known principally from Aristotle, who criticizes his method of squaring the circle. He also upset Aristotle by asserting that obscene language does not exist. Diogenes Laërtius and the Suda 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.
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." [ citation needed ] Aristotle criticized this method, 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.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 made appears to have made little progress.
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."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.
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Eudoxus of Cnidus was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his works are lost, though some fragments are preserved in Hipparchus' commentary on Aratus's poem on astronomy. Sphaerics by Theodosius of Bithynia may be based on a work by Eudoxus.
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Glaucon son of Ariston, was an ancient Athenian and Plato's older brother. He is primarily known as a major conversant with Socrates in the Republic, and the interlocutor during the Allegory of the Cave. He is also referenced briefly in the beginnings of two dialogues of Plato, the Parmenides and Symposium.
This page is a list of topics in ancient philosophy.
Bryson of Achaea was an ancient Greek philosopher.
On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.
The name Antiphon the Sophist is used to refer to the writer of several Sophistic treatises. He probably lived in Athens in the last two decades of the 5th century BC, but almost nothing is known of his life.