Bryson of Heraclea

Bryson of Heraclea (Greek: Βρύσων Ἡρακλεώτης, gen.: Βρύσωνος; fl. late 5th-century BCE) was an ancient Greek mathematician and sophist who contributed to solving the problem 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 only calculated π to a few digits. 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.

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."[9] 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.[10]


  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. ^ Robert Kilwardby, De ortu scientiarum, LIII, §512, pp. 272f.
  10. ^ Robert Kilwardby, De ortu scientiarum, LIII, §512, pp. 273.


  • 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.

External links

Abu al-Fadl Ja'far ibn 'Ali al-Dimashqi

Abū al-Faḍl Jaʻfar ibn ʻAlī al-Dimashqī (Arabic: أبو الفضل جعفر بن علي الدمشقي‎; fl. 12th-century) was a prosperous Muslim merchant from Damascus. He is best known for being the author of Kitab al-Isharah ila Mahasin at-Tijarah wa Marifat Jayyid al-A’rad wa Kadiiha wa Ghush-ush al-Mudallisin fiha (A Guide to the Merits of Commerce and to Recognition of Both Fine and Defective Merchandise and the Swindles of Those Who Deal Dishonestly).

Almost nothing is known about al-Dimashqi’s life. He was among the Muslim writers who were influenced by Greek sources, particularly by the neo-Pythagorean Bryson of Heraclea.

Ancient economic thought

In the history of economic thought, ancient economic thought refers to the ideas from people before the Middle Ages.

Economics in the classical age is defined in the modern analysis as a factor of ethics and politics, only becoming an object of study as a separate discipline during the 18th century.

Bryson (given name)

Bryson is a given name which may refer to:

Bryson of Achaea (fl. c. 330 BC), ancient Greek philosopher from the Megarian school of philosophy

Bryson of Heraclea (c. 450-c. 390 BC), ancient Greek mathematician and sophist

Bryson Goodwin (born 1985), Australian Rugby League player

Bryson Graham (1952-1993), English rock drummer

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Bryson Albright (born 1994), American football linebacker and free agent

Bryson Kelly (born 1989), American football defensive lineman for the Washington Valor of the Arena Football League

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Bryson Fonville (born 1994), American professional basketball playerFictional characters named Bryson include:

Bryson Bale, in the Marvel Comics Universe

Bryson of Achaea

Bryson of Achaea (or Bryson the Achaean; Greek: Βρύσων ὁ Ἀχαιός, gen.: Βρύσωνος; fl. 330 BC) was an ancient Greek philosopher.

Very little information is known about him. He was said to have been a pupil of Stilpo and Clinomachus, which would mean that he was a philosopher of the Megarian school. He was said to have taught Crates the Cynic, Pyrrho the Skeptic, and Theodorus the Atheist. Diogenes Laërtius includes him among a list of philosophers who left no writings.He is probably not the same person as Bryson of Heraclea, the sophist and mathematician who seems to have lived in the time of Socrates. The Suda, in its entry on Socrates, may be confusing the two Brysons when it refers to Bryson of Heraclea:

Bryson of Heraclea introduced eristic dialectic after Euclides, whereas Clinomachus augmented it, and whereas many came on account of it, it came to an end with Zeno of Citium, for he gave it the name Stoic, after its location, this having occurred in the 105th Olympiad; but some [say that] Bryson was a student not of Socrates but of Euclides

Economic history of the world

The economic history of the world is a record of the economic activities (i.e. the production, distribution and consumption of goods and services) of all humans, spanning both recorded history and evidenced prehistory.

Index of ancient philosophy articles

This page is a list of topics in ancient philosophy.

List of Greek mathematicians

In historical times, Greek civilization has played one of the major roles in the history and development of mathematics. To this day, a number of Greek mathematicians are considered for their innovations and influence on mathematics.

Sophistic works of Antiphon

The name Antiphon the Sophist (; Greek: Ἀντιφῶν) 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.It has been debated since antiquity whether the writer of these Sophistic treatises was in fact none other than Antiphon the Orator, or whether Antiphon the Sophist was indeed a separate person. This remains an active scholarly controversy; of recent editors, Gagarin, and Laks and Most, believe there to be only one Antiphon, whereas G. J. Pendrick argues for the existence of two separate individuals.The most important of these treatises was On Truth, whose surviving fragments cover many different subjects, from astronomy and mathematics to morality and ethics. Fragments have also been preserved of the treatises On Concord and Politicus; these fragments have sometimes been attributed to the Orator rather than to the Sophist.It is also not known for certain whether the treatise on the Interpretation of Dreams under the name of Antiphon was written by Antiphon the Sophist, or whether this was written by yet another different Antiphon. The editions of Pendrick and of Laks and Most proceed on the basis that this treatise was written by the same Antiphon as the Sophistic works.

Timeline of mathematics

This is a timeline of pure and applied mathematics history.



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