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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,[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:

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]


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

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

The History of Pi
O'Connor, John J.; Robertson, Edmund F., "Bryson of Heraclea", MacTutor History of Mathematics archive, University of St Andrews.


Ancient Greek and Hellenistic mathematics (Euclidean geometry)
Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heliodorus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus
Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Sphaerics The Quadrature of the Parabola The Sand Reckoner
Angle trisection Doubling the cube Squaring the circle Problem of Apollonius
Circles of Apollonius
Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center
In Elements
Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon
Apollonius's theorem
Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus
Cyrene Library of Alexandria Platonic Academy
Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction

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