# Bryson of Heraclea

**Bryson of Heraclea** (Greek: Βρύσων, *gen*.: Βρύσωνος;; 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]} |
” |

## Notes

- ↑ 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
- ↑ Robert Kilwardby,
*De ortu scientiarum*, LIII, §512, pp. 272f. - ↑ Robert Kilwardby,
*De ortu scientiarum*, LIII, §512, pp. 273.

## References

- 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.