# Bryson of Heraclea

Last updated

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.

## 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 made appears to have made little progress.[ citation needed ] 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]

## Notes

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. Aristotle, Posterior Analytics, 75b37-76a3.
10. Robert Kilwardby, De ortu scientiarum, LIII, §512, pp. 272f.
11. Robert Kilwardby, De ortu scientiarum, LIII, §512, pp. 273.

## Related Research Articles

Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

Democritus was an Ancient Greek pre-Socratic philosopher primarily remembered today for his formulation of an atomic theory of the universe.

Pythagoras of Samos was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.

The number π is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, and is spelled out as "pi". It is also referred to as Archimedes' constant.

Thales of Miletus was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition, and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy.

Chrysippus of Soli was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of Cleanthes in the Stoic school. When Cleanthes died, around 230 BC, Chrysippus became the third head of the school. A prolific writer, Chrysippus expanded the fundamental doctrines of Zeno of Citium, the founder of the school, which earned him the title of Second Founder of Stoicism.

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.

Zeno of Elea was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell described as "immeasurably subtle and profound".

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

The Method of Mechanical Theorems, also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the law of the lever, which was first demonstrated by Archimedes, and of the center of mass, which he had found for many special shapes.

Speusippus was an ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Academy, near age 60, and remained its head for the next eight years. However, following a stroke, he passed the chair to Xenocrates. Although the successor to Plato in the Academy, Speusippus frequently diverged from Plato's teachings. He rejected Plato's Theory of Forms, and whereas Plato had identified the Good with the ultimate principle, Speusippus maintained that the Good was merely secondary. He also argued that it is impossible to have satisfactory knowledge of any thing without knowing all the differences by which it is separated from everything else.

Epicharmus of Kos or Epicharmus Comicus or Epicharmus Comicus Syracusanus, thought to have lived between c. 550 and c. 460 BC, was a Greek dramatist and philosopher who is often credited with being one of the first comic writers, having originated the Doric or Sicilian comedic form.

In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant π:

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416.

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.