Concept
Squaring the circle
Overview
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a 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.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (
π
{\displaystyle \pi }
) is a transcendental number.
That is,
π
{\displaystyle \pi }
is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if
π
{\displaystyle \pi }
were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.
Despite the proof that it is impossible, attempts to square the circle have been common in mathematical crankery. The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. The term quadrature of the circle is sometimes used as a synonym for squaring the circle. It may also refer to approximate or numerical methods for finding the area of a circle. In general, quadrature or squaring may also be applied to other plane figures.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
For generally, not he that hath skill in geometry, or any other science speculative, but only he that understandeth what conduceth to the good and government of the people, is called a wise man. Now that no author of sedition can be wise in this acceptation of the word, is sufficiently proved, in that it hath been already demonstrated, that no pretence of sedition can be right or just.
Thomas Hobbes, The English works of Thomas Hobbes of Malmesbury, Volume 04
States what Squaring the circle is; the author is tied to Squaring the circle in the knowledge graph.
These words Appetite, and Aversion we have from the Latines; and they both of them signifie the motions, one of approaching, the other of retiring. So also do the Greek words for the same, which are orme and aphorme. For nature it selfe does often presse upon men those truths, which afterwards, when they look for somewhat beyond Nature, they stumble at.
Thomas Hobbes, Leviathan
States what Squaring the circle is; the author is tied to Squaring the circle in the knowledge graph.
Geometry, however, among the Greeks passed into a stage of abstraction in which lines, planes, etc., in the sense in which they are understood in our elementary texts, took the place of actually measured surfaces, and also took on the deductive form of presentation that has served as a model for all mathematical presentation since Euclid. Mensuration smacked too much of the exchange, and before the time of…
John Dewey, Creative Intelligence
States what Squaring the circle is.
Philosophers who discussed Squaring the circle
Related ideas
- Aristophanes
- Golden ratio
- Isaac Newton
- Ancient Greek mathematics
- James Joyce
- Method of exhaustion
- Anaxagoras
- Leonardo da Vinci
- Archimedes
- Indian mathematics
- Augustus De Morgan
- Geometry
- Metaphor
- Old Testament
- Paradiso
- Euclid's Elements
- Proclus
- Lewis Carroll
- Thomas Hobbes
- Thomas Mann
- Victorian era
- Eudemus of Rhodes
- Babylonian mathematics
- Alexander Pope
Place in the tradition
Part of Anaxagoras.