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# Squaring the circle

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 Title: Squaring the circle Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Squaring the circle

Squaring the circle: the areas of this square and this circle are both equal to π. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.
Some apparent partial solutions gave false hope for a long time. In this figure, the shaded figure is the Lune of Hippocrates. Its area is equal to the area of the triangle ABC (found by Hippocrates of Chios).

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. More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail 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) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.

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 to mean the same thing as squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.

## Contents

• History 1
• Impossibility 2
• Modern approximative constructions 3
• Squaring or quadrature as integration 4
• Claims of circle squaring 5
• Connection with the longitude problem 5.1
• Other modern claims 5.2
• In literature 6
• References 8

## History

Methods to approximate the area of a given circle with a square were known already to Babylonian mathematicians. The Egyptian Rhind papyrus of 1800 BC gives the area of a circle as (64/81) d 2, where d is the diameter of the circle, and pi approximated to 256/81, a number that appears in the older Moscow Mathematical Papyrus and used for volume approximations (i.e. hekat). Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras. Archimedes showed that the value of pi lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). See Numerical approximations of π for more on the history.

The first known Greek to be associated with the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios squared certain lunes, in the hope that it would lead to a solution — see Lune of Hippocrates. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up. The problem was even mentioned in Aristophanes's play The Birds.

It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge). James Gregory attempted a proof of its impossibility in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of pi. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility.

The famous Victorian-age mathematician, logician and author, Charles Lutwidge Dodgson (better known under the pseudonym "Lewis Carroll") also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write including one called "Plain Facts for Circle-Squarers". In the introduction to "A New Theory of Parallels", Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating:

The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.

Perhaps the most famous and effective ridiculing of circle squaring appears in

• Squaring the circle at the MacTutor History of Mathematics archive
• Squaring the Circle at cut-the-knot
• Circle Squaring at MathWorld, includes information on procedures based on various approximations of pi
• "Squaring the Circle" at "Convergence"
• The Quadrature of the Circle and Hippocrates' Lunes at Convergence
• How to Unroll a Circle Pi accurate to eight decimal places, using straightedge and compass.
• Squaring the Circle and Other Impossibilities, lecture by Robin Wilson, at Gresham College, 16 January 2008 (available for download as text, audio or video file).
• Grime, James. "Squaring the Circle". Numberphile.