Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proved that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.

Squaring the circle is a problem proposed by ancient geometers. It is the challenge to construct 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, which begins roughly with the earliest-recorded Greek poetry of Homer (7th century BC), and continues through the rise of Christianity and the fall of the Western Roman Empire (5th century AD... A geometer is a mathematician whose area of study is geometry. ... For other uses, see Square. ... Circle illustration This article is about the shape and mathematical concept of circle. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... This article is about a logical statement. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ...

In 1882, the task was proven to be impossible, as a consequence of the fact that pi (π) is a transcendental, rather than 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 if π were transcendental then the construction would be impossible, but that π is transcendental was not proven until 1882. Approximate squaring to any given non-perfect accuracy, on the other hand, is possible in a finite number of steps, as a consequence of the fact that there are rational numbers arbitrarily close to π. When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

The term quadrature of the circle is sometimes used synonymously, or may refer to approximate or numerical methods for finding the area of a circle. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ...

Some apparent partial solutions gave false hope for a long time. In this figure, the area of the shaded figure is equal to the area of the triangle ABC (found by Hippocrates of Chios).

Image File history File links Hipocrat_arcs. ... Image File history File links Hipocrat_arcs. ... Hippocrates of Chios was an ancient Greek mathematician (geometer) and astronomer, who lived c. ...

History

Methods to approximate the area of a given circle with a square were known already to Babylonian mathematicians. The Egyptian Rhind papyrus of 1800BC gives the area of a circle as 64 / 81d², where d is the diameter of the circle. Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras.^[1] Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts and perhaps our best indication of what ancient Egyptian mathematics might have been like near 2000 BC. They are both written on papyrus. ... This article is under construction. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ...

A partial history by Florian Cajori of attempts at the problem.^[2]

The first person to be associated with the problem in Greece 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. 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^[3]. The problem was even mentioned in Aristophanes's play Birds. Image File history File links Size of this preview: 395 × 600 pixelsFull resolution (440 × 668 pixel, file size: 84 KB, MIME type: image/png) p. ... Image File history File links Size of this preview: 395 × 600 pixelsFull resolution (440 × 668 pixel, file size: 84 KB, MIME type: image/png) p. ... Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ... Anaxagoras Anaxagoras (Greek: Αναξαγόρας, c. ... Hippocrates of Chios was an ancient Greek mathematician (geometer) and astronomer, who lived c. ... A lune is either of two figures, both shaped roughly like a crescent Moon. ... Antiphon the Sophist lived in Athens probably in the last two decades of the 5th century BC. There is an ongoing controversy over whether he is one and the same with Antiphon of the Athenian deme Rhamnus in Attica (480–411 BC), the earliest of the ten Attic orators. ... Eudemus of Rhodes (Ευδημος) was an ancient Greek philosopher, who lived from ca. ... Sketch of Aristophanes Aristophanes (Greek: , ca. ...

It is believed that Oenopides was the first person 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 incorrect, it was the first paper to attempt to solve the problem using algebraic properties of π. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility. Oenopides of Chios was an ancient Greek mathematician (geometer) and astronomer, who lived around 450 BCE. He was born shortly after 500 BC on the island of Chios, but mostly worked in Athens. ... James Gregory For other people with the same name, see James Gregory. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i. ...

Impossibility

A solution of the problem of squaring the circle by compass and straightedge demands construction of the number , and the impossibility of this undertaking follows from the fact that π is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. If one solves the problem of the quadrature of the circle using only compass and straightedge, then one has also found an algebraic value of π, which is impossible. Johann Heinrich Lambert conjectured that π was transcendental in 1768 in the same paper he proved its irrationality, even before the existence of transcendental numbers was proved. It wasn't until 1882 that Ferdinand von Lindemann proved its transcendence. In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ... Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 – September 25, 1777), was a mathematician, physicist and astronomer. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i. ...

It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.

Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky space (hyperbolic geometric space). Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Gauss-Bolyai-Lobachevsky space is a non-Euclidean space with a negative Gaussian curvature — that is, a hyperbolic geometry. ...

Note that the transcendence of π implies the impossibility of exactly "circling" the square, as well as of squaring the circle.

Modern approximative constructions

Though squaring the circle is an impossible problem using only compass and straightedge, approximations to squaring the circle can be given by constructing lengths close to π. It takes only minimal knowledge of elementary geometry to convert any given rational approximation of π into a corresponding compass-and-straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proved unsolvable, some mathematicians have applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly (and informally) as constructions that are particularly simple among other imaginable constructions that give similar precision.

Among the modern approximate constructions was one by E. W. Hobson in 1913. This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is accurate to 4 decimals. Ernest William Hobson (27 October 1856 - 19 April 1933) was an English mathematician, now remembered mostly for his books, some of which broke new ground in their coverage in English of topics from mathematical analysis. ...

Indian mathematician Srinivasa Ramanujan in 1913, C. D. Olds in 1963, Martin Gardner in 1966, and Benjamin Bold in 1982 all gave geometric constructions for Srinivasa Ramanujan Iyengar (Tamil: ) (22 December 1887 – 26 April 1920) was an Indian mathematician who is widely regarded as one of the greatest mathematical minds in recent history. ... Martin Gardner (b. ...

which is accurate to 6 decimal places of π.

Srinivasa Ramanujan in 1914 gave a ruler and compass construction which was equivalent to taking the approximate value for π to be

giving a remarkable 8 decimal places of π.

In 1991, Robert Dixon gave constructions for

and

(Kochanski's approximation), though these were only accurate to 4 decimal places of π.

Squaring or quadrature as integration

The problem of finding the area under a curve, known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. For example Newton wrote to Oldenberg in 1676 "I believe M. Leibnitz will not dislike ye Theorem towards ye beginning of my letter of pag. 4 for squaring Curve lines Geometrically."[1] After Newton and Leibniz invented calculus, they still referred to this integration problem as squaring a curve. Look up integration in Wiktionary, the free dictionary. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Henry Oldenburg (c. ... “Leibniz” redirects here. ...

"Squaring the circle" as a metaphor

The futility of undertaking exercises aimed at finding the quadrature of the circle has brought this term into use in totally unrelated contexts, where it is simply used to mean a hopeless, meaningless, or vain undertaking. For example, in Spanish, the expression "descubriste la cuadratura del círculo" ("you discovered the quadrature of the circle") is often used to derisively dismiss claims that someone has found a simple solution to a particularly hard or intractable problem. For other uses, see Hope (disambiguation). ...

Aleister Crowley used the metaphor in a different sense, to represent the goal of magick and mysticism. He implicitly associated his system of Thelema with π. For more information, see Abrahadabra. Aleister Crowley, born Edward Alexander Crowley, (12 October 1875 – 1 December 1947; the surname is pronounced // i. ... This article refers to the magical system of Aleister Crowley and Thelema. ... Thelema is the English transliteration of the Ancient Greek noun : will, from the verb θέλω: to will, wish, purpose. ... Abrahadabra is a word that first appears in The Book of the Law, the central sacred text of Thelema. ...

Claims of circle squaring, and the longitude problem

The mathematical proof that the quadrature of the circle is impossible using only compass and straightedge has not proved to be a hindrance to the many people who have invested years in this problem anyway. Having squared the circle is a famous crank assertion. (See also pseudomathematics.) For example, in his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... Crank is a pejorative term for a person who holds some belief which the vast majority of his contemporaries would consider false, clings to this belief in the face of all counterarguments or evidence presented to him. ... Pseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models. ... “Hobbes” redirects here. ...

During the 18th and 19th century, the notion that the problem of squaring the circle was somehow related to the longitude problem seems to have become prevalent among would-be circle squarers. Using "cyclometer" for circle-squarer, Augustus de Morgan wrote in 1872: The longitude prize was a prize offered by the British government through an Act of Parliament in 1714 for the precise determination of a ships longitude. ... The tone or style of this article or section may not be appropriate for Wikipedia. ...

Montucla says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country.^[4]

Exactly why this connection was made is not clear. De Morgan goes on to say that "[t]he longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted." In his book, de Morgan also mentions receiving many threatening letters from would-be circle squarers, accusing him of trying to "cheat them out their prize." Jean-Étienne Montucla. ...

See also

• The two other classical problems of antiquity were doubling the cube and "trisecting the angle", described in the compass and straightedge article. Unlike squaring the circle, these two problems can be solved by the slightly more powerful construction method of origami, as described at mathematics of paper folding.
• For a more modern related problem, see Tarski's circle-squaring problem.
• Vitruvian man by Leonardo da Vinci and its connection with squaring the circle.
• The Indiana Pi Bill, an 1897 attempt by the Indiana state legislature to dictate a solution to the problem by legislative fiat.

Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. ... A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... This article is about paper folding. ... The art of paper folding or origami has received a considerable amount of mathematical study. ... Tarskis circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a circle (including its interior) in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. ... Leonardo da Vincis Vitruvian Man (1492). ... “Da Vinci” redirects here. ... The Indiana Pi Bill is the popular name for Indiana House of Representatives bill #246 of 1897, which is one of the most famous historical attempts to (erroneously) define scientific truth by legislative fiat. ... The Indiana General Assembly is the state legislature, or legislative branch, of the state government of Indiana. ...

References

1. ^ O'Connor, John J. and Robertson, Edmund F. (2000). The Indian Sulbasutras, MacTutor History of Mathematics archive, St Andrews University.
2. ^ Florian Cajori, A History of Mathematics, second edition, p.143, New York: The Macmillan Company, 1919.
3. ^ Heath, Thomas (1981). History of Greek Mathematics. Courier Dover Publications. 
4. ^ Augustus de Morgan (1872) A Budget of Paradoxes, pp. 96.

The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... University of St Andrews The University of St Andrews was founded between 1410-1413 and is the oldest university in Scotland and the third oldest in the United Kingdom. ... The tone or style of this article or section may not be appropriate for Wikipedia. ...

External links

• 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 π
• Squaring the Circle at Convergence
• The Quadrature of the Circle and Hippocrates' Lunes at Convergence