![Squaring the circle](https://upload.wikimedia.org/wikipedia/commons/a/a7/Squaring_the_circle.svg)
Squaring the circle is a problem in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
first proposed in
Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
. It is the challenge of constructing a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
with the
area of a circle
In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
One method of deriving this formula, which origi ...
by using only a finite number of steps with a
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
. The difficulty of the problem raised the question of whether specified
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s of
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
concerning the existence of
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
and
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s implied the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the
Lindemann–Weierstrass theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:
In other words, the extension field \mathbb(e^, \dots, e^) has transcen ...
, which proves that
pi (
) is a
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
.
That is,
is not the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of any
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
coefficients. It had been known for decades that the construction would be impossible if
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
pseudomathematics
Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved to be unsolvable or re ...
(i.e. the work of mathematical cranks). 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, but it may also refer to approximate or numerical methods for finding the
area of a circle
In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
One method of deriving this formula, which origi ...
.
History
Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. These methods can be summarized by stating the
approximation to that they produce. In around 2000 BCE, the
Babylonian mathematicians used the approximation and at approximately the same time the
ancient Egyptian mathematicians
Ancient history is a time period from the beginning of writing and recorded human history to as far as late antiquity. The span of recorded history is roughly 5,000 years, beginning with the Sumerian cuneiform script. Ancient history cove ...
used Over 1000 years later, the
Old Testament
The Old Testament (often abbreviated OT) is the first division of the Christian biblical canon, which is based primarily upon the 24 books of the Hebrew Bible or Tanakh, a collection of ancient religious Hebrew writings by the Israelites. The ...
''
Books of Kings
The Book of Kings (, '' Sēfer Məlāḵīm'') is a book in the Hebrew Bible, found as two books (1–2 Kings) in the Old Testament of the Christian Bible. It concludes the Deuteronomistic history, a history of Israel also including the books ...
'' used the simpler approximation Ancient
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
, as recorded in the ''
Shatapatha Brahmana
The Shatapatha Brahmana ( sa, शतपथब्राह्मणम् , Śatapatha Brāhmaṇam, meaning 'Brāhmaṇa of one hundred paths', abbreviated to 'SB') is a commentary on the Śukla (white) Yajurveda. It is attributed to the Vedic ...
'' and ''
Shulba Sutras The ''Shulva Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.
Purpose and origins
The ...
'', used several different approximations
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
proved a formula for the area of a circle, according to which
. In
Chinese mathematics
Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
, in the third century CE,
Liu Hui
Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century
Zu Chongzhi
Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3. ...
found
, an approximation known as
Milü
Milü (; "close ratio"), also known as Zulü ( Zu's ratio), is the name given to an approximation to (pi) found by Chinese mathematician and astronomer Zu Chongzhi in the 5th century. Using Liu Hui's algorithm (which is based on the areas of r ...
.
The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from
Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
. Greek mathematicians found compass and straightedge constructions to convert any
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
into a square of equivalent area.
[ They used this construction to compare areas of polygons geometrically, rather than by the numerical computation of area that would be more typical in modern mathematics. As ]Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes:
The first known Greek to study the problem was Anaxagoras
Anaxagoras (; grc-gre, Ἀναξαγόρας, ''Anaxagóras'', "lord of the assembly"; 500 – 428 BC) was a Pre-Socratic Greek philosopher. Born in Clazomenae at a time when Asia Minor was under the control of the Persian Empire, ...
, who worked on it while in prison. Hippocrates of Chios
Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer.
He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates
In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune (mathematics), lune bounded by Circular arc, arcs of two circles, the smaller of which has as its diameter a Chord (geometry), chord spanning a right angle on the l ...
, that could be squared. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the method of exhaustion
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 bet ...
). Since any polygon can be squared,[The construction of a square equal in area to a given polygon is Proposition 14 of Euclid's ''Elements'', Book II.] he argued, the circle can be squared. In contrast, Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle would never be used up. Contemporaneously with Antiphon, Bryson of Heraclea argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
. The more general goal of carrying out all geometric constructions using only a compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
has often been attributed to Oenopides
Oenopides of Chios ( el, Οἰνοπίδης ὁ Χῖος; born c. 490 BCE) was an ancient Greek geometer and astronomer, who lived around 450 BCE.
Biography
Only limited information are known about the early life of Oenopides except his birt ...
, but the evidence for this is circumstantial.
The problem of finding the area under an arbitrary curve, now known as integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, or quadrature in numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, 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
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
wrote to Oldenburg Oldenburg may also refer to:
Places
*Mount Oldenburg, Ellsworth Land, Antarctica
*Oldenburg (city), an independent city in Lower Saxony, Germany
**Oldenburg (district), a district historically in Oldenburg Free State and now in Lower Saxony
*Olde ...
in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically". In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve retains the idea of using only restricted geometric methods.
James Gregory attempted a proof of the impossibility of squaring the circle 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 . Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subjec ...
proved in 1761 that is an irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
. It was not until 1882 that Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
succeeded in proving more strongly that is a transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
, and by doing so also proved the impossibility of squaring the circle with compass and straightedge.
After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts. As well, several later mathematicians including Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
developed compass and straightedge constructions that approximate the problem accurately in few steps.
Two other classical problems of antiquity, famed for their impossibility, were doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
and trisecting the angle
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
. Like squaring the circle, these cannot be solved by compass and straightedge. However, they have a different character than squaring the circle, in that their solution involves the root of a cubic equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
, rather than being transcendental. Therefore, more powerful methods than compass and straightedge constructions, such as neusis construction
In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematicians.
Geometric construction
The neusis construction consists of fitting a line e ...
or mathematical paper folding, can be used to construct solutions to these problems.
Impossibility
The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number , the length of the side of a square whose area equals that of a unit circle. If were a constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is con ...
, it would follow from standard compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
constructions that would also be constructible. In 1837, Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
In a paper from 1837, Wantzel pr ...
showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients. Thus, constructible lengths must be algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. If the circle could be squared using only compass and straightedge, then would have to be an algebraic number. It was not until 1882 that Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
proved the transcendence of and so showed the impossibility of this construction. Lindemann's idea was to combine the proof of transcendence of Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
, shown by Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
in 1873, with Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circum ...
This identity immediately shows that is an irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
, because a rational power of a transcendental number remains transcendental. Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:
In other words, the extension field \mathbb(e^, \dots, e^) has transcen ...
on linear independence of algebraic powers of , to show that is transcendental and therefore that squaring the circle is impossible.
Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
makes squaring the circle possible in some sense. For example, Dinostratus' theorem
In geometry, Dinostratus' theorem describes a property of Hippias' trisectrix, that allows for the squaring the circle if the trisectrix can be used in addition to straightedge and compass. The theorem is named after the Greek mathematician Dinost ...
uses the quadratrix of Hippias
The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve (a curve created through motion). Its discovery is attributed to the G ...
to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it. The Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
can be used for another similar construction. Although the circle cannot be squared in Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, it sometimes can be in hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
under suitable interpretations of the terms. The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains ''regular quadrilaterals'', shapes with four equal sides and four equal angles sharper than right angles. There exist in the hyperbolic plane (countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Symmetrically, there is no method for starting with an arbitrary circle and constructing a regular quadrilateral of equal area, and for sufficiently large circles no such quadrilateral exists.
Approximate constructions
Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to .
It takes only elementary geometry to convert any given rational approximation of into a corresponding compass and straightedge construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision.
Construction by Kochański
One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit Adam Adamandy Kochański, producing an approximation diverging from in the 5th decimal place. Although much more precise numerical approximations to were already known, Kochański's construction has the advantage of being quite simple. In the left diagram
In the same work, Kochański also derived a sequence of increasingly accurate rational approximations
Constructions using 355/113
Jacob de Gelder published in 1849 a construction based on the approximation
This value is accurate to six decimal places and has been known in China since the 5th century as Zu Chongzhi
Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3. ...
's fraction, and in Europe since the 17th century.
Gelder did not construct the side of the square; it was enough for him to find the value
The illustration shows de Gelder's construction.
In 1914, Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
gave another geometric construction for the same approximation.
Constructions using the golden ratio
An approximate construction by E. W. Hobson
Ernest William Hobson Fellow of the Royal Society, FRS (27 October 1856 – 19 April 1933) was an England, English mathematician, now remembered mostly for his books, some of which broke new ground in their coverage in English of topics fro ...
in 1913 is accurate to three decimal places. Hobson's construction corresponds to an approximate value of
where is the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
, .
The same approximate value appears in a 1991 construction by Robert Dixon. In 2022 Frédéric Beatrix presented a geometrographic construction in 13 steps.
Second construction by Ramanujan
In 1914, Ramanujan gave a construction which was equivalent to taking the approximate value for to be
giving eight decimal places of .
He describes the construction of line segment OS as follows.
Incorrect constructions
In his old age, the English philosopher Thomas Hobbes
Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book ''Leviathan'', in which he expounds an influent ...
convinced himself that he had succeeded in squaring the circle, a claim refuted by John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
as part of the Hobbes–Wallis controversy The Hobbes–Wallis controversy was a polemic debate that continued from the mid-1650s well into the 1670s, between the philosopher Thomas Hobbes and the mathematician and clergyman John Wallis. It was sparked by ''De corpore'', a philosophical work ...
. During the 18th and 19th century, the false notions that the problem of squaring the circle was somehow related to the longitude problem
The history of longitude describes the centuries-long effort by astronomers, cartographers and navigators to discover a means of determining the longitude of any given place on Earth.
The measurement of longitude is important to both cartography ...
, and that a large reward would be given for a solution, became prevalent among would-be circle squarers. In 1851, John Parker published a book ''Quadrature of the Circle'' in which he claimed to have squared the circle. His method actually produced an approximation of accurate to six digits.
The Victorian-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by his pseudonym Lewis Carroll
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ...
, 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:
A ridiculing of circle squaring appears in Augustus De Morgan's book ''A Budget of Paradoxes'', published posthumously by his widow in 1872. Having originally published the work as a series of articles in '' The Athenæum'', he was revising it for publication at the time of his death. Circle squaring declined in popularity after the nineteenth century, and it is believed that De Morgan's work helped bring this about.
Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined as equal to 3.2. Goodwin then proposed the Indiana Pi Bill
The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most notorious attempts to establish mathematical truth by legislative fiat. Despite its name, the main result claimed by the b ...
in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press.
The mathematical crank Carl Theodore Heisel also claimed to have squared the circle in his 1934 book, "Behold! : the grand problem no longer unsolved: the circle squared beyond refutation." Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
referred to the book as a "classic crank book."
In literature
The problem of squaring the circle has been mentioned over a wide range of literary eras, with a variety of metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
ical meanings. Its literary use dates back at least to 414 BC, when the play '' The Birds'' by Aristophanes
Aristophanes (; grc, Ἀριστοφάνης, ; c. 446 – c. 386 BC), son of Philippus, of the deme
In Ancient Greece, a deme or ( grc, δῆμος, plural: demoi, δημοι) was a suburb or a subdivision of Athens and other city-states ...
was first performed. In it, the character Meton of Athens
Meton of Athens ( el, Μέτων ὁ Ἀθηναῖος; ''gen''.: Μέτωνος) was a Greek mathematician, astronomer, geometer, and engineer who lived in Athens in the 5th century BC. He is best known for calculations involving the eponymou ...
mentions squaring the circle, possibly to indicate the paradoxical nature of his utopian city.
Dante's ''Paradise
In religion, paradise is a place of exceptional happiness and delight. Paradisiacal notions are often laden with pastoral imagery, and may be cosmogonical or eschatological or both, often compared to the miseries of human civilization: in paradis ...
'', canto XXXIII, lines 133–135, contain the verse:
As the geometer his mind applies
To square the circle, nor for all his wit
Finds the right formula, howe'er he tries
For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise. Dante's image also calls to mind a passage from Vitruvius
Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...
, famously illustrated later in Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
's ''Vitruvian Man
The ''Vitruvian Man'' ( it, L'uomo vitruviano; ) is a drawing by the Italian Renaissance artist and scientist Leonardo da Vinci, dated to . Inspired by the writings by the ancient Roman architect Vitruvius, the drawing depicts a nude man in two s ...
'', of a man simultaneously inscribed in a circle and a square. Dante uses the circle as a symbol for God, and may have mentioned this combination of shapes in reference to the simultaneous divine and human nature of Jesus. Earlier, in canto XIII, Dante calls out Greek circle-squarer Bryson as having sought knowledge instead of wisdom.
Several works of 17th-century poet Margaret Cavendish
Margaret Lucas Cavendish, Duchess of Newcastle-upon-Tyne (1623 – 15 December 1673) was an English philosopher, poet, scientist, fiction writer and playwright.
Her husband, William Cavendish, 1st Duke of Newcastle-upon-Tyne, was Cavalier, R ...
elaborate on the circle-squaring problem and its metaphorical meanings, including a contrast between unity of truth and factionalism, and the impossibility of rationalizing "fancy and female nature". By 1742, when Alexander Pope
Alexander Pope (21 May 1688 O.S. – 30 May 1744) was an English poet, translator, and satirist of the Enlightenment era who is considered one of the most prominent English poets of the early 18th century. An exponent of Augustan literature, ...
published the fourth book of his ''Dunciad
''The Dunciad'' is a landmark, mock-heroic, narrative poem by Alexander Pope published in three different versions at different times from 1728 to 1743. The poem celebrates a goddess Dulness and the progress of her chosen agents as they bring ...
'', attempts at circle-squaring had come to be seen as "wild and fruitless":
Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.
Similarly, the Gilbert and Sullivan
Gilbert and Sullivan was a Victorian era, Victorian-era theatrical partnership of the dramatist W. S. Gilbert (1836–1911) and the composer Arthur Sullivan (1842–1900), who jointly created fourteen comic operas between 1871 and 1896, of which ...
comic opera ''Princess Ida
''Princess Ida; or, Castle Adamant'' is a comic opera with music by Arthur Sullivan and libretto by W. S. Gilbert. It was their eighth operatic collaboration of fourteen. ''Princess Ida'' opened at the Savoy Theatre on 5 January 1884, for a ru ...
'' features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding perpetual motion
Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
. One of these goals is "And the circle – they will square it/Some fine day."
The sestina
A sestina (, from ''sesto'', sixth; Old Occitan: ''cledisat'' ; also known as ''sestine'', ''sextine'', ''sextain'') is a fixed verse, fixed verse form consisting of six stanzas of six lines each, normally followed by a three-line envoi. The wor ...
, a poetic form first used in the 12th century by Arnaut Daniel
Arnaut Daniel (; fl. 1180–1200) was an Occitan troubadour of the 12th century, praised by Dante as "the best smith" (''miglior fabbro'') and called a "grand master of love" (''gran maestro d'amore'') by Petrarch. In the 20th century he was lau ...
, has been said to metaphorically square the circle in its use of a square number of lines (six stanzas of six lines each) with a circular scheme of six repeated words. writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth. A similar metaphor was used in "Squaring the Circle", a 1908 short story by O. Henry, about a long-running family feud. In the title of this story, the circle represents the natural world, while the square represents the city, the world of man.
In later works, circle-squarers such as Leopold Bloom in James Joyce
James Augustine Aloysius Joyce (2 February 1882 – 13 January 1941) was an Irish novelist, poet, and literary critic. He contributed to the modernist avant-garde movement and is regarded as one of the most influential and important writers of ...
's novel ''Ulysses
Ulysses is one form of the Roman name for Odysseus, a hero in ancient Greek literature.
Ulysses may also refer to:
People
* Ulysses (given name), including a list of people with this name
Places in the United States
* Ulysses, Kansas
* Ulysse ...
'' and Lawyer Paravant in Thomas Mann
Paul Thomas Mann ( , ; ; 6 June 1875 – 12 August 1955) was a German novelist, short story writer, social critic, philanthropist, essayist, and the 1929 Nobel Prize in Literature laureate. His highly symbolic and ironic epic novels and novella ...
's ''The Magic Mountain
''The Magic Mountain'' (german: Der Zauberberg, links=no, ) is a novel by Thomas Mann, first published in German in November 1924. It is widely considered to be one of the most influential works of twentieth-century German literature.
Mann st ...
'' are seen as sadly deluded or as unworldly dreamers, unaware of its mathematical impossibility and making grandiose plans for a result they will never attain.
See also
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References
Further reading and external links
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{{DEFAULTSORT:Squaring The Circle
Compass and straightedge constructions
Euclidean plane geometry
Unsolvable puzzles
History of geometry
Pseudomathematics