Editing Squaring the circle (section)

==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 [[Approximations of π|approximation to {{pi}}]] that they produce. In around 2000 BCE, the [[Babylonian mathematics|Babylonian mathematicians]] used the approximation {{nowrap|<math>\pi\approx\tfrac{25}{8}=3.125</math>,}} and at approximately the same time the [[ancient Egyptian mathematics|ancient Egyptian mathematicians]] used {{nowrap|<math>\pi\approx\tfrac{256}{81}\approx 3.16</math>.}} Over 1000 years later, the [[Old Testament]] ''[[Books of Kings]]'' used the simpler approximation {{nowrap|<math>\pi\approx3</math>.{{r|b3p}}}} Ancient [[Indian mathematics]], as recorded in the ''[[Shatapatha Brahmana]]'' and ''[[Shulba Sutras]]'', used several different approximations {{nowrap|to <math>\pi</math>.{{r|plofker}}}} [[Archimedes]] proved a formula for the area of a circle, according to which <math>3\,\tfrac{10}{71}\approx 3.141<\pi<3\,\tfrac{1}{7}\approx 3.143</math>.{{r|b3p}} In [[Chinese mathematics]], in the third century CE, [[Liu Hui]] found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century [[Zu Chongzhi]] found <math>\pi\approx 355/113\approx 3.141593</math>, an approximation known as [[Milü]].{{r|china}}

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 mathematicians found compass and straightedge constructions to convert any [[polygon]] into a square of equivalent area.<ref name=square-polygons/> 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]] wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes:
{{bi|left=1.6|Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. For if a parallelogram is found equal to any rectilinear figure, it is worthy of investigation whether one can prove that rectilinear figures are equal to figures bound by circular arcs.<ref>Translation from {{harvtxt|Knorr|1986}}, p. 25</ref>}}

[[File:Hipocrat arcs.svg|thumb|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 {{math|ABC}} (found by [[Hippocrates of Chios]]).]]

The first known Greek to study the problem was [[Anaxagoras]], who worked on it while in prison. [[Hippocrates of Chios]] attacked the problem by finding a shape bounded by circular arcs, the [[lune of Hippocrates]], 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]]). Since any polygon can be squared,<ref name=square-polygons>The construction of a square equal in area to a given polygon is Proposition 14 of [[Euclid's Elements|Euclid's ''Elements'']], Book II.</ref> he argued, the circle can be squared. In contrast, [[Eudemus of Rhodes|Eudemus]] argued that magnitudes can be divided up without limit, so the area of the circle would never be used up.{{r|heath}} 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]].{{r|bryson}} The more general goal of carrying out all geometric constructions using only a [[Straightedge and compass construction|compass and straightedge]] has often been attributed to [[Oenopides]], but the evidence for this is circumstantial.{{r|knorr}}

The problem of finding the area under an arbitrary curve, now known as [[Integral|integration]] in [[calculus]], or [[numerical quadrature|quadrature]] in [[numerical analysis]], was known as ''squaring'' before the invention of calculus.{{r|guicciardini}}  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, [[Isaac Newton|Newton]] wrote to [[Henry Oldenburg|Oldenburg]] in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically".{{r|newton-cotes}} 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.

A 1647 attempt at squaring the circle, ''Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum'' by [[Grégoire de Saint-Vincent]], was heavily criticized by [[Vincent Léotaud]].{{r|leotaud}} Nevertheless, de Saint-Vincent succeeded in his quadrature of the [[hyperbola]], and in doing so was one of the earliest to develop the [[natural logarithm]].{{r|burn}} [[James Gregory (astronomer and mathematician)|James Gregory]], following de Saint-Vincent, attempted another 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 <math>\pi</math>.{{r|gregory|crippa}} [[Johann Heinrich Lambert]] proved in 1761 that <math>\pi</math> is an [[irrational number]].{{r|lambert1|lambert2}} It was not until 1882 that [[Ferdinand von Lindemann]] succeeded in proving more strongly that {{pi}} is a [[transcendental number]], and by doing so also proved the impossibility of squaring the circle with compass and straightedge.{{r|lindemann|fritsch}}

After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by [[Pseudomathematics|pseudomathematical]] attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts.{{r|dudley}} As well, several later mathematicians including [[Srinivasa Ramanujan]] developed compass and straightedge constructions that approximate the problem accurately in few steps.{{r|ramanujan1|ramanujan2}}

Two other classical problems of antiquity, famed for their impossibility, were [[doubling the cube]] and [[trisecting the angle]]. 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]], rather than being transcendental. Therefore, more powerful methods than compass and straightedge constructions, such as [[neusis construction]] or [[mathematics of paper folding|mathematical paper folding]], can be used to construct solutions to these problems.{{r|origami1|origami2}}