Editing Square (section)

===Area and quadrature <span class="anchor" id="Squaring the circle"></span>===
{{also|Area|Quadrature (geometry)|Squaring the circle}}
[[File:Pythagorean.svg|thumb|upright=0.8|The [[Pythagorean theorem]]: the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse.]]
[[File:Squaring the Circle J.svg|thumb|A circle and square with the same area|upright=0.6]]
Conventionally, since ancient times, most units of [[area]] have been defined in terms of various squares, typically a square with a standard unit of [[length]] as its side, for example a [[square meter]] or [[square inch]].<ref name=Treese>{{cite book |last=Treese |first=Steven A. |year=2018 |chapter=Historical Area |title=History and Measurement of the Base and Derived Units |publisher=Springer |doi=10.1007/978-3-319-77577-7_5 |pages=301–390 |isbn=978-3-319-77576-0 }}</ref> The area of an arbitrary rectangle can then be simply computed as the product of its length and its width, and more complicated shapes can be measured by conceptually breaking them up into unit squares or into arbitrary triangles.{{r|Treese}}

In [[Euclidean geometry|ancient Greek deductive geometry]], the area of a planar shape was measured and compared by constructing a square with the same area by using only a finite number of steps with [[compass and straightedge]], a process called ''[[Quadrature (geometry)|quadrature]]'' or ''squaring''. [[Euclid's Elements|Euclid's ''Elements'']] shows how to do this for rectangles, parallelograms, triangles, and then more generally for [[simple polygon]]s by breaking them into triangular pieces.<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book II, Proposition 14. [http://aleph0.clarku.edu/~djoyce/elements/bookII/propII14.html Online English version] by [[David E. Joyce]].</ref> Some shapes with curved sides could also be squared, such as the [[lune of Hippocrates]]<ref>{{cite journal|title=The problem of squarable lunes|journal=[[The American Mathematical Monthly]]|volume=107|issue=7|year=2000|pages=645–651|jstor=2589121|first=M. M.|last=Postnikov|author-link=Mikhail Postnikov|doi=10.2307/2589121}}</ref> and the [[Quadrature of the Parabola|parabola]].<ref>{{cite journal
 | last = Berendonk | first = Stephan
 | doi = 10.1007/s00591-016-0173-0
 | issue = 1
 | journal = Mathematische Semesterberichte
 | mr = 3629442
 | pages = 1–13
 | title = Ways to square the parabola—a commented picture gallery
 | volume = 64
 | year = 2017}}</ref>

This use of a square as the defining shape for area measurement also occurs in the Greek formulation of the [[Pythagorean theorem]]: squares constructed on the two sides of a [[right triangle]] have equal total area to a square constructed on the [[hypotenuse]].<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book I, Proposition 47. [http://aleph0.clarku.edu/~djoyce/elements/bookI/propI47.html Online English version] by [[David E. Joyce]].</ref> Stated in this form, the theorem would be equally valid for other shapes on the sides of the triangle, such as equilateral triangles or semicircles,<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 31. [http://aleph0.clarku.edu/~djoyce/elements/bookVI/propVI31.html Online English version] by [[David E. Joyce]].</ref> but the Greeks used squares. In modern mathematics, this formulation of the theorem using areas of squares has been replaced by an algebraic formulation involving [[Square (algebra)|squaring numbers]]: the lengths of the sides and hypotenuse of the right triangle obey the equation <math>a^2+b^2=c^2</math>.<ref>{{cite book|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|page=xi|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|year=2019}}</ref>

Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully to [[squaring the circle|square the circle]], constructing a square with the same area as a given circle, again using finitely many steps with a compass and straightedge. In 1882, the task was proven to be impossible as a consequence of the [[Lindemann–Weierstrass theorem]]. This theorem proves that [[pi]] ({{pi}}) is a [[transcendental number]] rather than an [[algebraic number|algebraic irrational number]]; that is, it is not the [[root of a function|root]] of any [[polynomial]] with [[rational number|rational]] coefficients. A construction for squaring the circle could be translated into a polynomial formula for {{pi}}, which does not exist.<ref>{{cite journal
 | last = Kasner | first = Edward | author-link = Edward Kasner
 | date = July 1933
 | issue = 1
 | journal = [[The Scientific Monthly]]
 | jstor = 15685
 | pages = 67–71
 | title = Squaring the circle
 | volume = 37}}</ref>
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