Editing Squaring the circle (section)

== 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 <math>\pi</math>.
It takes only elementary geometry to convert any given rational approximation of <math>\pi</math> into a corresponding [[compass-and-straightedge construction|compass and straightedge construction]], 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 ===
{{multiple image
| image1 = Kochanski-1.svg
| caption1 = [[Adam Adamandy Kochański|Kochański's]] approximate construction
| image2 = Quadratur des kreises.svg 
| caption2 = Continuation with equal-area circle and square; <math>r</math> denotes the initial radius
| total_width = 600
}}
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 <math>\pi</math> in the 5th decimal place. Although much more precise numerical approximations to <math>\pi</math> were already known, Kochański's construction has the advantage of being quite simple.{{r|kochanski1}} In the left diagram
<math display=block>|P_3 P_9|=|P_1 P_2|\sqrt{\frac{40}{3}-2\sqrt{3}}\approx 3.141\,5{\color{red}33\,338}\cdot|P_1 P_2|\approx \pi r.</math> 
In the same work, Kochański also derived a sequence of increasingly accurate rational approximations {{nowrap|for <math>\pi</math>.{{r|kochanski2}}}}
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=== Constructions using 355/113 ===
{{multiple image
|image1=01 Squaring the circle-Gelder-1849.svg|caption1=Jacob de Gelder's 355/113 construction
|image2=Approximately squaring the circle.svg|caption2=Ramanujan's 355/113 construction
|total_width=600}}
Jacob de Gelder published in 1849 a construction based on the approximation
<math display=block>\pi\approx\frac{355}{113} = 3.141\;592{\color{red}\;920\;\ldots}</math>
This value is accurate to six decimal places and has been known in China since the 5th century as [[Milü]], and in Europe since the 17th century.{{r|hobson}}

Gelder did not construct the side of the square; it was enough for him to find the value 
<math display=block>\overline{AH}= \frac{4^2}{7^2+8^2}.</math>
The illustration shows de Gelder's construction.

In 1914, Indian mathematician [[Srinivasa Ramanujan]] gave another geometric construction for the same approximation.{{r|ramanujan1|ramanujan2}}
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=== Constructions using the golden ratio ===
{{multiple image
| image1            = 01 Squaring the circle-Hobson.svg
| caption1          = Hobson's golden ratio construction
| image2            = 01 Squaring the circle-Dixon.svg
| caption2          = Dixon's golden ratio construction
| total_width       = 600
| image3            = Squaring the circle like a medieval Master Mason, by Frederic Beatrix.png
| caption3          = Beatrix's 13-step construction
}}

An approximate construction by [[E. W. Hobson]] in 1913{{r|hobson}} is accurate to three decimal places. Hobson's construction corresponds to an approximate value of
<math display="block">\frac{6}{5}\cdot \left( 1 + \varphi\right) = 3.141\;{\color{red}640\;\ldots},</math> where <math>\varphi</math> is the [[golden ratio]], <math>\varphi=(1+\sqrt5)/2</math>.

The same approximate value appears in a 1991 construction by [[Robert Dixon (mathematician)|Robert Dixon]].{{r|dixon}} In 2022 Frédéric Beatrix presented a [[Geometrography|geometrographic]] construction in 13 steps.{{r|beatrix}}
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=== Second construction by Ramanujan ===
{{multiple image
| image1 = Squaring_the_circle-Ramanujan-1914.svg
| caption1 = Squaring the circle, approximate construction according to Ramanujan of 1914, with continuation of the construction (dashed lines, mean proportional red line), see [[:File:01-Squaring_the_circle-Ramanujan-1914.gif|animation]].
| image2 = Squaring the circle-Ramanujan-1913.png
| caption2 = Sketch of "Manuscript book 1 of Srinivasa Ramanujan" p. 54, Ramanujan's 355/113 construction
| total_width = 500
}} 

In 1914, Ramanujan gave a construction which was equivalent to taking the approximate value for <math>\pi</math> to be
<math display=block>\left(9^2 + \frac{19^2}{22}\right)^\frac14 = \sqrt[4]{\frac{2143}{22}} = 3.141\;592\;65{\color{red}2\;582\;\ldots}</math>
giving eight decimal places of <math>\pi</math>.{{r|ramanujan1|ramanujan2}} 
He describes the construction of line segment OS as follows.{{r|ramanujan1}}

{{bi|left=1.6|Let AB (Fig.2) be a diameter of a circle whose centre is O. Bisect the arc ACB at C and trisect AO at T. Join BC and cut off from it CM and MN equal to AT. Join AM and AN and cut off from the latter AP equal to AM. Through P draw PQ parallel to MN and meeting AM at Q. Join OQ and through T draw TR, parallel to OQ and meeting AQ at R. Draw AS perpendicular to AO and equal to AR, and join OS. Then the mean proportional between OS and OB will be very nearly equal to a sixth of the circumference, the error being less than a twelfth of an inch when the diameter is 8000 miles long.}}
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