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What i can't understand is that I'm reading book " a History of Mathematics by boyer" and it says Archimedes made possible to construct a triangle equal in area to that of a circle by help of spirals. Etc.. And then it says " with a simple geometric transformation you can produce a square out of it. But, wait.

isnt the squaring of circle concidered impossible all the time. I am very confused. Plz help me understand.

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  • $\begingroup$ If archimedes could produce triangle equal in area of a circle. And from that triangle u can construct a square. Then isn't this the solution of squaring the circle. $\endgroup$
    – Asim
    Apr 1, 2020 at 7:01

2 Answers 2

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The problem of "squaring the circle" without qualifications usually refers to doing it with only compass and straightedge, which is indeed impossible. On the contrary, solutions that use further tools not emulatable with compass and straightedge, like spirals, were known even in antiquity.

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  • $\begingroup$ But i tried to google it and it looks like its the idea of squaring the circle which is impossible everyone is talking about. in exact term only approximation is possible. But if archimedes showed that is possible geometrically. Then how does it not exact and only approximare $\endgroup$
    – Asim
    Apr 1, 2020 at 7:07
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    $\begingroup$ @Asim The solution with spirals is exact, but the spiral cannot be constructed within the implied compass and straightedge framework. $\endgroup$
    – Parcly Taxel
    Apr 1, 2020 at 7:08
  • $\begingroup$ So u CAN produce a square equal to area of a circle in exact way? $\endgroup$
    – Asim
    Apr 1, 2020 at 7:16
  • $\begingroup$ @Asim Yes. Just not in the purest way of the ancient geometers - not by compass and straightedge alone! $\endgroup$
    – Parcly Taxel
    Apr 1, 2020 at 7:17
  • $\begingroup$ What makes me confuse is that when i googled the terms " squaring the circle" the results came out that made an impression that it is totally impossible to make an square which has exactly same area of a given circle. ( even with any method, let aside compass and straightedge) $\endgroup$
    – Asim
    Apr 1, 2020 at 7:19
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We certainly can square the circle with the right devices.

  • Make the circle the base of a cylinder.
  • Mark off spot on a string.
  • Wrap the string around the cylinder along the edge of its base and mark off a second spot one lap from the original mark.
  • Sraighten the string so now the two marks are separated by the circumference.
  • Using compass and straightedge methods, construct a rectangle whose dimensions are half the circumference and the radius, and square this rectangle.

Archimedes is better known for approximating the circumference rather than rendering an exact construction:

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes.[1] This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant".[2] Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that

223/71 < π < 22/7 (that is 3.1408 < π < 3.1429).[3]

Cited references:

1. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka, p. 169.

2. Ibid., p. 170.

3. Ibid., pp. 175, 205.

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