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I am unclear about what "Squaring the Circle" is, let alone how people tried to solve it.

Please tell me if "Squaring the Circle" means finding square and circle with same area OR finding square and circle with same perimeter/circumference.

Below is my thought-process...

In the book The Quadrivium, page 32 it says

Shown opposite is the extraordinary fact that the size of the Moon relates to the size of the Earth as does three to eleven. What this means is that if we draw down the Moon to the Earth, as shown, then a heavenly curcle through the Moon will have a circumference equal to the perimeter of a square around the Earth. This is called 'squaring of the circle.'

enter image description here

Then page 78 talks about square and circle having same area

A circle and square can also be married by having the same area, and a double rainbow, with bows at $41.5^{\circ}$ and $52.5^{\circ}$

enter image description here

Rhonda
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    Squaring the circle – Edward Evans May 14 '16 at 14:07
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    Historically, Squaring the Circle refers to area, not perimeter. But mathematically both problems come down to constructing a segment of length $\pi$ using straightedge and compass, so in that sense they are equivalent. – lulu May 14 '16 at 14:08
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    Note: the first reference is just an approximation (and not a very good one). If you suppose that the radius of the earth is $1$ (in some units) then we'd get that the perimeter of the square is $8$ and the circumference of the circle is $2\pi\frac {14}{11}\sim 7.9967813$. So...it's close, but certainly not equal. – lulu May 14 '16 at 14:12

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From Wikipedia:

Squaring the circle is a problem proposed by ancient geometers.This is one of the three geometric problems of antiquity. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

In $1882$, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number.

The solution of the problem of squaring the circle by compass and straightedge demands construction of the number ${\displaystyle \scriptstyle {\sqrt {\pi }}}$ , and the impossibility of this undertaking follows from the fact that pi is a transcendental (non-algebraic and therefore non-constructible) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible.

However, approximations to circle squaring are given by constructing lengths close to $pi=3.1415926....$ Ramanujan ($1913-1914$), Olds ($1963$), Gardner ($1966$), and (Bold 1982,) give geometric constructions for $355/113=3.1415929....$

Pedro
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Vidyanshu Mishra
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    "an algebraic irrational number". Only algebraic numbers can be created but not all algebraic numbers can be created. cube root of 2 is impossible to construct. Only algebraic numbers that solve polynomials of degree $2^n$ can be constructed. – fleablood Nov 30 '16 at 20:08
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    Your answer contains almost identical excerpts of en.wikipedia.org/wiki/Squaring_the_circle, but you fail to cite this source, and have failed to add any value to such article. This is unacceptable. I am adding the source, and I hope this is not repeated. Regards, – Pedro Dec 05 '16 at 17:16
  • Thanks @PedroTamaroff, I assure you that this will not happen in future. I have already gone through a rough discussion on it. Thanks for reminding me. – Vidyanshu Mishra Dec 05 '16 at 17:18
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To be brief: I give you a piece of paper with a circle drawn on it (equivalently, and perhaps easier to work with, I give you a line segment which you think of as the radius of the circle). Your job is to start from this and, using a compass and straightedge, construct a square with the same area as the circle in finely many steps.

To make it even more concrete, I give you a line segment and you have to construct a second line segment that's exactly $\sqrt{\pi}$ times as long.

This can't be done: there are only finitely many types of things you're allowed to do with a compass and straightedge, and you can check that each ultimately results in taking a length and multiplying it by an algebraic number. Since $\sqrt{\pi}$ isn't algebraic, it can't be reached.

UPDATE: I guess this was more a response to some of the comments on the other answers than the actual question as stated. To answer that, I'll just repeat what Lulu said in the comments, namely that the term literally refers only to the problem of constructing a square of the same area as a circle, but because this (somewhat surprisingly!) turns out to be essentially the same problem as constructing a square with perimeter equal to the circumference of a circle there is no harm in using the term to reply to either problem.

Daniel McLaury
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  • But the quadrivium is refering to something else entirely-- finding a square with the same perimeter i.e. to find a line that is $\pi/2$ as long as a given line. Can these both be called "squaring the circle"? – fleablood Nov 30 '16 at 20:04
  • Also not all algebraic numbers can be constructed. We can't find the cube root of 2 for instance. (That's called doubling the cube-- finding a cube with twice the volume of a given cube). We can only construct algebraic numbers whose polynomials are of degree of a power of 2. – fleablood Nov 30 '16 at 20:06
  • Properly speaking squaring the circle refers to producing a square with the same area as the circle, but as you know both problems come down to constructing $\pi$ so it's not so terrible to conflate them. And, yes, you cannot construct arbitrary algebraic numbers, but it is true that every number you can construct is algebraic. – Daniel McLaury Nov 30 '16 at 20:44
  • The point of my comment was the op found a book that specifically referred to finding a square with the same parameter as squaring the circle and then later referred to find a square with the same area as "marrying the square to the circle". The question is, was the book wrong or right? I'm not sure. – fleablood Nov 30 '16 at 22:29
  • @fleablood: Edited to address. – Daniel McLaury Dec 02 '16 at 15:38
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I had always heard of squaring the circle to mean using a straightedge and compass to construct a square with the same area of a given circle (or vice versa).

In essence if you are given $r$ you are being asked to construct an $s$ so that $s^2 = \pi r^2$ or in other words, to construct $\sqrt \pi$. This is impossible.

What is being described here is constructing a square with the same perimeter. i.e. given $r$ construct $s$ so that $4s = 2\pi r$. i.e. construct $\pi/2$.

But these are equivalent because if we can construct $\pi/2$ we can construct $\pi$ and $ \sqrt{\pi}$ (see Compass-and-straightedge construction of the square root of a given line?)

Likewise if we can construct $\sqrt{\pi}$ we can construct $\pi$ (see Representing the multiplication of two numbers on the real line) and therefore $\pi/2$.

So I think it is fair to call both of these squaring a circle.

Community
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fleablood
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Very brief answer: The word "squaring" is an old-fashioned version of the word "quadrature". You should be able to take it from there.

Pirx
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