History of the Problems

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Three Famous Problems

### History of the problem

#### Problem Definition:

To construct a square which has the same area as the given circle.

This is equivalent to construct a line segment of length p or its square root.

********** circle_squaring_problem.dwg** *********

#### Origin of the Problem :

The circle is the simplest , and hence the first curve men has created. So naturally from the earliest days
in history, men have had a keen interest in the area and circumference of this curve.

#### Rule of the Game ( Platonian Rule ) :

##### Tools allowed

the compasses and unmarked straight-edge
##### Permitted use of these 2 tools

1. The drawing of a straight line of indefinite length through two given points.
2. The construction of a circle with center at a given point and passing through a second given point.

#### Summary of Early Attempts:

#### Proof of impossibility:

In 1880 Ferndinand von Lindemann (1852 - 1939)
proved that p is transcendental, that it is not the root of any polynomial equation
with rational coefficients. Thus he proved that it is not possible to square the circle with ruler and compass.

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Last Updated Nov 22, 2006

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