Are the Constructions Even Possible?
Many attempts were made to solve the problem of squaring the circle, but it is believed that the ancient Greek mathematicians knew that the construction was impossible. Although the Greeks could not prove that the construction is impossible, 18th and 19th century mathematicians proved that the construction is impossible.
Let us think about what we would need to be able to do in order to construct a square having an area equal to the area of a given circle. Suppose the circle has a radius of length one. Then the area of the circle is p, so the area of the square must also be p. In order for the area of the square to be p, each side of the square must have a length of Öp. In order to square the circle, we would need to construct a segment of length Öp. If we could prove that it is impossible to construct a segment of length Öp, then we would have proved that squaring the circle is impossible.
In 1761, Lambert proved that p is irrational. This alone was not enough to prove that the construction is impossible, but it was a step in the right direction. In 1880, using Euler’s equation (eip + 1 = 0) and the fact that e is transcendental, Lindemann was able to prove that p is transcendental. Since p is transcendental, it is not constructible. Therefore, it is not possible to construct a segment of length p or Öp.
See my references page for a link to a proof that p is transcendental.
