His idea used as basis the fact that the Greek mathematicians of his time knew how to square regular polygons. He proposed that if a regular polygon were to be inscribed in a circle, and if the number of sides of the polygon were doubled again and again, the polygon would eventually become indistinguishable from the circle. This solution was still incorrect, because making the polygon coincide with the circle required an infinite number of permutations.

The Europeans of the Renaissance, Leonardo da Vinci among them, tried in vain to solve the problem. They did not realize that it was impossible to square a circle using the straight edge and compass method.

The Scottish mathematician James Gregory (1638 - 1675) proposed that the ratio of the area of any sector of a circle to that of inscribed or circumscribed regular polygons cannot be expressed with a finite number of terms, thus concluding that squaring the circle is impossible.

Ferdinand von Lindemann (1852 - 1939) published in 1882 his proof that pi is not an algebraic, but a transcendental number (not a solution of any polynomial with rational coefficients). This essentially means that there is no way to determine if any physical object or quantity has an exact measure of pi (regardless of the unit system). Because squaring the circle involves the use of pi, Lindemann's proof ended the quest for the plane technique of squaring the circle.

Trisecting an Angle

Another classic mathematical problem is that of trisecting an angle, again with the restriction of using only an unmarked straight edge and a compass. Although there are certain angles that can be trisected with this method, the problem is to trisect an arbitrary angle. It has been proven that this is impossible. It can be solved, however, without the said restriction.

There is no certainty of when this specific problem first arose. It is known that Hippocrates in the 5th century B.C. considered the problem. For centuries the problem of trisecting an angle (using Euclidean constructions) was pondered upon by mathematicians, but these early mathematicians focused on plane methods of trisecting an arbitrary angle, and thus failed.

The first known mathematician who worked on the problem was the Greek Hippias (460 - 400 BC). He came up with the curve called the quadratix, which was originally used for squaring the circle, and was also used to solve the problem of angle trisection. Archimedes (287 - 212 BC) came up with a curve, the Archimedean spiral, and also used it to solve the problem. The Greek mathematician Nicomedes (280 - 210 BC) also worked on the problem. He came up with the curve known as the conchoid, and used it for angle trisection. These solutions, however, break the straight-edge-and-compass restriction.

It was only in 1837 that it was proven that there is no solution for the original problem of trisecting any angle with only an unmarked straight edge and a compass. The French mathematician Pierre Wantzel proved the impossibility of solving the problem under the straight edge and compass restriction.

Antiphon

Antiphon was an orator, writer, teacher, and philosopher. He was a contemporary of Socrates. He died, like Socrates, because of political troubles. There
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