Squaring the circle

Squaring the circle is a classic problem in geometry. The goal is to construct a square with the exact same area as a given circle. To make this a formal mathematical challenge, you must use only a compass and a straightedge. You are limited to a finite number of steps using these two tools. This problem asks if the rules of Euclidean geometry allow for such a construction. For centuries, mathematicians wondered if the basic axioms of geometry implied such a square could exist.

To understand the mechanism, one must look at the relationship between a circle's area and a square's area. The area of a circle depends on the number pi, which is the ratio of a circle's circumference to its diameter. If you want a square to have the same area, the length of its side must be the square root of pi times the radius. Therefore, squaring the circle requires the ability to construct a specific length related to pi. In geometry, a construction is possible only if the length is an algebraic number. This means the number must be a solution to a polynomial equation with rational coefficients.

Historically, mathematicians used different methods to find the area of a circle. Ancient Babylonian mathematicians used the approximation 3.125 around 2000 BCE. At a similar time, ancient Egyptian mathematicians used a different approximation. The Old Testament Books of Kings used an even simpler estimate. Ancient Indian mathematics, found in the Shulba Sutras, also used several approximations. Later, Archimedes proved a specific formula for the area of a circle. In the third century CE, Liu Hui used Archimedes' methods to find accurate approximations in China. By the fifth century, Zu Chongzhi found a very precise approximation called Milü.

The specific problem of constructing the square comes from Greek mathematics. Greek mathematicians knew how to turn any polygon into a square of equal area. They used these constructions to compare the areas of different shapes geometrically. This motivated the search for ways to compare circles to other shapes. Anaxagoras was the first known Greek to study this problem while in prison. Hippocrates of Chios tried to find a shape called a lune that could be squared. Antiphon the Sophist suggested a method called exhaustion, where you increase the sides of a polygon to fill a circle. However, Eudemus argued that magnitudes can be divided without limit, so the area would never be fully used up.

For a long time, many people believed the problem could be solved. In 1676, Isaac Newton wrote about the idea of squaring curve lines geometrically. In 1647, Grégoire de Saint-Vincent attempted a solution in his work, *Opus geometricum*. While he failed to square the circle, he succeeded in the quadrature of the hyperbola. This work helped lead to the development of the natural logarithm. In 1667, James Gregory attempted a proof of impossibility using the algebraic properties of pi. Although his proof was faulty, it was a significant step toward the truth.

The mystery was finally solved in 1882 by Ferdinand von Lindemann. He proved that pi is a transcendental number. A transcendental number is a number that is not the root of any polynomial equation with rational coefficients. This proof relied on the Lindemann–Weierstrass theorem. Earlier, in 1761, Johann Heinrich Lambert had proved that pi is an irrational number. But being irrational was not enough to prove the construction was impossible. Because pi is transcendental, the length required to square the circle cannot be constructed with a compass and straightedge. This settled the debate for professional mathematicians.

Even though an exact solution is impossible, mathematicians still create approximate constructions. These methods produce results that are very close to the true area. In 1685, Adam Adamandy Kochański created a simple construction that is accurate to the fifth decimal place. In 1849, Jacob de Gelder used the Milü approximation to create a construction accurate to six decimal places. The famous mathematician Srinivasa Ramanujan also developed a geometric construction for this approximation in 1914. Other mathematicians have used the golden ratio to find near-perfect approximations.

Squaring the circle relates to other famous impossible problems from antiquity. These include doubling the cube and trisecting an angle. Like squaring the circle, these cannot be solved with only a compass and straightedge. However, they are different because they involve the roots of cubic equations rather than transcendental numbers. In modern math, the term "quadrature" is often used when calculus is allowed. In the past, squaring was the term used for finding the area under a curve. Today, we recognize that while the perfect square is unreachable, the pursuit of its approximation has driven much of mathematical discovery.