10/02/2021

In Folha, the 'impossible task' of squaring the circle.

O matemático e escritor Lewis Carroll era um dos fascinados pelo problema da quadratura do círculo. Foto: Wikimedia Commons

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

The expression "squaring the circle" has entered the language as a synonym for an impossible task, but the problem—constructing a square with an area equal to that of a given circle using a ruler and compass—has obsessed professionals and amateurs for more than 25 centuries.

No one expressed this obsession better than Charles L. Dodgson, better known as Lewis Carroll, mathematician and author of "Alice in Wonderland." Regarding correspondence with a "quadrature expert," he wrote in 1888: "This deluded visionary filled me with the great ambition of achieving a feat never accomplished by mankind: to convince a 'squarer of the circle' of his error! The value my friend used for π was 3.2: such a large error that I thought I could easily show him that he was wrong. More than twenty letters were exchanged before I sadly convinced myself that I had no chance."

The calculation of the area of a circle dates back to the dawn of civilization. Archimedes (287–212 BC) proved that it is proportional to the square of the circle's radius: the constant of proportionality is represented by the Greek letter π (pi) and its value is 3.1415926535…. Much earlier, the famous Rhind papyrus (Egypt, circa 1800 BC) already contained the approximate value 256/81=3.1604938….

The problem of squaring the circle has the rare distinction of being mentioned in a play: "The Birds" by the Greek Aristophanes (446–386 BC). It is believed that the first to require that the solution use only a ruler and compass was another Greek, Oinopides, who lived around 450 BC. By the 17th century, there were already suspicions that with this requirement the problem was impossible: the Scotsman James Gregory (1638–1675) published a "proof" in 1667, although it was wrong.

In 1837, the Frenchman Pierre Wantzel (1814–1848) showed that quantities that can be constructed with ruler and compass must be solutions of certain polynomial equations with integer coefficients, and deduced from this that the other two classical geometric problems—duplication of the cube and trisection of the angle—are impossible.

To read the full text, visit the newspaper's website.