Geometric Problems of Classical Greece: Squaring the Circle – 02/09/2021 – Marcelo Viana

The phrase “square of the circle” has been introduced into the language as a synonym for an impossible task, but the problem – building a square with a ruler and compass with an area equal to that of a particular circle – has posed professionals and amateurs alike for longer than possessed 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 the correspondence with a “quadrature expert” he wrote in 1888: “This deluded visionary filled me with the great ambition to achieve an achievement that mankind has never achieved: to convince a“ circle ”of its mistake! The value the friend used for π was 3.2: an error so large that I thought I could easily show that I was wrong. More than twenty letters were exchanged before I was sadly convinced that I had no chance. “

The calculation of the area of ​​the circle goes back to the beginning of civilization. Archimedes (287–212 BC) has proven that it is proportional to the square of the radius of the circle: the constant of proportionality is represented by the Greek letter π (pi) and its value is 3.1415926535 …. Long before that, the famous Rhind papyrus (Egypt, around 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 one piece: “The birds” of the Greek Aristophanes (446 – 386 BC). It is believed that the first to ask for the solution to using just a ruler and compass was another Greek, Oinópides, who lived around 450 BC. Chr. Lived. As early as the 17th century there was suspicion that this requirement made the problem impossible: The Scot James Gregory (1638–1675) published a “proof” in 1667, although this was wrong.

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

The Swiss Johann Heinrich Lambert (1728 – 1777) had shown in 1761 that π is an irrational number (it cannot be written as a fraction of whole numbers) and assumed that it is transcendent, that is, it is not a solution to a polynomial equation. This conjecture was proven in 1882 by the German Ferdinand von Lindemann (1852 – 1939), who showed once and for all that it is impossible to square the circle. No ‘square circle’ was discouraged by this detail …

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