The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com].
The challenge is to construct a square with area equal to that of a given circle using only the methods of classical geometry. Thus, only a ruler and compass may be used in the construction and the process must terminate in a finite number of steps.
A circle of unit radius has area π. We must construct a line segment whose length is the square root of π. With this as the side, the square has area π and the job is done. But it is not possible to construct line segments of arbitrary length using the permitted method.
Squaring the circle has attracted attention from outstanding mathematicians for millennia. In the Rhind papyrus, an ancient Egyptian mathematical script, Ahmes the scribe gives a method to construct a square of area nearly equal to that of a circle: cut one-ninth off the diameter and take the remainder as the side of the square. This corresponds to a value 3.16 for π, close to the true value 3.14159 … .
The ancient Greek philosopher Anaxagoras wrote on the problem while he was in prison. Archimedes managed to construct the desired square using the spiral curve that is named after him, but his method goes beyond the limitations of Euclidean geometry, so it does not qualify as a solution. Many others tried to crack the problem, without success. It proved so difficult that the phrase “squaring the circle” became a metaphor for attempting the impossible.
In 1837 the French mathematician Pierre Wantzel (1814–1848) used the theory of fields to show that some ancient geometric problems could not be solved. They included the trisection of angles and the duplication of the cube. A length is constructable only if it can be written using the basic operations of addition, subtraction, multiplication and division together with the extraction of square roots. As an example, if we take a square with unit side, the theorem of Pythagoras shows the length of the diagonal to be the square root of 2.
Wantzel proved that the trisection of an angle and the duplication of a cube (constructing a cube with twice the volume of a given one) are impossible within the strict confines of Euclidean geometry. His proof depended on showing the impossibility of constructing cubed roots of given lengths. But it took another fifty years before the question of squaring the circle was resolved. In 1882, Ferdinand Lindemann showed that the number π is transcendental: it is not the solution of any simple polynomial equation and is therefore not constructable.
The problem of squaring the circle has attracted the attention of amateur mathematicians as well as professionals. Indeed, thousands of false “proofs” have been produced by amateurs (see cover pages of Carl Theodore Heisel’s book above). In 1775, “circle-squarers” had become so numerous that the Scientific Academy in Paris passed a resolution that no alleged proofs would be examined. Purported solutions of the problem continue to appear from time to time.
Leopold Bloom had a keen interest in the matter. There are at least three references in Ulysses to the quadrature of the circle. In the Ithaca episode, we read that Bloom had planned to devote the summer of 1886 to square the circle “and win that million”. Later in the same episode, Joyce writes again of Bloom’s dream of winning a government award of £1 million for the solution. In reality, no such prize was ever on offer.
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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/