Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
It had been known for decades that the construction would be impossible if π were transcendental (not an algebraic irrational number or the root of any polynomial with rational coefficients), which was first proved in 1882 by Lindemann–Weierstrass theorem. The expression ‘squaring the circle’ is sometimes used as a metaphor for trying to do the impossible.
Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π. The term ‘quadrature of the circle’ is sometimes used to mean the same thing as squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.
Methods to approximate the area of a given circle with a square, which can be thought of as a precursor problem to squaring the circle, were known already to Babylonian mathematicians. Indian mathematicians also found an approximate method, though less accurate, documented in the Shulba Sutras. Archimedes proved the formula for the area of a circle (A = πr2, where r is the radius of the circle).
The first known Greek to be associated with the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios squared certain lunes (a crescent-shaped figure formed on a sphere or plane by two arcs intersecting at two points), in the hope that it would lead to a solution. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up. The problem was even mentioned in Aristophanes’s play ‘The Birds.’
It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge). Scottish mathematician James Gregory attempted a proof of its impossibility in ‘Vera Circuli et Hyperbolae Quadratura’ (‘The True Squaring of the Circle and of the Hyperbola’) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π. It was not until 1882 that German mathematician Ferdinand von Lindemann rigorously proved its impossibility.
The Victorian-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by the pseudonym Lewis Carroll, also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write including one called ‘Plain Facts for Circle-Squarers.’ In the introduction to ‘A New Theory of Parallels,’ Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating:
‘The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.’
A ridiculing of circle-squaring appears in British mathematician Augustus de Morgan’s A Budget of Paradoxes published posthumously by his widow in 1872. Having originally published the work as a series of articles in the Athenæum, he was revising it for publication at the time of his death. Circle squaring was very popular in the nineteenth century, but hardly anyone indulges in it today and it is believed that de Morgan’s work helped bring this about.
The two other classical problems of antiquity, famed for their impossibility, were ‘doubling the cube’ and ‘trisecting the angle.’ Like squaring the circle, these cannot be solved by compass-and-straightedge methods. However, unlike squaring the circle, they can be solved by the slightly more powerful construction method of origami.
The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number √π. If √π is constructible, it follows from standard constructions that π would also be constructible. In 1837, French mathematician Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients. Thus, constructible lengths must be algebraic numbers.
If the problem of the quadrature of the circle could be solved using only compass and straightedge, then π would have to be an algebraic number. Swiss polymath Johann Heinrich Lambert conjectured that π was not algebraic, that is, a transcendental number, in 1761.
Transcendental numbers are a type of irrational number (real numbers that cannot be written as a complete ratio of two integers), but unlike other irrational numbers they also cannot be found as a result of an algebraic equation with integer coefficients (constants placed before and multiplying the variable). He did this in the same paper in which he proved its irrationality, even before the general existence of transcendental numbers had been proven. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of π and so showed the impossibility of this construction.
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