Squaring the circle

It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge.

Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks).

Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area.

As Proclus wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes: The first known Greek to study the problem was Anaxagoras, who worked on it while in prison.

[8] The more general goal of carrying out all geometric constructions using only a compass and straightedge has often been attributed to Oenopides, but the evidence for this is circumstantial.

[10] Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge.

For example, Newton wrote to Oldenburg in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag.

A 1647 attempt at squaring the circle, Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum by Grégoire de Saint-Vincent, was heavily criticized by Vincent Léotaud.

[18][19] After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts.

[20] As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps.

In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients.

After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision.

[21] In his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle, a claim refuted by John Wallis as part of the Hobbes–Wallis controversy.

[38][39][40] The Victorian-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by his 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:[41] A ridiculing of circle squaring appears in Augustus De Morgan's book 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 declined in popularity after the nineteenth century, and it is believed that De Morgan's work helped bring this about.

[20] Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle.

[42] The mathematical crank Carl Theodore Heisel also claimed to have squared the circle in his 1934 book, "Behold!

"[44] The problem of squaring the circle has been mentioned over a wide range of literary eras, with a variety of metaphorical meanings.

In it, the character Meton of Athens mentions squaring the circle, possibly to indicate the paradoxical nature of his utopian city.

[46] Dante's Paradise, canto XXXIII, lines 133–135, contain the verse: As the geometer his mind applies To square the circle, nor for all his wit Finds the right formula, howe'er he tries Qual è ’l geométra che tutto s’affige per misurar lo cerchio, e non ritrova, pensando, quel principio ond’elli indige, For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise.

[48] Dante uses the circle as a symbol for God, and may have mentioned this combination of shapes in reference to the simultaneous divine and human nature of Jesus.

[45][48] Earlier, in canto XIII, Dante calls out Greek circle-squarer Bryson as having sought knowledge instead of wisdom.

[45] Several works of 17th-century poet Margaret Cavendish elaborate on the circle-squaring problem and its metaphorical meanings, including a contrast between unity of truth and factionalism, and the impossibility of rationalizing "fancy and female nature".

[45] By 1742, when Alexander Pope published the fourth book of his Dunciad, attempts at circle-squaring had come to be seen as "wild and fruitless":[39] Mad Mathesis alone was unconfined, Too mad for mere material chains to bind, Now to pure space lifts her ecstatic stare, Now, running round the circle, finds it square.

Similarly, the Gilbert and Sullivan comic opera Princess Ida features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding perpetual motion.

[50] A similar metaphor was used in "Squaring the Circle", a 1908 short story by O. Henry, about a long-running family feud.

[51] In later works, circle-squarers such as Leopold Bloom in James Joyce's novel Ulysses and Lawyer Paravant in Thomas Mann's The Magic Mountain are seen as sadly deluded or as unworldly dreamers, unaware of its mathematical impossibility and making grandiose plans for a result they will never attain.