Oh, this again. Humanity’s eternal struggle with the obvious. Fine, let’s talk about the problem of constructing equal-area shapes . Try not to strain your delicate sensibilities with the sheer weight of mathematical futility.

For other uses, should your mind wander to lesser topics, see Squaring the circle (disambiguation) , Square the Circle (disambiguation) , and Squared circle (disambiguation) . And for the love of all that’s geometrically sound, do not confuse this with the utterly unrelated, yet equally frustrating, concept of a Square peg in a round hole . One is a mathematical impossibility, the other is just poor planning, though both tend to reveal a distinct lack of insight.

The image you see, of a square and a circle, both mockingly displaying an area equal to π , is a testament to what cannot be. In 1882, it was definitively, irrevocably, and quite frankly, predictably proven that this seemingly simple figure cannot be constructed in a finite number of steps with an idealized compass and straightedge . A shame, really, for all the poor souls who wasted centuries trying.

This particular conundrum is merely one facet of a grander, often misunderstood, mathematical constant: π . A constant that has haunted mathematicians for millennia, its value, 3.1415926535897932384626433…, stretching out into an eternity of non-repeating digits, much like my patience.

Consider its various entanglements:

Uses

• The fundamental Area of a circle , a concept seemingly simple enough, yet its precise measurement proved to be a Gordian knot for ancient minds.
• The Circumference , the boundary that defines the circle, forever linked to its diameter by this elusive number.
• Its ubiquitous Use in other formulae , demonstrating its pervasive influence across vast swathes of mathematics and physics, a constant reminder of its fundamental nature.

Properties

• Its Irrationality , a property that means it cannot be expressed as a simple fraction, a fact proven by Johann Heinrich Lambert in 1761, a significant hurdle for any exact construction.
• Its Transcendence , a far more profound characteristic, meaning it’s not the root of any non-zero polynomial equation with rational coefficients. This, as we shall see, was the final nail in the coffin for circle squarers.

Value

• The ancient, yet surprisingly effective, observation that it is Less than 22/7 , a fractional approximation that served humanity for centuries.
• Numerous Approximations throughout history, each a testament to human ingenuity and, often, stubbornness.
• The Milü , a particularly accurate approximation from Chinese mathematics, showing that precision wasn’t solely a Western pursuit.
• Madhava’s correction term , an early foray into infinite series by Indian mathematicians, pushing the boundaries of numerical accuracy.
• The bizarre pursuit of Memorization , where individuals dedicate vast swathes of their memory to reciting its digits, a curious hobby for those with an abundance of time.

People

A parade of individuals, some brilliant, others merely persistent, who grappled with this constant:

• Archimedes , whose method of exhaustion laid crucial groundwork.
• Liu Hui and Zu Chongzhi , Chinese mathematicians who achieved remarkable approximations.
• Aryabhata and Madhava of Sangamagrama , pivotal figures in Indian mathematics.
• Jamshīd al-Kāshī , Ludolph van Ceulen , François Viète , Seki Takakazu , Takebe Kenko , William Jones , John Machin , William Shanks , Srinivasa Ramanujan , John Wrench , the Chudnovsky brothers , and Yasumasa Kanada , all contributing to the ever-growing precision of π .

History

• The Chronology of computation of π , a timeline of humanity’s obsession.
• “A History of Pi”, a dedicated chronicle to this numerical saga.

In culture

• The infamous Indiana pi bill , a delightful example of legislative ignorance attempting to redefine mathematical truth.
• Pi Day , a celebration of the constant, perhaps a touch too enthusiastic for my taste.

Related topics

• Squaring the circle itself, the subject of our current, rather tedious, discussion.
• The Basel problem , a different, though equally engaging, mathematical challenge.
• The curious case of the Six nines in π , an anomaly that fuels speculation.
• Other topics related to π , because apparently, one constant isn’t enough.
• And for those who prefer to argue about notation, Tau , a rival constant, though frankly, the universe cares little for your preferences.

Problem of constructing equal-area shapes

Squaring the circle is a problem rooted deep in geometry , first posited in the fertile intellectual grounds of Greek mathematics . It presents a rather straightforward, yet ultimately unattainable, challenge: to construct a square whose area of a given circle is precisely equal to that of a given circle, using only a finite sequence of steps with an idealized compass and straightedge . One might wonder why such a specific constraint. The difficulty of this seemingly simple task quickly escalated, prompting profound inquiries into the very axioms of Euclidean geometry . It forced mathematicians to question whether the fundamental principles concerning the existence of lines and circles inherently guaranteed the existence of such a square. It’s a question of what can truly be built within the confines of a defined system, a surprisingly deep rabbit hole for something so seemingly trivial.

The grand, centuries-long pursuit culminated in 1882 when the task was finally, unequivocally, proven to be impossible. This definitive proof emerged as a direct consequence of the Lindemann–Weierstrass theorem , a mathematical sledgehammer that established the transcendence of pi (

π

{\displaystyle \pi }

). For those who prefer plain language over mathematical jargon, this means that π is not the root of any polynomial equation with rational coefficients. The implication for squaring the circle had been understood for decades: if π were transcendental, the construction would be fundamentally impossible. However, the concrete proof of π ’s transcendental nature remained elusive until Ferdinand von Lindemann delivered it in 1882, putting an end to countless futile attempts. Of course, approximate constructions, offering various degrees of non-perfect accuracy, have always existed, and many such methods have been devised, serving as a kind of mathematical consolation prize for those who refuse to accept definitive limits.

Despite the irrefutable proof of its impossibility, the allure of “squaring the circle” continued to captivate, and unfortunately, still does. It has, predictably, become a common fixture in the annals of mathematical crankery , attracting individuals who, despite all evidence, believe they’ve found the secret. The phrase “squaring the circle” has, perhaps deservedly, evolved into a popular metaphor for attempting the utterly impossible, or at least, the wildly impractical. The term “quadrature of the circle” is often used interchangeably, though it can also refer to more legitimate approximate or numerical methods for determining the area of a circle . More broadly, the concept of quadrature or squaring can be applied to other two-dimensional geometric figures, though none have generated quite the same level of persistent, misguided obsession.

History

The human fascination with the area of a circle is ancient, predating the specific Greek formulation of the squaring problem. Methods to calculate the approximate area of a given circle, precursors to the exact squaring challenge, were already well-established in a multitude of ancient cultures. These early approaches are best understood by examining the approximation to π they yielded. Around 2000 BCE, for instance, the Babylonian mathematicians utilized an approximation of

π ≈

25 8

= 3.125

{\displaystyle \pi \approx {\tfrac {25}{8}}=3.125}

. Roughly contemporaneously, the ancient Egyptian mathematicians employed a value of

π ≈

256 81

≈ 3.16

{\displaystyle \pi \approx {\tfrac {256}{81}}\approx 3.16}

. Over a millennium later, the Old Testament Books of Kings recorded a far simpler, though less accurate, approximation of

π ≈ 3

{\displaystyle \pi \approx 3}

, illustrating that practical utility sometimes trumped rigorous precision. Ancient Indian mathematics , as meticulously documented in texts like the Shatapatha Brahmana and Shulba Sutras , explored several distinct approximations for π , showcasing a varied and evolving understanding. It was Archimedes , through his brilliant method of exhaustion , who provided a rigorous formula for the area of a circle, establishing the bounds that

3

10 71

≈ 3.141 < π < 3

1 7

≈ 3.143

{\displaystyle 3,{\tfrac {10}{71}}\approx 3.141<\pi <3,{\tfrac {1}{7}}\approx 3.143}

. This level of precision, achieved without modern calculus, was a monumental intellectual achievement. In Chinese mathematics , the third century CE saw Liu Hui derive even more accurate approximations, employing a methodology remarkably similar to Archimedes’. Centuries later, in the fifth century, Zu Chongzhi refined this further, arriving at

π ≈ 355

/

113 ≈ 3.141593

{\displaystyle \pi \approx 355/113\approx 3.141593}

, an approximation so precise it became known as Milü and remained the most accurate for centuries.

The shift from merely approximating the area to demanding an exact construction of a square with equivalent area originates, as most foundational geometric problems do, from Greek mathematics . These early Greek mathematicians had already mastered compass and straightedge constructions to transform any polygon into a square of an equivalent area. This wasn’t merely an academic exercise; it was a practical tool for geometrically comparing the areas of various polygons, a stark contrast to the numerical area computations that are commonplace in modern mathematics. As the philosopher Proclus observed many centuries later, this success with polygons naturally spurred the search for analogous methods for non-polygonal shapes, particularly the enigmatic circle:

“Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. For if a parallelogram is found equal to any rectilinear figure, it is worthy of investigation whether one can prove that rectilinear figures are equal to figures bound by circular arcs.”

This quote, delivered with an almost poetic sense of historical inevitability, highlights the logical progression of their inquiries. The success with polygons fueled a perfectly reasonable, if ultimately doomed, ambition to extend this mastery to the curved.

Indeed, some seemingly promising, yet ultimately misleading, partial solutions offered false hope for an extended period. For instance, the shaded figure known as the lune of Hippocrates (discovered by Hippocrates of Chios ) demonstrably has an area equal to that of a triangle. Such discoveries, where curved shapes could be squared, must have tantalized those seeking a universal solution.

The earliest known Greek to delve into this problem was Anaxagoras , who, rather inconveniently, worked on it while imprisoned—a testament to the enduring power of intellectual curiosity, even in confinement. Hippocrates of Chios made significant inroads by identifying specific shapes bounded by circular arcs, like the aforementioned lune of Hippocrates , which could be squared. This was a crucial distinction, demonstrating that not all curved figures were inherently intractable. Antiphon the Sophist proposed a method involving inscribing regular polygons within a circle and progressively doubling their number of sides, believing that this process would eventually “fill up” the area of the circle. This is, in essence, an early conceptualization of the method of exhaustion . Since any polygon can be squared, he reasoned, the circle could, by extension, also be squared. However, his contemporary, Eudemus of Rhodes , countered that magnitudes could be divided infinitely, implying that the circle’s area would never be entirely “used up” by finite polygonal approximations, a surprisingly modern insight. Around the same time, Bryson of Heraclea advanced an argument based on the existence of circles both larger and smaller than the target, suggesting there must be one of equal area. This principle, while intuitive, foreshadows the modern intermediate value theorem in its logical structure. The broader commitment to performing all geometric constructions solely with a compass and straightedge is often attributed to Oenopides , though the evidence supporting this claim remains largely circumstantial.

The challenge of determining the area beneath an arbitrary curve, a problem now known as integration in calculus or quadrature in numerical analysis , was historically referred to as “squaring” long before the sophisticated tools of calculus were conceived. In that era, without the luxury of analytical methods, it was universally assumed that any “squaring” operation had to be achieved through explicit geometric constructions, specifically using only a compass and straightedge. For example, Isaac Newton , in a letter to Henry Oldenburg in 1676, expressed his confidence: “I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically.” This illustrates the prevailing mindset. In contemporary mathematics, however, the terms have diverged: “quadrature” typically permits the use of calculus-based methods, whereas “squaring the curve” (or circle, in this context) strictly adheres to the original, more restrictive geometric constraints.

A notable attempt at squaring the circle occurred in 1647 with Grégoire de Saint-Vincent ’s Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum. This work faced considerable criticism, particularly from Vincent Léotaud . Despite this, de Saint-Vincent achieved a significant success in the quadrature of the hyperbola , a feat that made him one of the earliest pioneers in the development of the natural logarithm . Following in de Saint-Vincent’s wake, James Gregory made his own attempt to prove the impossibility of squaring the circle in his 1667 treatise, Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola). While his proof ultimately contained flaws, it holds historical significance as the first academic paper to approach the problem by examining the algebraic properties of π . Later, in 1761, Johann Heinrich Lambert provided a crucial piece of the puzzle by definitively proving that π is an irrational number . This was a major step, but not the final one. The definitive proof had to wait until 1882, when Ferdinand von Lindemann successfully demonstrated the stronger claim that π is a transcendental number . This monumental achievement simultaneously and irrevocably proved the impossibility of squaring the circle using only compass and straightedge.

After Lindemann’s impossibility proof, the matter was considered definitively settled by the professional mathematical community. Its subsequent historical trajectory is largely characterized by a predictable, and frankly tiresome, proliferation of pseudomathematical attempts at circle-squaring constructions, predominantly by amateurs, and the equally predictable, yet necessary, debunking of these misguided efforts. However, not all post-proof work was futile; several legitimate mathematicians, including the brilliant Srinivasa Ramanujan , developed compass and straightedge constructions that achieved remarkably accurate approximations in a minimal number of steps, proving that even in impossibility, elegance can be found.

It’s worth noting that squaring the circle belongs to a trio of classical problems from antiquity, all famed for their compass-and-straightedge impossibility. The other two are doubling the cube and trisecting the angle . While equally unsolvable with the restricted tools, their nature differs from squaring the circle. Their solutions involve roots of cubic equations , making them algebraic problems, unlike the transcendental nature of squaring the circle. Consequently, more powerful geometric methods beyond the standard compass and straightedge, such as neusis construction or mathematical paper folding , can be employed to construct solutions to these problems. This subtle distinction often escapes the less discerning enthusiast.

Impossibility

The very heart of the problem of squaring the circle, when confined to the strictures of compass and straightedge construction, lies in the requirement to construct a specific length:

π

{\displaystyle {\sqrt {\pi }}}

. This length represents the side of a square whose area would be precisely equal to that of a unit circle. If this value,

π

{\displaystyle {\sqrt {\pi }}}

, were a constructible number , then by established geometric principles, π itself would also be constructible. However, in 1837, Pierre Wantzel delivered a critical blow, demonstrating that any length constructible with a compass and straightedge must be a solution to a certain type of polynomial equation with rational coefficients. This means that constructible lengths are, by definition, algebraic numbers . Therefore, for the circle to be squared using these classical tools, π would necessarily have to be an algebraic number.

The final, undeniable proof arrived in 1882, courtesy of Ferdinand von Lindemann . He achieved the monumental task of proving the transcendence of π , thereby conclusively demonstrating the impossibility of the construction. Lindemann’s ingenious approach built upon the earlier work of Charles Hermite , who in 1873 had proven the transcendence of Euler’s number

e

{\displaystyle e}

. Lindemann then cleverly leveraged Euler’s identity , the elegant equation

e

i π

= − 1.

{\displaystyle e^{i\pi }=-1.}

This identity, in itself, immediately reveals that π must be an irrational number since a rational power of a transcendental number (like e) would remain transcendental, and -1 is most certainly not transcendental. Lindemann, however, extended this argument further, employing the powerful Lindemann–Weierstrass theorem concerning the linear independence of algebraic powers of

e

{\displaystyle e}

. This ultimately allowed him to establish that π is indeed transcendental, sealing the fate of the circle-squaring problem once and for all. It was, for anyone paying attention, a rather satisfying conclusion to centuries of misdirected effort.

Of course, if one is willing to “bend the rules” – a euphemism for introducing supplemental tools, allowing an infinite number of compass-and-straightedge operations, or relocating the problem to certain non-Euclidean geometries – then squaring the circle becomes possible, albeit in a modified, often disingenuous, sense. For example, Dinostratus’ theorem demonstrates that if the quadratrix of Hippias is somehow already given (a rather significant “if,” as constructing this curve itself is not possible with just compass and straightedge), then a square and circle of equal areas can be constructed from it. Similarly, the Archimedean spiral offers another avenue for such a construction under similar relaxed conditions. However, it’s crucial to understand that neusis construction , while more powerful than pure compass and straightedge, still cannot square the circle, as it is limited to constructing specific algebraic ratios and cannot produce the transcendental ratio required for this particular problem.

And then there’s the delightful complication of non-Euclidean geometry . While the circle cannot be squared in conventional Euclidean space , it can sometimes be achieved in hyperbolic geometry , under very specific interpretations of the terms. The hyperbolic plane, you see, doesn’t contain traditional squares (quadrilaterals with four right angles and four equal sides). Instead, it features “regular quadrilaterals”—shapes possessing four equal sides and four equal angles, though these angles are distinctly sharper than right angles. Within this peculiar geometric landscape, an infinite (and countable ) number of pairs of constructible circles and constructible regular quadrilaterals of equal area exist. However, these are constructed simultaneously. There is no method to begin with an arbitrary regular quadrilateral and then construct a circle of equal area. Symmetrically, the inverse is also true: there’s no way to start with an arbitrary circle and construct a regular quadrilateral of equal area. Furthermore, for sufficiently large circles in hyperbolic geometry, no such quadrilateral even exists. It’s almost as if the universe enjoys adding layers of complexity just to spite our simple demands.

Approximate constructions

Since squaring the circle exactly with a compass and straightedge is, as established, a fool’s errand, the next best thing, for those who simply must draw something, is an approximation. These constructions provide lengths that are merely close to π . It doesn’t take a genius to realize that any given rational approximation of π can be translated into a corresponding compass and straightedge construction . The catch, and there’s always a catch, is that such constructions tend to be incredibly convoluted and lengthy, often offering a meager increase in accuracy for a disproportionate amount of effort. After the exact problem was officially declared unsolvable (much to the relief of anyone with a modicum of sense), some mathematicians channeled their ingenuity into devising approximations that were, at least, aesthetically simple, even if their precision was still, by definition, imperfect.

Construction by Kochański

One of the more historically significant approximate compass-and-straightedge constructions emerged from a 1685 paper by the Polish Jesuit Adam Adamandy Kochański . This particular method yields an approximation of π that deviates in the 5th decimal place. While far more precise numerical approximations of π were already available at the time, Kochański’s construction earned its place in history due to its remarkable simplicity. Observe the left diagram:

π

≈

|

P

3

P

9

|

|

P

1

P

2

|

=

40 3

− 2

3

= 3.141

5

33

338

…

{\displaystyle {\begin{aligned}\pi &\approx {\frac {|P_{3}P_{9}|}{|P_{1}P_{2}|}}={\sqrt {{\frac {40}{3}}-2{\sqrt {3}}}}\[3mu]&=3.141;5{\color {red}33,338;\ldots }\end{aligned}}}

In the very same work, Kochański also meticulously derived a sequence of progressively more accurate rational approximations for π , demonstrating that even within the confines of approximation, there was a drive for refinement.

Constructions using 355/113

Jacob de Gelder, in 1849, unveiled a construction based on the approximation:

π

≈

355 113

= 3.141

592

920

…

{\displaystyle {\begin{aligned}\pi &\approx {\frac {355}{113}}\[3mu]&=3.141;592{\color {red};920;\ldots }\end{aligned}}}

This particular value is impressively accurate to six decimal places, a level of precision that was first achieved in China in the 5th century (known as Milü ) and only reached Europe by the 17th century. De Gelder, rather than constructing the side of the square directly, focused on deriving the value

A H

¯

=

4

2

7

2

8

2

.

{\displaystyle {\overline {AH}}={\frac {4^{2}}{7^{2}+8^{2}}}.}

The accompanying illustration clearly depicts de Gelder’s construction. Years later, in 1914, the brilliant Indian mathematician Srinivasa Ramanujan provided an alternative geometric construction that yielded the exact same highly accurate approximation, proving that elegance in approximation wasn’t confined to a single approach.

Constructions using the golden ratio

An approximate construction, meticulously detailed by E. W. Hobson in 1913, manages to achieve accuracy up to three decimal places. Hobson’s construction is mathematically equivalent to employing the approximate value of

π

≈

6 5

⋅

(

1 + φ

)

= 3.141

640

…

,

{\displaystyle {\begin{aligned}\pi &\approx {\frac {6}{5}}\cdot \left(1+\varphi \right)\[3mu]&=3.141;{\color {red}640;\ldots },\end{aligned}}}

where φ is the renowned golden ratio , defined as

φ

1 2

(

1 +

5

)

{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}}

. This same approximate value resurfaces in a 1991 construction by Robert Dixon . More recently, in 2022, Frédéric Beatrix presented a geometrographic construction that accomplishes this approximation in a mere 13 steps, a testament to the ongoing pursuit of elegant, if imperfect, solutions.

Second construction by Ramanujan

In 1914, Srinivasa Ramanujan , a man whose intuition often outpaced formal proof, presented yet another construction. This one was equivalent to adopting an extraordinarily precise approximate value for π :

π

≈

(

9

2

19

2

22

)

1 4

=

2143 22

4

= 3.141

592

65

2

582

…

{\displaystyle {\begin{aligned}\pi &\approx \left(9^{2}+{\frac {19^{2}}{22}}\right)^{\frac {1}{4}}={\sqrt[{4}]{\frac {2143}{22}}}\[3mu]&=3.141;592;65{\color {red}2;582;\ldots }\end{aligned}}}

This remarkable formula yields accuracy to eight decimal places of π , a truly astonishing feat for a geometric construction. Ramanujan, with his characteristic blend of genius and cryptic precision, described the construction of the line segment OS in his own words:

“Let AB (Fig.2) be a diameter of a circle whose centre is O. Bisect the arc ACB at C and trisect AO at T. Join BC and cut off from it CM and MN equal to AT. Join AM and AN and cut off from the latter AP equal to AM. Through P draw PQ parallel to MN and meeting AM at Q. Join OQ and through T draw TR, parallel to OQ and meeting AQ at R. Draw AS perpendicular to AO and equal to AR, and join OS. Then the mean proportional between OS and OB will be very nearly equal to a sixth of the circumference, the error being less than a twelfth of an inch when the diameter is 8000 miles long.”

Reading that, you almost feel the universe sighing in exasperation.

Incorrect constructions

In his twilight years, the otherwise astute English philosopher Thomas Hobbes famously convinced himself that he had, against all mathematical wisdom, succeeded in squaring the circle. This rather embarrassing claim was swiftly and decisively refuted by John Wallis , becoming a notable point of contention in the Hobbes–Wallis controversy . It just goes to show that even great minds can succumb to delusion when faced with an enticing, impossible problem. Throughout the 18th and 19th centuries, a pair of persistent, yet utterly false, notions took root among aspiring circle-squarers: that the problem was somehow intertwined with the elusive longitude problem (a practical and highly remunerative challenge), and that a substantial financial reward awaited anyone who could provide a solution. This combination of misplaced hope and imagined riches fueled a veritable army of would-be mathematical saviors.

In 1851, John Parker published a book, Quadrature of the Circle, wherein he audaciously claimed to have squared the circle. His method, predictably, did no such thing; it merely produced an approximation of π accurate to a mere six digits—hardly a revolutionary achievement, and certainly not an exact squaring.

The Victorian -era polymath Charles Lutwidge Dodgson, better known by his literary nom de plume Lewis Carroll , harbored a keen interest in discrediting illogical circle-squaring theories. In a diary entry from 1855, Dodgson listed various books he intended to write, among them one titled “Plain Facts for Circle-Squarers.” In the introduction to “A New Theory of Parallels,” Dodgson recounted his rather futile attempt to illuminate the logical errors of a pair of circle-squarers, lamenting:

“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.”

One can almost hear the cosmic sigh of resignation.

A particularly scathing and enduring ridiculing of circle squaring can be found in Augustus De Morgan ’s A Budget of Paradoxes, a work published posthumously by his widow in 1872. De Morgan had originally serialized the content as articles in The Athenæum and was in the process of revising it for book publication at the time of his death. Circle squaring, perhaps thanks in part to De Morgan’s incisive wit and relentless logic, began a slow decline in popularity after the nineteenth century, though never truly vanishing entirely.

Heisel’s 1934 book

Even after the impossibility had been firmly established, the human capacity for self-delusion persisted. In 1894, the amateur mathematician Edwin J. Goodwin boldly claimed to have developed a method to square the circle. His technique, unsurprisingly, did not accurately square the circle; instead, it yielded an incorrect area of the circle, effectively redefining π as equal to 3.2. Goodwin then proceeded to propose the infamous Indiana pi bill in the Indiana state legislature, seeking to legally mandate the use of his method in education without having to pay him royalties. The bill, a testament to legislative oversight, passed the state house without a single objection. Fortunately, amidst a growing chorus of ridicule from the press and the timely intervention of Purdue University professor C.A. Waldo, the bill was ultimately tabled in the Senate and never brought to a vote. A narrow escape from mathematical absurdity being enshrined in law.

Not to be outdone in the realm of mathematical crankery , Carl Theodore Heisel also proclaimed his success in squaring the circle in his 1934 book, Behold!: the grand problem no longer unsolved: the circle squared beyond refutation. Paul Halmos , a respected mathematician, succinctly and accurately referred to the book as a “classic crank book.” Some delusions, it seems, simply refuse to die.

In literature

The problem of squaring the circle, with its tantalizing impossibility, has permeated a vast spectrum of literary eras, acquiring a diverse array of metaphorical meanings along the way. Its literary presence dates back to at least 414 BC, when Aristophanes ’ play, The Birds , was first performed. In it, the character Meton of Athens makes a reference to squaring the circle, quite possibly to underscore the inherently paradoxical and utopian nature of his envisioned city.

Dante Alighieri ’s Paradise , specifically canto XXXIII, lines 133–135, immortalizes the problem with the evocative 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 serves as a potent symbol for a task that lies utterly beyond human comprehension, a challenge he poignantly compares to his own profound inability to fully grasp the divine splendor of Paradise. Dante’s imagery also subtly recalls a passage from Vitruvius , famously brought to life centuries later in Leonardo da Vinci ’s iconic Vitruvian Man , depicting a human figure simultaneously inscribed within both a circle and a square. Dante, using the circle as an established symbol for God, may have invoked this combination of shapes to allude to the dual divine and human nature of Jesus. Earlier in Paradise, in canto XIII, Dante pointedly criticizes the Greek circle-squarer Bryson of Heraclea for pursuing mere knowledge over true wisdom, a distinction many still fail to grasp.

The prolific 17th-century poet Margaret Cavendish, Duchess of Newcastle-upon-Tyne , explored the circle-squaring problem and its rich metaphorical implications across several of her works. She used it to highlight contrasts between the unity of truth and the fragmentation of factionalism, and perhaps most intriguingly, to symbolize the inherent impossibility of fully rationalizing “fancy and female nature”—a rather pointed observation, if I do say so myself. By 1742, when Alexander Pope released the fourth book of his satirical masterpiece, The Dunciad , attempts at circle-squaring had firmly cemented their reputation as “wild and fruitless” endeavors:

“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 that humorously enumerates the impossible aspirations of the women’s university established by the titular character, including the pursuit of perpetual motion . Among these unattainable goals, the lyrics declare: “And the circle – they will square it/Some fine day.” A charming, if naive, sentiment.

The sestina , a complex poetic form first employed in the 12th century by Arnaut Daniel , has been interpreted as metaphorically squaring the circle. This is due to its distinctive structure: a square number of lines (six stanzas, each comprising six lines) combined with a circular, repeating scheme of six specific words. Spanos (1978) suggests that this form evokes a symbolic meaning where the circle represents heaven, and the square, the earth. A strikingly similar metaphor was central to “Squaring the Circle,” a 1908 short story by O. Henry , which chronicles a protracted family feud. In this narrative, the circle symbolizes the natural world, while the square represents the city, the realm of human construction and conflict.

In more contemporary works, figures obsessed with squaring the circle, such as Leopold Bloom in James Joyce ’s monumental novel Ulysses and Lawyer Paravant in Thomas Mann ’s The Magic Mountain , are often portrayed as tragically deluded or as unworldly dreamers. They remain blissfully unaware of the mathematical impossibility of their quest, yet they concoct grandiose plans for a result that will forever elude their grasp. It’s a perennial human flaw, this pursuit of the impossible, often at the expense of understanding the possible.

See also

• Mrs. Miniver’s problem – A problem concerning the areas of intersecting circles, a far more solvable conundrum.
• Round square copula – A philosophical examination of oxymorons, an intellectual exercise that, unlike squaring the circle, can actually be completed.
• Squircle – A fascinating shape that exists somewhere between a square and a circle, a compromise for those who can’t choose.
• Tarski’s circle-squaring problem – A problem concerning the cutting and reassembling of a disk into a square, a different kind of challenge entirely.