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transcendental ratio required for this problem 28 approximate constructions edit although squaring the circle exactly with compass and straightedge is impossible approximations to squaring the circle can be given by constructing lengths close to π displaystyle pi it takes only elementary geometry to convert any given rational approximation of π displaystyle pi into a corresponding compass and straightedge construction but such constructions tend to be very long winded in comparison to the accuracy they achieve 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 construction by kochański edit kochański s approximate construction continuation with equal area circle and square r displaystyle r denotes the initial radius one of many early historical approximate compass and straightedge constructions is from a 1685 paper by polish jesuit adam adamandy kochański producing an approximation diverging from π displaystyle pi in the 5th decimal place although much more precise numerical approximations to π displaystyle pi were already known kochański s construction has the advantage of being quite simple 29 in 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 same work kochański also derived a sequence of increasingly accurate rational approximations for π displaystyle pi 30 constructions using 355 113 edit jacob de gelder s 355 113 construction ramanujan s 355 113 construction jacob de gelder published in 1849 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 value is accurate to six decimal places and has been known in china since the 5th century as milü and in europe since the 17th century 31 gelder did not construct the side of the square it was enough for him to find the value a h 4 2 7 2 8 2 displaystyle overline ah frac 4 2 7 2 8 2 the illustration shows de gelder s construction in 1914 indian mathematician srinivasa ramanujan gave another geometric construction for the same approximation 21 22 constructions using the golden ratio edit hobson s golden ratio construction dixon s golden ratio construction beatrix s 13 step construction an approximate construction by e w hobson in 1913 31 is accurate to three decimal places hobson s construction corresponds to an 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 displaystyle varphi is the golden ratio φ 1 2 1 5 displaystyle varphi tfrac 1 2 bigl 1 sqrt 5 bigr the same approximate value appears in a 1991 construction by robert dixon 32 in 2022 frédéric beatrix presented a geometrographic construction in 13 steps 33 second construction by ramanujan edit squaring the circle approximate construction according to ramanujan of 1914 with continuation of the construction dashed lines mean proportional red line see animation sketch of manuscript book 1 of srinivasa ramanujan p 54 ramanujan s 355 113 construction in 1914 ramanujan gave a construction which was equivalent to taking the approximate value for π displaystyle pi to be 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 giving eight decimal places of π displaystyle pi 21 22 he describes the construction of line segment os as follows 21 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 incorrect constructions edit 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 34 during the 18th and 19th century the false notions that the problem of squaring the circle was somehow related to the longitude problem and that a large reward would be given for a solution became prevalent among would be circle squarers 35 36 in 1851 john parker published a book quadrature of the circle in which he claimed to have squared the circle his method actually produced an approximation of π displaystyle pi accurate to six digits 37 38 39 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 40 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 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 heisel s 1934 book 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 the technique he developed did not accurately square the circle and provided an incorrect area of the circle which essentially redefined π displaystyle pi as equal to 3 2 goodwin then proposed the indiana pi bill in the indiana state legislature allowing the state to use his method in education without paying royalties to him the bill passed with no objections in the state house but the bill was tabled and never voted on in the senate amid increasing ridicule from the press 41 the mathematical crank carl theodore heisel also claimed to have squared the circle in his 1934 book behold the grand problem no longer unsolved the circle squared beyond refutation 42 paul halmos referred to the book as a classic crank book 43 in non euclidean geometry edit although the circle cannot be squared in the euclidean plane it sometimes can be in hyperbolic geometry under suitable interpretations of the terms the hyperbolic plane does not contain squares quadrilaterals with four right angles and four equal sides but instead it contains regular quadrilaterals shapes with four equal sides and four equal angles sharper than right angles there exist in the hyperbolic plane countably infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area which however are constructed simultaneously there is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area symmetrically there is no method for starting with an arbitrary circle and constructing a regular quadrilateral of equal area and for sufficiently large circles no such quadrilateral exists 44 45 under the same interpretation a countable infinity of differently sized circles can also be squared in spherical geometry among these is a circle whose area is one sixth of the total spherical area whose corresponding regular quadrilateral is just the projected face of an inscribed cube here too the constructions involve specifically sized circles and regular quadrilaterals and do not lend themselves to any general circle squaring method 46 in literature edit the problem of squaring the circle has been mentioned over a wide range of literary eras with a variety of metaphorical meanings 47 its literary use dates back at least to 414 bc when the play the birds by aristophanes was first performed in it the character meton of athens mentions squaring the circle possibly to indicate the paradoxical nature of his utopian city 48 vitruvian man 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 49 dante s image also calls to mind a passage from vitruvius famously illustrated later in leonardo da vinci s vitruvian man of a man simultaneously inscribed in a circle and a square 50 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 47 50 earlier in canto xiii dante calls out greek circle squarer bryson as having sought knowledge instead of wisdom 47 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 47 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 38 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 one of these goals is and the circle they will square it some fine day 51 the sestina a poetic form first used in the 12th century by arnaut daniel has been said to metaphorically square the circle in its use of a square number of lines six stanzas of six lines each with a circular scheme of six repeated words spanos 1978 writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth 52 a similar metaphor was used in squaring the circle a 1908 short story by o henry about a long running family feud in the title of this story the circle represents the natural world while the square represents the city the world of man 53 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 54 55 see also edit mrs miniver s problem problem on areas of intersecting circles round square copula philosophical treatment of oxymorons pages displaying short descriptions of redirect targets squircle shape between a square and a circle tarski s circle squaring problem problem of cutting and reassembling a disk into a square references edit ammer christine square the circle dictionary com the american heritage dictionary of idioms houghton mifflin company retrieved 16 april 2012 1 2 bailey d h borwein j m borwein p b plouffe s 1997 the quest for pi the mathematical intelligencer 19 1 50 57 doi 10 1007 bf03024340 mr 1439159 s2cid 14318695 plofker kim 2009 mathematics in india princeton university press p 27 isbn 978 0691120676 lam lay yong ang tian se 1986 circle measurements in ancient china historia mathematica 13 4 325 340 doi 10 1016 0315 0860 86 90055 8 mr 0875525 reprinted in berggren j l borwein jonathan m borwein peter eds 2004 pi a source book springer pp 20 35 isbn 978 0387205717 1 2 the construction of a square equal in area to a given polygon is proposition 14 of euclid s elements book ii translation from knorr 1986 p 25 heath thomas 1921 history of greek mathematics the clarendon press see in particular anaxagoras pp 172 174 lunes of hippocrates pp 183 200 later work including antiphon eudemus and aristophanes pp 220 235 bos henk j m 2001 the legitimation of geometrical procedures before 1590 redefining geometrical exactness descartes transformation of the early modern concept of construction sources and studies in the history of mathematics and physical sciences new york springer pp 23 36 doi 10 1007 978 1 4613 0087 8_2 isbn 978 1 4612 6521 4 mr 1800805 knorr wilbur richard 1986 the ancient tradition of geometric problems boston birkhäuser pp 15 16 isbn 0 8176 3148 8 mr 0884893 guicciardini niccolò 2009 isaac newton on mathematical certainty and method transformations vol 4 mit press p 10 isbn 9780262013178 cotes roger 1850 correspondence of sir isaac newton and professor cotes including letters of other eminent men robson eleanor stedall jacqueline eds 2009 the oxford handbook of the history of mathematics oxford university press p 554 isbn 9780199213122 burn r p 2001 alphonse antonio de sarasa and logarithms historia mathematica 28 1 17 doi 10 1006 hmat 2000 2295 gregory james 1667 vera circuli et hyperbolæ quadratura the true squaring of the circle and of the hyperbola padua giacomo cadorino available at eth bibliothek zürich switzerland crippa davide 2019 james gregory and the impossibility of squaring the central conic sections the impossibility of squaring the circle in the 17th century springer international publishing pp 35 91 doi 10 1007 978 3 030 01638 8_2 isbn 978 3 030 01637 1 s2cid 132820288 lambert johann heinrich 1761 mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques memoir on some remarkable properties of circular transcendental and logarithmic quantities histoire de l académie royale des sciences et des belles lettres de berlin in french 17 published 1768 265 322 laczkovich m 1997 on lambert s proof of the irrationality of π the american mathematical monthly 104 5 439 443 doi 10 1080 00029890 1997 11990661 jstor 2974737 mr 1447977 1 2 lindemann f 1882 über die zahl π on the number π mathematische annalen in german 20 2 213 225 ...