1982:
2167:
1774:
6041:
1786:
2183:
1958:
1970:
2515:
1488:
44:
1407:) 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.
6028:
882:
1473:
2441:
2416:, 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:
2567:, of a man simultaneously inscribed in a circle and a square. 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. Earlier, in canto XIII, Dante calls out Greek circle-squarer Bryson as having sought knowledge instead of wisdom.
1459:, 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.
2356:
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,
2419:
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
973:
in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically". In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve
1735:
877:
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
2448:
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
2331:
2357:
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.
2473:
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.
2066:
800:
1567:
1851:
2218:
547:
Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. These methods can be summarized by stating the
964:
before the invention of calculus. 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,
2574:
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". By 1742, when
869:
into a square of equivalent area. They used this construction to compare areas of polygons geometrically, rather than by the numerical computation of area that would be more typical in modern mathematics. As
1935:
647:
2628:
writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth. A similar metaphor was used in "Squaring the Circle", a 1908 short story by
2437:, 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.
852:
1087:. Like squaring the circle, these cannot be solved by compass and straightedge. However, they have a different character than squaring the circle, in that their solution involves the root of a
599:
1187:
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
2553:
For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend
Paradise. Dante's image also calls to mind a passage from
3799:
Mathematics throughout the ages. Including papers from the 10th and 11th
Novembertagung on the History of Mathematics held in Holbæk, October 28–31, 1999 and in Brno, November 2–5, 2000
2140:
2001:
1005:(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
1399:
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
513:
were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.
1153:
1129:
1302:
685:
1801:
2624:, 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.
1068:
After
Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by
2086:
1415:
Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to
2172:
Squaring the circle, approximate construction according to
Ramanujan of 1914, with continuation of the construction (dashed lines, mean proportional red line), see
1375:
Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain
2467:
2403:
2351:
2213:
1757:
1562:
1542:
1453:
1433:
1370:
1322:
1233:
1209:
1181:
1047:
1023:
725:
511:
479:
455:
735:
2632:, 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.
1511:
1350:
1256:
2659:
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.
4797:
3953:
917:
believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the
2571:
2481:
also claimed to have squared the circle in his 1934 book, "Behold! : the grand problem no longer unsolved: the circle squared beyond refutation."
1072:
attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts. As well, several later mathematicians including
3614:
1863:
1730:{\displaystyle |P_{3}P_{9}|=|P_{1}P_{2}|{\sqrt {{\frac {40}{3}}-2{\sqrt {3}}}}\approx 3.141\,5{\color {red}33\,338}\cdot |P_{1}P_{2}|\approx \pi r.}
1387:
to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it. The
929:
argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern
3930:
6070:
4554:, with the satire of college women presented by Gilbert and Sullivan. He writes that "Vivie naturally knew better than to try to square circles."
2381:, and that a large reward would be given for a solution, became prevalent among would-be circle squarers. In 1851, John Parker published a book
5901:
2173:
520:(i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.
3642:
2326:{\displaystyle \left(9^{2}+{\frac {19^{2}}{22}}\right)^{\frac {1}{4}}={\sqrt{\frac {2143}{22}}}=3.141\;592\;65{\color {red}2\;582\;\ldots }}
4720:
998:
925:
argued that magnitudes can be divided up without limit, so the area of the circle would never be used up. Contemporaneously with
Antiphon,
3002:
5501:
4748:
1328:, because a rational power of a transcendental number remains transcendental. Lindemann was able to extend this argument, through the
5979:
4345:
Tubbs, Robert (December 2020). "Squaring the circle: A literary history". In Tubbs, Robert; Jenkins, Alice; Engelhardt, Nina (eds.).
5282:
4818:
2609:
features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding
4790:
4738:
2420:
demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.
611:
5826:
5556:
4362:
3777:
3697:
3182:
2974:
1456:
1160:
934:
861:
The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from
5521:
20:
4628:
Similarly, the story "Squaring the Circle" is permeated with the integrating image: nature is a circle, the city a square.
2377:. During the 18th and 19th century, the false notions that the problem of squaring the circle was somehow related to the
1091:, rather than being transcendental. Therefore, more powerful methods than compass and straightedge constructions, such as
874:
wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes:
5889:
5292:
2689:
817:
1981:
563:
5956:
4783:
3612:[Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass].
2876:
2813:
1107:
The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number
24:
1520:
One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit
6075:
4621:
4262:
4188:
4063:
3509:
3395:
3238:
3094:
3047:
3012:
1403:, shapes with four equal sides and four equal angles sharper than right angles. There exist in the hyperbolic plane (
1329:
430:
140:
4315:
1235:
and so showed the impossibility of this construction. Lindemann's idea was to combine the proof of transcendence of
283:
3439:
2433:
1268:
374:
3610:"Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas"
3958:
2735:
28:
2095:
119:
5925:
5858:
5491:
5371:
4209:
3826:
3730:
2961:. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 23–36.
2752:
2431:, published posthumously by his widow in 1872. Having originally published the work as a series of articles in
810:
found even more accurate approximations using a method similar to that of
Archimedes, and in the fifth century
4310:
345:
51:. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized
4326:
3983:
3609:
3288:
3210:
3139:
2374:
237:
2493:
The problem of squaring the circle has been mentioned over a wide range of literary eras, with a variety of
135:
6085:
6065:
5994:
5751:
5639:
4814:
3906:
2146:
1096:
604:
166:
1998:
in 1913 is accurate to three decimal places. Hobson's construction corresponds to an approximate value of
1564:
were already known, Kochański's construction has the advantage of being quite simple. In the left diagram
982:
885:
Some apparent partial solutions gave false hope for a long time. In this figure, the shaded figure is the
156:
6080:
5703:
5634:
4546:
3211:"Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques"
2799:
4733:
4173:
The
Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays
192:
6090:
5330:
5146:
2959:
Redefining
Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction
1076:
developed compass and straightedge constructions that approximate the problem accurately in few steps.
930:
35:
5121:
3213:[Memoir on some remarkable properties of circular transcendental and logarithmic quantities].
1521:
1478:
1065:, and by doing so also proved the impossibility of squaring the circle with compass and straightedge.
187:
5865:
5836:
5196:
5051:
3165:
Crippa, Davide (2019). "James
Gregory and the impossibility of squaring the central conic sections".
2924:
2668:
1737:
In the same work, Kochański also derived a sequence of increasingly accurate rational approximations
1236:
1001:, following de Saint-Vincent, attempted another proof of the impossibility of squaring the circle in
1134:
1110:
5999:
5733:
5276:
3722:
664:
407:
52:
1853:
This value is accurate to six decimal places and has been known in China since the 5th century as
212:
5961:
5937:
5811:
5746:
5687:
5624:
5614:
5350:
5269:
5131:
5041:
4921:
3892:
3568:
3284:
3206:
2412:-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by his pseudonym
2385:
in which he claimed to have squared the circle. His method actually produced an approximation of
1380:
1212:
1054:
1026:
207:
5231:
2195:
In 1914, Ramanujan gave a construction which was equivalent to taking the approximate value for
5984:
5932:
5831:
5659:
5609:
5594:
5589:
5360:
5161:
5096:
5086:
5036:
3337:
2710:
2511:
mentions squaring the circle, possibly to indicate the paradoxical nature of his utopian city.
1376:
556:
227:
3084:
3069:
Correspondence of Sir Isaac Newton and Professor Cotes: Including letters of other eminent men
3067:
2071:
2061:{\displaystyle {\frac {6}{5}}\cdot \left(1+\varphi \right)=3.141\;{\color {red}640\;\ldots },}
549:
161:
6032:
5884:
5728:
5669:
5546:
5469:
5417:
5219:
5126:
4976:
4642:
4400:
4053:
4010:
3910:
3801:. Dějiny Matematiky/History of Mathematics. Vol. 17. Prague: Prometheus. pp. 7–20.
3293:
3153:
3037:
2866:
2805:
1384:
1062:
536:
528:
458:
4564:
Spanos, Margaret (1978). "The Sestina: An Exploration of the Dynamics of Poetic Structure".
4421:
3427:
2188:
Sketch of "Manuscript book 1 of Srinivasa Ramanujan" p. 54, Ramanujan's 355/113 construction
795:{\displaystyle 3\,{\tfrac {10}{71}}\approx 3.141<\pi <3\,{\tfrac {1}{7}}\approx 3.143}
527:
is sometimes used as a synonym for squaring the circle. It may also refer to approximate or
6009:
5949:
5913:
5756:
5579:
5526:
5496:
5486:
5395:
5258:
5151:
5066:
5021:
5001:
4846:
4831:
4372:
4286:
4272:
4230:
4154:
4113:
4084:
3857:
3806:
3665:
3591:
3548:
3469:
3358:
3267:
3112:
3022:
2984:
2895:
2858:
2835:
2773:
2600:
2478:
2378:
1156:
1084:
953:
918:
914:
906:
894:
516:
Despite the proof that it is impossible, attempts to square the circle have been common in
367:
2452:
2388:
2336:
2198:
1860:
Gelder did not construct the side of the square; it was enough for him to find the value
1742:
1547:
1527:
1438:
1418:
1355:
1307:
1218:
1194:
1166:
1032:
1008:
710:
496:
464:
440:
8:
5989:
5908:
5896:
5877:
5841:
5761:
5679:
5664:
5654:
5604:
5599:
5541:
5302:
5166:
5156:
5056:
5026:
4966:
4941:
4866:
4856:
4841:
4688:
4551:
3886:
3679:
3423:
2944:
2940:
2936:
2674:
1941:
1396:
1263:
1092:
1073:
910:
886:
803:
695:
493:
coefficients. It had been known for decades that the construction would be impossible if
262:
252:
222:
4292:
Behold! : the grand problem the circle squared beyond refutation no longer unsolved
4025:
921:). Since any polygon can be squared, he argued, the circle can be squared. In contrast,
6045:
6004:
5944:
5872:
5738:
5713:
5531:
5506:
5474:
5312:
5071:
5016:
4981:
4876:
4651:
4591:
4583:
4566:
4535:
4506:
4464:
4456:
4417:
4376:
4234:
4101:
4005:
3975:
3861:
3835:
3747:
3536:
3518:
3486:
3362:
3310:
3255:
3188:
2777:
2655:
2424:
1496:
1391:
can be used for another similar construction. Although the circle cannot be squared in
1388:
1335:
1241:
957:
926:
690:
482:
415:
217:
986:
6040:
5718:
5629:
5479:
5405:
5378:
5141:
4961:
4951:
4886:
4806:
4617:
4595:
4510:
4468:
4380:
4358:
4258:
4238:
4184:
4059:
3979:
3865:
3773:
3751:
3703:
3693:
3391:
3366:
3314:
3192:
3178:
3090:
3043:
3008:
2970:
2872:
2849:
2830:
2809:
2558:
2182:
2166:
1846:{\displaystyle \pi \approx {\frac {355}{113}}=3.141\;592{\color {red}\;920\;\ldots }}
1773:
1325:
1080:
1069:
1050:
994:
922:
862:
517:
399:
395:
3656:
3637:
2781:
1785:
5920:
5791:
5708:
5464:
5452:
5400:
5136:
4575:
4502:
4498:
4448:
4409:
4350:
4250:
4218:
4204:
4176:
4140:
4093:
3967:
3845:
3765:
3739:
3651:
3577:
3528:
3504:
3482:
3478:
3383:
3346:
3302:
3251:
3247:
3170:
3121:
2962:
2894:
The construction of a square equal in area to a given polygon is Proposition 14 of
2844:
2761:
2739:
2610:
2528:
2499:
2369:
convinced himself that he had succeeded in squaring the circle, a claim refuted by
1435:. It takes only elementary geometry to convert any given rational approximation of
1188:
865:. Greek mathematicians found compass and straightedge constructions to convert any
532:
403:
109:
2957:
Bos, Henk J. M. (2001). "The legitimation of geometrical procedures before 1590".
232:
5561:
5551:
5445:
5171:
4760:
4526:
4413:
4368:
4354:
4290:
4268:
4254:
4226:
4150:
4109:
3853:
3802:
3687:
3661:
3587:
3544:
3354:
3263:
3018:
2980:
2854:
2769:
2645:
2508:
2470:
1544:
in the 5th decimal place. Although much more precise numerical approximations to
1392:
1259:
1131:, the length of the side of a square whose area equals that of a unit circle. If
970:
933:. The more general goal of carrying out all geometric constructions using only a
490:
419:
360:
337:
306:
289:
267:
4394:
Amati, Matthew (2010). "Meton's star-city: Geometry and utopia in Aristophanes'
3174:
2966:
2711:"Square the Circle. Dictionary.com. The American Heritage® Dictionary of Idioms"
1957:
979:
Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum
47:
Squaring the circle: the areas of this square and this circle are both equal to
5821:
5816:
5644:
5536:
5516:
5344:
4901:
4871:
4168:
3633:
3605:
2928:
2613:. One of these goals is "And the circle – they will square it/Some fine day."
2576:
2514:
2150:
1969:
1184:
1088:
1079:
Two other classical problems of antiquity, famed for their impossibility, were
656:
247:
4764:
4703:
4452:
3849:
6059:
5781:
5649:
5619:
5440:
5248:
5191:
4765:"2000 years unsolved: Why is doubling cubes and squaring circles impossible?"
4049:
3707:
2747:
2743:
2636:
2621:
2563:
2519:
2413:
2409:
2366:
1404:
701:
652:
332:
114:
4775:
3335:
has been known for about a century—but who was the man who discovered it?".
1798:
Jacob de Gelder published in 1849 a construction based on the approximation
5723:
5511:
5186:
4946:
4936:
4693:
4609:
4145:
4128:
3882:
3683:
3125:
2998:
2605:
2585:, attempts at circle-squaring had come to be seen as "wild and fruitless":
2504:
2089:
1995:
1191:. If the circle could be squared using only compass and straightedge, then
966:
4180:
3971:
3215:
Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin
5337:
5225:
4931:
4916:
4711:
4306:
3063:
2795:
2650:
2640:
2581:
2482:
2370:
811:
410:. The difficulty of the problem raised the question of whether specified
257:
242:
197:
171:
63:
4655:
4544:
Dolid contrasts Vivie Warren, a fictional female mathematics student in
4539:
4460:
4129:"Charles L. Dodgson's geometric approach to arctangent relations for pi"
4030:. Cambridge University Library: Royal Observatory. 1737–1779. p. 48
3540:
2497:
meanings. Its literary use dates back at least to 414 BC, when the play
1487:
1372:
is transcendental and therefore that squaring the circle is impossible.
43:
5323:
5242:
5181:
5176:
5116:
5101:
5046:
5031:
4986:
4926:
4911:
4891:
4861:
4826:
4587:
4222:
4105:
3743:
3490:
3350:
3306:
3259:
2765:
944:
The problem of finding the area under an arbitrary curve, now known as
902:
729:
486:
429:
In 1882, the task was proven to be impossible, as a consequence of the
909:
attacked the problem by finding a shape bounded by circular arcs, the
5076:
5061:
5011:
4906:
4896:
4881:
4851:
3582:
3563:
3532:
3523:
2554:
990:
938:
202:
4579:
4097:
2865:
Berggren, J. L.; Borwein, Jonathan M.; Borwein, Peter, eds. (2004).
989:. Nevertheless, de Saint-Vincent succeeded in his quadrature of the
5213:
4991:
4836:
4435:
Herzman, Ronald B.; Towsley, Gary B. (1994). "Squaring the circle:
3232:
Laczkovich, M. (1997). "On Lambert's proof of the irrationality of
2683:
2629:
2494:
949:
945:
391:
3840:
3820:
Fukś, Henryk (2012). "Adam Adamandy Kochański's approximations of
5786:
5111:
5106:
5006:
4996:
4971:
2617:
1211:
would have to be an algebraic number. It was not until 1882 that
871:
866:
807:
3110:
Burn, R.P. (2001). "Alphonse Antonio de Sarasa and logarithms".
2692: – Problem of cutting and reassembling a disk into a square
2353:. He describes the construction of line segment OS as follows.
1944:
gave another geometric construction for the same approximation.
1854:
881:
855:
5081:
4956:
4249:(Third ed.). New York: Springer-Verlag. pp. 236–239.
2440:
1379:
makes squaring the circle possible in some sense. For example,
423:
311:
3794:
1930:{\displaystyle {\overline {AH}}={\frac {4^{2}}{7^{2}+8^{2}}}.}
1472:
732:
proved a formula for the area of a circle, according to which
5457:
5435:
5091:
4245:
Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2004).
2943:; Later work, including Antiphon, Eudemus, and Aristophanes,
2145:
The same approximate value appears in a 1991 construction by
974:
retains the idea of using only restricted geometric methods.
411:
3167:
The Impossibility of Squaring the Circle in the 17th Century
2734:
642:{\displaystyle \pi \approx {\tfrac {256}{81}}\approx 3.16}
4524:
Dolid, William A. (1980). "Vivie Warren and the Tripos".
993:, and in doing so was one of the earliest to develop the
4721:"An Investigation of Historical Geometric Constructions"
3954:"Squaring the Circle: Hobbes on Philosophy and Geometry"
1099:, can be used to construct solutions to these problems.
539:
or squaring may also be applied to other plane figures.
4349:. Springer International Publishing. pp. 169–185.
3042:. Transformations. Vol. 4. MIT Press. p. 10.
2679:
Pages displaying short descriptions of redirect targets
1947:
434:
72:
4244:
3148:
The true squaring of the circle and of the hyperbola …
2864:
775:
744:
622:
574:
4749:"The Quadrature of the Circle and Hippocrates' Lunes"
4731:
4680:
3169:. Springer International Publishing. pp. 35–91.
2455:
2391:
2339:
2221:
2201:
2098:
2074:
2004:
1866:
1804:
1745:
1570:
1550:
1530:
1499:
1441:
1421:
1358:
1338:
1310:
1271:
1244:
1221:
1197:
1169:
1137:
1113:
1035:
1011:
820:
738:
713:
667:
614:
566:
499:
467:
443:
4082:
Schepler, Herman C. (1950). "The chronology of pi".
4640:Pendrick, Gerard (1994). "Two notes on "Ulysses"".
4347:
The Palgrave Handbook of Literature and Mathematics
3083:Robson, Eleanor; Stedall, Jacqueline, eds. (2009).
2156:
847:{\displaystyle \pi \approx 355/113\approx 3.141593}
4719:Harper, Suzanne; Driskell, Shannon (August 2010).
3931:"Squaring the circle like a medieval master mason"
3772:(Fourth ed.). W H Freeman. pp. 520–528.
3692:. John Wiley & Sons. pp. 62–63, 113–115.
3564:"Angle trisection with origami and related topics"
2620:, a poetic form first used in the 12th century by
2532:, canto XXXIII, lines 133–135, contain the verse:
2461:
2397:
2345:
2325:
2207:
2134:
2080:
2060:
1929:
1845:
1751:
1729:
1556:
1536:
1505:
1447:
1427:
1364:
1344:
1316:
1296:
1250:
1227:
1203:
1175:
1147:
1123:
1041:
1017:
846:
794:
719:
679:
641:
594:{\displaystyle \pi \approx {\tfrac {25}{8}}=3.125}
593:
505:
473:
449:
4485:Kay, Richard (July 2005). "Vitruvius and Dante's
3626:
3463:Castellanos, Dario (April 1988). "The ubiquitous
3086:The Oxford Handbook of the History of Mathematics
3039:Isaac Newton on Mathematical Certainty and Method
2890:
2888:
1937:The illustration shows de Gelder's construction.
6057:
4671:The Big Deal: Card Games in 20th-Century Fiction
3507:(2005). "Trisections and totally real origami".
2671: – Problem on areas of intersecting circles
2595:Now, running round the circle, finds it square.
2485:referred to the book as a "classic crank book."
1493:Continuation with equal-area circle and square;
889:. Its area is equal to the area of the triangle
3941:(2). UNSW School of Mathematics and Statistics.
941:, but the evidence for this is circumstantial.
901:The first known Greek to study the problem was
4718:
4434:
4305:
4027:Board of Longitude / Vol V / Confirmed Minutes
3331:Fritsch, Rudolf (1984). "The transcendence of
2885:
1763:
1332:on linear independence of algebraic powers of
406:by using only a finite number of steps with a
4805:
4791:
3888:Squaring the Circle: A History of the Problem
3678:
3643:Bulletin of the American Mathematical Society
3458:
3456:
3454:
3418:
3416:
3414:
3412:
3082:
368:
4732:O'Connor, J J; Robertson, E F (April 1999).
4480:
4478:
3795:"Squaring the circle in XVI–XVIII centuries"
3615:Journal de Mathématiques Pures et Appliquées
3279:
3277:
3056:
3035:
3029:
2677: – Philosophical treatment of oxymorons
1524:, producing an approximation diverging from
1462:
1410:
4673:(PhD). University of Montréal. p. 196.
4077:
4075:
3462:
3004:The Ancient Tradition of Geometric Problems
2686: – Shape between a square and a circle
2593:Now to pure space lifts her ecstatic stare,
2469:as equal to 3.2. Goodwin then proposed the
2423:A ridiculing of circle squaring appears in
555:that they produce. In around 2000 BCE, the
4798:
4784:
4616:. Chelsea House Publishers. p. 1848.
4203:
3813:
3786:
3723:"Squaring circles in the hyperbolic plane"
3451:
3409:
3326:
3324:
3231:
2549:pensando, quel principio ond’elli indige,
2540:Finds the right formula, howe'er he tries
2507:was first performed. In it, the character
2360:
2317:
2313:
2304:
2300:
2135:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
2049:
2043:
1837:
1833:
1827:
375:
361:
4475:
4428:
4144:
4018:
4004:
3877:
3875:
3839:
3764:
3655:
3581:
3522:
3428:"Modular equations and approximations to
3422:
3378:
3376:
3283:
3274:
2848:
2591:Too mad for mere material chains to bind,
2538:To square the circle, nor for all his wit
1676:
1667:
773:
742:
16:Problem of constructing equal-area shapes
4639:
4633:
4340:
4338:
4336:
4334:
4175:. New York: Copernicus. pp. 29–31.
4081:
4072:
4048:
4042:
3998:
3555:
3497:
3225:
3199:
3089:. Oxford University Press. p. 554.
2635:In later works, circle-squarers such as
2513:
2439:
2365:In his old age, the English philosopher
1857:, and in Europe since the 17th century.
1057:succeeded in proving more strongly that
880:
705:, used several different approximations
426:implied the existence of such a square.
398:. It is the challenge of constructing a
42:
4739:MacTutor History of Mathematics archive
4557:
4197:
4167:
4161:
3928:
3922:
3891:. Cambridge University Press. pp.
3792:
3604:
3598:
3503:
3330:
3321:
3205:
3138:
3132:
3076:
2794:
2149:. In 2022 Frédéric Beatrix presented a
977:A 1647 attempt at squaring the circle,
603:and at approximately the same time the
6071:Compass and straightedge constructions
6058:
5827:Latin translations of the 12th century
4755:. Mathematical Association of America.
4727:. Mathematical Association of America.
4668:
4662:
4563:
4299:
4285:
4279:
4126:
4120:
3881:
3872:
3770:Euclidean and Non-Euclidean Geometries
3758:
3714:
3672:
3632:
3382:
3373:
3164:
3158:
3144:Vera Circuli et Hyperbolæ Quadratura …
3007:. Boston: Birkhäuser. pp. 15–16.
2831:"Circle measurements in ancient China"
2828:
2804:. Princeton University Press. p.
2788:
2625:
2547:per misurar lo cerchio, e non ritrova,
1779:Jacob de Gelder's 355/113 construction
1183:would also be constructible. In 1837,
5557:Straightedge and compass construction
4779:
4746:
4686:
4614:Twentieth-century American literature
4608:
4602:
4523:
4517:
4393:
4387:
4344:
4331:
3919:Reprinted by Dover Publications, 1991
3905:
3899:
3561:
3062:
2997:
2950:
2923:
2911:
2730:
2728:
2702:
2545:Qual è ’l geométra che tutto s’affige
1457:compass and straightedge construction
1304:This identity immediately shows that
1003:Vera Circuli et Hyperbolae Quadratura
5522:Incircle and excircles of a triangle
3951:
3945:
3824:: reconstruction of the algorithm".
3819:
3720:
3390:. Springer-Verlag. pp. xi–xii.
3154:ETH Bibliothek (Zürich, Switzerland)
3109:
3103:
2991:
2917:
2829:Lam, Lay Yong; Ang, Tian Se (1986).
2822:
1948:Constructions using the golden ratio
905:, who worked on it while in prison.
21:Squaring the circle (disambiguation)
4759:
4484:
4058:. St. Martin's Press. p. 178.
2956:
2570:Several works of 17th-century poet
13:
4681:Further reading and external links
4207:(1985). "The legal values of pi".
2725:
2589:Mad Mathesis alone was unconfined,
1963:Hobson's golden ratio construction
25:Square the Circle (disambiguation)
14:
6102:
4701:
4439:33 and the poetics of geometry".
3510:The American Mathematical Monthly
3239:The American Mathematical Monthly
2708:
2579:published the fourth book of his
2309:
2045:
1975:Dixon's golden ratio construction
1832:
1672:
6039:
6026:
4092:(3): 165–170, 216–228, 279–283.
3440:Quarterly Journal of Mathematics
2690:Tarski's circle-squaring problem
2557:, famously illustrated later in
2536:As the geometer his mind applies
2488:
2181:
2165:
2157:Second construction by Ramanujan
1980:
1968:
1956:
1791:Ramanujan's 355/113 construction
1784:
1772:
1486:
1471:
1159:, it would follow from standard
1102:
3959:Journal of the History of Ideas
3657:10.1090/s0002-9904-1918-03088-7
3150:]. Padua: Giacomo Cadorino.
2333:giving eight decimal places of
1994:An approximate construction by
660:used the simpler approximation
605:ancient Egyptian mathematicians
29:Squared circle (disambiguation)
5859:A History of Greek Mathematics
5372:The Quadrature of the Parabola
4747:Otero, Daniel E. (July 2010).
4503:10.1080/02666286.2005.10462116
4210:The Mathematical Intelligencer
3827:The Mathematical Intelligencer
3731:The Mathematical Intelligencer
3483:10.1080/0025570X.1988.11977350
3252:10.1080/00029890.1997.11990661
3036:Guicciardini, Niccolò (2009).
2935:See in particular Anaxagoras,
2904:
2753:The Mathematical Intelligencer
2121:
2105:
1987:Beatrix's 13-step construction
1940:In 1914, Indian mathematician
1711:
1686:
1630:
1605:
1597:
1572:
1148:{\displaystyle {\sqrt {\pi }}}
1124:{\displaystyle {\sqrt {\pi }}}
1:
2696:
1330:Lindemann–Weierstrass theorem
1297:{\displaystyle e^{i\pi }=-1.}
1053:. It was not until 1882 that
937:has often been attributed to
680:{\displaystyle \pi \approx 3}
431:Lindemann–Weierstrass theorem
5640:Intersecting secants theorem
4422:10.5184/classicalj.105.3.213
4414:10.5184/classicalj.105.3.213
4355:10.1007/978-3-030-55478-1_10
4255:10.1007/978-1-4757-4217-6_27
4127:Abeles, Francine F. (1993).
3917:. Blackwell. pp. 44–47.
2930:History of Greek Mathematics
2871:. Springer. pp. 20–35.
2850:10.1016/0315-0860(86)90055-8
2750:(1997). "The quest for pi".
1877:
1215:proved the transcendence of
985:, was heavily criticized by
854:, an approximation known as
418:concerning the existence of
7:
5635:Intersecting chords theorem
5502:Doctrine of proportionality
4316:L'Enseignement mathématique
3291:[On the number π].
3175:10.1007/978-3-030-01638-8_2
2967:10.1007/978-1-4613-0087-8_2
2662:
1764:Constructions using 355/113
806:, in the third century CE,
651:Over 1000 years later, the
10:
6107:
5331:On the Sphere and Cylinder
5284:On the Sizes and Distances
4311:"How to Write Mathematics"
3929:Beatrix, Frédéric (2022).
3797:. In Fuchs, Eduard (ed.).
3221:(published 1768): 265–322.
2713:. Houghton Mifflin Company
2153:construction in 13 steps.
1513:denotes the initial radius
1097:mathematical paper folding
931:intermediate value theorem
542:
36:Square peg in a round hole
33:
18:
6033:Ancient Greece portal
6022:
5972:
5850:
5837:Philosophy of mathematics
5807:
5800:
5774:
5752:Ptolemy's table of chords
5696:
5678:
5577:
5570:
5426:
5388:
5205:
4813:
4807:Ancient Greek mathematics
4453:10.1017/S0362152900013015
3850:10.1007/s00283-012-9312-1
3721:Jagy, William C. (1995).
2375:Hobbes–Wallis controversy
1463:Construction by Kochański
1411:Approximate constructions
1395:, it sometimes can be in
983:Grégoire de Saint-Vincent
913:, that could be squared.
557:Babylonian mathematicians
167:Madhava's correction term
6076:Euclidean plane geometry
5704:Aristarchus's inequality
5277:On Conoids and Spheroids
4547:Mrs. Warren's Profession
3952:Bird, Alexander (1996).
3793:Więsław, Witold (2001).
3689:A History of Mathematics
3638:"Pierre Laurent Wantzel"
3207:Lambert, Johann Heinrich
2939:; Lunes of Hippocrates,
2405:accurate to six digits.
2383:Quadrature of the Circle
2081:{\displaystyle \varphi }
1481:approximate construction
1377:non-Euclidean geometries
1161:compass and straightedge
935:compass and straightedge
525:quadrature of the circle
408:compass and straightedge
346:Other topics related to
53:compass and straightedge
34:Not to be confused with
5812:Ancient Greek astronomy
5625:Inscribed angle theorem
5615:Greek geometric algebra
5270:Measurement of a Circle
3569:Elemente der Mathematik
3562:Fuchs, Clemens (2011).
3388:A Budget of Trisections
2649:and Lawyer Paravant in
2477:The mathematical crank
2361:Incorrect constructions
1522:Adam Adamandy Kochański
1213:Ferdinand von Lindemann
1055:Ferdinand von Lindemann
1027:Johann Heinrich Lambert
559:used the approximation
6046:Mathematics portal
5832:Non-Euclidean geometry
5787:Mouseion of Alexandria
5660:Tangent-secant theorem
5610:Geometric mean theorem
5595:Exterior angle theorem
5590:Angle bisector theorem
5294:On Sizes and Distances
4687:Bogomolny, Alexander.
4669:Goggin, Joyce (1997).
4146:10.1006/hmat.1993.1013
3883:Hobson, Ernest William
3338:Results in Mathematics
3126:10.1006/hmat.2000.2295
2933:. The Clarendon Press.
2669:Mrs. Miniver's problem
2523:
2463:
2445:
2399:
2347:
2327:
2209:
2136:
2082:
2062:
1931:
1847:
1753:
1731:
1558:
1538:
1507:
1449:
1429:
1401:regular quadrilaterals
1366:
1346:
1318:
1298:
1252:
1229:
1205:
1177:
1149:
1125:
1043:
1019:
898:
848:
796:
721:
681:
643:
595:
507:
475:
451:
404:area of a given circle
71:mathematical constant
56:
5734:Pappus's area theorem
5670:Theorem of the gnomon
5547:Quadratrix of Hippias
5470:Circles of Apollonius
5418:Problem of Apollonius
5396:Constructible numbers
5220:Archimedes Palimpsest
4734:"Squaring the circle"
4704:"Squaring the Circle"
4689:"Squaring the Circle"
4643:James Joyce Quarterly
4401:The Classical Journal
4287:Heisel, Carl Theodore
4181:10.1007/0-387-28952-6
4011:A Budget of Paradoxes
3972:10.1353/jhi.1996.0012
3911:"Squaring the circle"
3766:Greenberg, Marvin Jay
3294:Mathematische Annalen
2999:Knorr, Wilbur Richard
2517:
2464:
2443:
2429:A Budget of Paradoxes
2400:
2348:
2328:
2210:
2137:
2083:
2063:
1932:
1848:
1754:
1732:
1559:
1539:
1508:
1455:into a corresponding
1450:
1430:
1385:quadratrix of Hippias
1367:
1347:
1319:
1299:
1253:
1230:
1206:
1178:
1150:
1126:
1063:transcendental number
1044:
1020:
884:
849:
797:
722:
693:, as recorded in the
682:
644:
596:
508:
476:
459:transcendental number
452:
120:Use in other formulae
46:
5950:prehistoric counting
5747:Ptolemy's inequality
5688:Apollonius's theorem
5527:Method of exhaustion
5497:Diophantine equation
5487:Circumscribed circle
5304:On the Moving Sphere
4771:– via YouTube.
4714:– via YouTube.
4133:Historia Mathematica
4085:Mathematics Magazine
3470:Mathematics Magazine
3113:Historia Mathematica
2836:Historia Mathematica
2801:Mathematics in India
2601:Gilbert and Sullivan
2479:Carl Theodore Heisel
2462:{\displaystyle \pi }
2453:
2398:{\displaystyle \pi }
2389:
2346:{\displaystyle \pi }
2337:
2219:
2208:{\displaystyle \pi }
2199:
2096:
2072:
2002:
1864:
1802:
1752:{\displaystyle \pi }
1743:
1568:
1557:{\displaystyle \pi }
1548:
1537:{\displaystyle \pi }
1528:
1497:
1448:{\displaystyle \pi }
1439:
1428:{\displaystyle \pi }
1419:
1381:Dinostratus' theorem
1365:{\displaystyle \pi }
1356:
1336:
1317:{\displaystyle \pi }
1308:
1269:
1242:
1228:{\displaystyle \pi }
1219:
1204:{\displaystyle \pi }
1195:
1176:{\displaystyle \pi }
1167:
1157:constructible number
1135:
1111:
1085:trisecting the angle
1042:{\displaystyle \pi }
1033:
1029:proved in 1761 that
1018:{\displaystyle \pi }
1009:
919:method of exhaustion
915:Antiphon the Sophist
907:Hippocrates of Chios
895:Hippocrates of Chios
818:
736:
720:{\displaystyle \pi }
711:
665:
612:
564:
506:{\displaystyle \pi }
497:
474:{\displaystyle \pi }
465:
450:{\displaystyle \pi }
441:
433:, which proves that
64:a series of articles
19:For other uses, see
6086:History of geometry
6066:Squaring the circle
6036: •
5842:Neusis construction
5762:Spiral of Theodorus
5655:Pythagorean theorem
5600:Euclidean algorithm
5542:Lune of Hippocrates
5411:Squaring the circle
5167:Theon of Alexandria
4842:Aristaeus the Elder
4552:George Bernard Shaw
4006:De Morgan, Augustus
3686:(11 January 2011).
2675:Round square copula
1942:Srinivasa Ramanujan
1397:hyperbolic geometry
1163:constructions that
1093:neusis construction
1074:Srinivasa Ramanujan
911:lune of Hippocrates
887:lune of Hippocrates
804:Chinese mathematics
696:Shatapatha Brahmana
388:Squaring the circle
328:Squaring the circle
263:Chudnovsky brothers
253:Srinivasa Ramanujan
6081:Unsolvable puzzles
5729:Menelaus's theorem
5719:Irrational numbers
5532:Parallel postulate
5507:Euclidean geometry
5475:Apollonian circles
5017:Isidore of Miletus
4223:10.1007/BF03024180
3986:on 16 January 2022
3744:10.1007/BF03024895
3351:10.1007/BF03322501
3307:10.1007/bf01446522
2766:10.1007/BF03024340
2709:Ammer, Christine.
2656:The Magic Mountain
2572:Margaret Cavendish
2524:
2459:
2446:
2444:Heisel's 1934 book
2425:Augustus De Morgan
2395:
2343:
2323:
2321:
2205:
2132:
2078:
2058:
2053:
1927:
1843:
1841:
1749:
1727:
1680:
1554:
1534:
1503:
1445:
1425:
1389:Archimedean spiral
1362:
1342:
1314:
1294:
1248:
1225:
1201:
1173:
1145:
1121:
1070:pseudomathematical
1039:
1015:
958:numerical analysis
927:Bryson of Heraclea
899:
844:
792:
784:
753:
717:
691:Indian mathematics
677:
639:
631:
591:
583:
503:
471:
447:
416:Euclidean geometry
394:first proposed in
218:Ludolph van Ceulen
57:
6091:Pseudomathematics
6053:
6052:
6018:
6017:
5770:
5769:
5757:Ptolemy's theorem
5630:Intercept theorem
5480:Apollonian gasket
5406:Doubling the cube
5379:The Sand Reckoner
4364:978-3-030-55477-4
4247:Pi: a source book
4205:Singmaster, David
3779:978-0-7167-9948-1
3699:978-0-470-52548-7
3505:Alperin, Roger C.
3384:Dudley, Underwood
3289:"Über die Zahl π"
3184:978-3-030-01637-1
2976:978-1-4612-6521-4
2910:Translation from
2868:Pi: A Source Book
2559:Leonardo da Vinci
2379:longitude problem
2292:
2286:
2271:
2256:
2119:
2013:
1922:
1880:
1819:
1659:
1657:
1644:
1506:{\displaystyle r}
1345:{\displaystyle e}
1326:irrational number
1251:{\displaystyle e}
1189:algebraic numbers
1143:
1119:
1081:doubling the cube
1051:irrational number
995:natural logarithm
863:Greek mathematics
783:
752:
630:
582:
550:approximation to
529:numerical methods
518:pseudomathematics
396:Greek mathematics
385:
384:
6098:
6044:
6043:
6031:
6030:
6029:
5805:
5804:
5792:Platonic Academy
5739:Problem II.8 of
5709:Crossbar theorem
5665:Thales's theorem
5605:Euclid's theorem
5575:
5574:
5492:Commensurability
5453:Axiomatic system
5401:Angle trisection
5366:
5356:
5318:
5308:
5298:
5288:
5264:
5254:
5237:
4800:
4793:
4786:
4777:
4776:
4772:
4761:Polster, Burkard
4756:
4743:
4728:
4715:
4698:
4675:
4674:
4666:
4660:
4659:
4637:
4631:
4630:
4606:
4600:
4599:
4561:
4555:
4543:
4521:
4515:
4514:
4491:Word & Image
4482:
4473:
4472:
4432:
4426:
4425:
4391:
4385:
4384:
4342:
4329:
4324:
4303:
4297:
4296:
4283:
4277:
4276:
4242:
4201:
4195:
4194:
4165:
4159:
4158:
4148:
4124:
4118:
4117:
4079:
4070:
4069:
4046:
4040:
4039:
4037:
4035:
4022:
4016:
4015:
4002:
3996:
3995:
3993:
3991:
3982:. Archived from
3949:
3943:
3942:
3926:
3920:
3918:
3907:Dixon, Robert A.
3903:
3897:
3896:
3879:
3870:
3869:
3843:
3823:
3817:
3811:
3810:
3790:
3784:
3783:
3762:
3756:
3755:
3727:
3718:
3712:
3711:
3684:Merzbach, Uta C.
3676:
3670:
3669:
3659:
3630:
3624:
3623:
3602:
3596:
3595:
3585:
3559:
3553:
3552:
3533:10.2307/30037438
3526:
3501:
3495:
3494:
3466:
3460:
3449:
3448:
3436:
3431:
3420:
3407:
3401:
3380:
3371:
3370:
3334:
3328:
3319:
3318:
3281:
3272:
3271:
3235:
3229:
3223:
3222:
3203:
3197:
3196:
3162:
3156:
3151:
3136:
3130:
3129:
3107:
3101:
3100:
3080:
3074:
3073:
3060:
3054:
3053:
3033:
3027:
3026:
2995:
2989:
2988:
2954:
2948:
2934:
2921:
2915:
2908:
2902:
2892:
2883:
2882:
2862:
2852:
2826:
2820:
2819:
2792:
2786:
2785:
2732:
2723:
2722:
2720:
2718:
2706:
2680:
2611:perpetual motion
2468:
2466:
2465:
2460:
2404:
2402:
2401:
2396:
2352:
2350:
2349:
2344:
2332:
2330:
2329:
2324:
2322:
2293:
2291:
2279:
2278:
2273:
2272:
2264:
2262:
2258:
2257:
2252:
2251:
2242:
2237:
2236:
2214:
2212:
2211:
2206:
2185:
2169:
2141:
2139:
2138:
2133:
2128:
2120:
2115:
2087:
2085:
2084:
2079:
2067:
2065:
2064:
2059:
2054:
2036:
2032:
2014:
2006:
1984:
1972:
1960:
1936:
1934:
1933:
1928:
1923:
1921:
1920:
1919:
1907:
1906:
1896:
1895:
1886:
1881:
1876:
1868:
1852:
1850:
1849:
1844:
1842:
1820:
1812:
1788:
1776:
1760:
1758:
1756:
1755:
1750:
1736:
1734:
1733:
1728:
1714:
1709:
1708:
1699:
1698:
1689:
1681:
1660:
1658:
1653:
1645:
1637:
1635:
1633:
1628:
1627:
1618:
1617:
1608:
1600:
1595:
1594:
1585:
1584:
1575:
1563:
1561:
1560:
1555:
1543:
1541:
1540:
1535:
1512:
1510:
1509:
1504:
1490:
1475:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1371:
1369:
1368:
1363:
1351:
1349:
1348:
1343:
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1284:
1283:
1264:Euler's identity
1257:
1255:
1254:
1249:
1234:
1232:
1231:
1226:
1210:
1208:
1207:
1202:
1182:
1180:
1179:
1174:
1154:
1152:
1151:
1146:
1144:
1139:
1130:
1128:
1127:
1122:
1120:
1115:
1060:
1048:
1046:
1045:
1040:
1024:
1022:
1021:
1016:
892:
853:
851:
850:
845:
834:
801:
799:
798:
793:
785:
776:
754:
745:
728:
726:
724:
723:
718:
688:
686:
684:
683:
678:
650:
648:
646:
645:
640:
632:
623:
602:
600:
598:
597:
592:
584:
575:
553:
533:area of a circle
531:for finding the
512:
510:
509:
504:
480:
478:
477:
472:
456:
454:
453:
448:
390:is a problem in
377:
370:
363:
349:
341:
213:Jamshīd al-Kāshī
110:Area of a circle
96:
95:
92:
89:
86:
75:
59:
58:
50:
6106:
6105:
6101:
6100:
6099:
6097:
6096:
6095:
6056:
6055:
6054:
6049:
6038:
6027:
6025:
6014:
5980:Arabian/Islamic
5968:
5957:numeral systems
5846:
5796:
5766:
5714:Heron's formula
5692:
5674:
5566:
5562:Triangle center
5552:Regular polygon
5429:and definitions
5428:
5422:
5384:
5364:
5354:
5316:
5306:
5296:
5286:
5262:
5252:
5235:
5201:
5172:Theon of Smyrna
4817:
4809:
4804:
4683:
4678:
4667:
4663:
4638:
4634:
4624:
4607:
4603:
4580:10.2307/2855144
4562:
4558:
4527:The Shaw Review
4522:
4518:
4483:
4476:
4433:
4429:
4392:
4388:
4365:
4343:
4332:
4304:
4300:
4284:
4280:
4265:
4202:
4198:
4191:
4169:Gardner, Martin
4166:
4162:
4125:
4121:
4098:10.2307/3029284
4080:
4073:
4066:
4055:A History of Pi
4047:
4043:
4033:
4031:
4024:
4023:
4019:
4003:
3999:
3989:
3987:
3950:
3946:
3927:
3923:
3904:
3900:
3880:
3873:
3821:
3818:
3814:
3791:
3787:
3780:
3763:
3759:
3725:
3719:
3715:
3700:
3677:
3673:
3634:Cajori, Florian
3631:
3627:
3603:
3599:
3560:
3556:
3502:
3498:
3464:
3461:
3452:
3434:
3429:
3421:
3410:
3398:
3381:
3374:
3332:
3329:
3322:
3282:
3275:
3233:
3230:
3226:
3204:
3200:
3185:
3163:
3159:
3152:Available at:
3137:
3133:
3108:
3104:
3097:
3081:
3077:
3061:
3057:
3050:
3034:
3030:
3015:
2996:
2992:
2977:
2955:
2951:
2922:
2918:
2909:
2905:
2893:
2886:
2879:
2827:
2823:
2816:
2793:
2789:
2733:
2726:
2716:
2714:
2707:
2703:
2699:
2678:
2665:
2599:Similarly, the
2597:
2594:
2592:
2590:
2551:
2548:
2546:
2542:
2539:
2537:
2509:Meton of Athens
2491:
2471:Indiana pi bill
2454:
2451:
2450:
2421:
2390:
2387:
2386:
2373:as part of the
2363:
2358:
2338:
2335:
2334:
2308:
2287:
2277:
2263:
2247:
2243:
2241:
2232:
2228:
2227:
2223:
2222:
2220:
2217:
2216:
2200:
2197:
2196:
2193:
2192:
2191:
2190:
2189:
2186:
2178:
2177:
2170:
2159:
2151:geometrographic
2124:
2114:
2097:
2094:
2093:
2073:
2070:
2069:
2044:
2022:
2018:
2005:
2003:
2000:
1999:
1992:
1991:
1990:
1989:
1988:
1985:
1977:
1976:
1973:
1965:
1964:
1961:
1950:
1915:
1911:
1902:
1898:
1897:
1891:
1887:
1885:
1869:
1867:
1865:
1862:
1861:
1831:
1811:
1803:
1800:
1799:
1796:
1795:
1794:
1793:
1792:
1789:
1781:
1780:
1777:
1766:
1744:
1741:
1740:
1738:
1710:
1704:
1700:
1694:
1690:
1685:
1671:
1652:
1636:
1634:
1629:
1623:
1619:
1613:
1609:
1604:
1596:
1590:
1586:
1580:
1576:
1571:
1569:
1566:
1565:
1549:
1546:
1545:
1529:
1526:
1525:
1518:
1517:
1516:
1515:
1514:
1498:
1495:
1494:
1491:
1483:
1482:
1476:
1465:
1440:
1437:
1436:
1420:
1417:
1416:
1413:
1393:Euclidean space
1357:
1354:
1353:
1352:, to show that
1337:
1334:
1333:
1309:
1306:
1305:
1276:
1272:
1270:
1267:
1266:
1260:Charles Hermite
1243:
1240:
1239:
1220:
1217:
1216:
1196:
1193:
1192:
1168:
1165:
1164:
1138:
1136:
1133:
1132:
1114:
1112:
1109:
1108:
1105:
1058:
1034:
1031:
1030:
1010:
1007:
1006:
987:Vincent Léotaud
960:, was known as
890:
879:
830:
819:
816:
815:
774:
743:
737:
734:
733:
712:
709:
708:
706:
666:
663:
662:
661:
621:
613:
610:
609:
608:
573:
565:
562:
561:
560:
551:
545:
498:
495:
494:
466:
463:
462:
442:
439:
438:
381:
347:
339:
307:Indiana pi bill
290:A History of Pi
268:Yasumasa Kanada
93:
90:
87:
84:
82:
73:
48:
39:
32:
17:
12:
11:
5:
6104:
6094:
6093:
6088:
6083:
6078:
6073:
6068:
6051:
6050:
6023:
6020:
6019:
6016:
6015:
6013:
6012:
6007:
6002:
5997:
5992:
5987:
5982:
5976:
5974:
5973:Other cultures
5970:
5969:
5967:
5966:
5965:
5964:
5954:
5953:
5952:
5942:
5941:
5940:
5930:
5929:
5928:
5918:
5917:
5916:
5906:
5905:
5904:
5894:
5893:
5892:
5882:
5881:
5880:
5870:
5869:
5868:
5854:
5852:
5848:
5847:
5845:
5844:
5839:
5834:
5829:
5824:
5822:Greek numerals
5819:
5817:Attic numerals
5814:
5808:
5802:
5798:
5797:
5795:
5794:
5789:
5784:
5778:
5776:
5772:
5771:
5768:
5767:
5765:
5764:
5759:
5754:
5749:
5744:
5736:
5731:
5726:
5721:
5716:
5711:
5706:
5700:
5698:
5694:
5693:
5691:
5690:
5684:
5682:
5676:
5675:
5673:
5672:
5667:
5662:
5657:
5652:
5647:
5645:Law of cosines
5642:
5637:
5632:
5627:
5622:
5617:
5612:
5607:
5602:
5597:
5592:
5586:
5584:
5572:
5568:
5567:
5565:
5564:
5559:
5554:
5549:
5544:
5539:
5537:Platonic solid
5534:
5529:
5524:
5519:
5517:Greek numerals
5514:
5509:
5504:
5499:
5494:
5489:
5484:
5483:
5482:
5477:
5467:
5462:
5461:
5460:
5450:
5449:
5448:
5443:
5432:
5430:
5424:
5423:
5421:
5420:
5415:
5414:
5413:
5408:
5403:
5392:
5390:
5386:
5385:
5383:
5382:
5375:
5368:
5358:
5348:
5345:Planisphaerium
5341:
5334:
5327:
5320:
5310:
5300:
5290:
5280:
5273:
5266:
5256:
5246:
5239:
5229:
5222:
5217:
5209:
5207:
5203:
5202:
5200:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5164:
5159:
5154:
5149:
5144:
5139:
5134:
5129:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5089:
5084:
5079:
5074:
5069:
5064:
5059:
5054:
5049:
5044:
5039:
5034:
5029:
5024:
5019:
5014:
5009:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4954:
4949:
4944:
4939:
4934:
4929:
4924:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4884:
4879:
4874:
4869:
4864:
4859:
4854:
4849:
4844:
4839:
4834:
4829:
4823:
4821:
4815:Mathematicians
4811:
4810:
4803:
4802:
4795:
4788:
4780:
4774:
4773:
4757:
4744:
4729:
4716:
4702:Grime, James.
4699:
4682:
4679:
4677:
4676:
4661:
4650:(1): 105–107.
4632:
4622:
4601:
4574:(3): 545–557.
4556:
4516:
4497:(3): 252–260.
4474:
4427:
4408:(3): 213–222.
4386:
4363:
4330:
4307:Paul R. Halmos
4298:
4278:
4263:
4196:
4189:
4160:
4139:(2): 151–159.
4119:
4071:
4064:
4050:Beckmann, Petr
4041:
4017:
3997:
3966:(2): 217–231.
3944:
3921:
3898:
3871:
3812:
3785:
3778:
3757:
3713:
3698:
3680:Boyer, Carl B.
3671:
3650:(7): 339–347.
3625:
3597:
3583:10.4171/EM/179
3576:(3): 121–131.
3554:
3517:(3): 200–211.
3496:
3450:
3408:
3404:The Trisectors
3396:
3372:
3345:(2): 164–183.
3320:
3301:(2): 213–225.
3273:
3246:(5): 439–443.
3224:
3198:
3183:
3157:
3140:Gregory, James
3131:
3102:
3095:
3075:
3055:
3048:
3028:
3013:
2990:
2975:
2949:
2916:
2903:
2884:
2878:978-0387205717
2877:
2843:(4): 325–340.
2821:
2815:978-0691120676
2814:
2787:
2744:Borwein, P. B.
2740:Borwein, J. M.
2724:
2700:
2698:
2695:
2694:
2693:
2687:
2681:
2672:
2664:
2661:
2587:
2577:Alexander Pope
2543:
2534:
2490:
2487:
2458:
2418:
2394:
2362:
2359:
2355:
2342:
2320:
2316:
2312:
2307:
2303:
2299:
2296:
2290:
2285:
2282:
2276:
2270:
2267:
2261:
2255:
2250:
2246:
2240:
2235:
2231:
2226:
2204:
2187:
2180:
2179:
2171:
2164:
2163:
2162:
2161:
2160:
2158:
2155:
2131:
2127:
2123:
2118:
2113:
2110:
2107:
2104:
2101:
2077:
2057:
2052:
2048:
2042:
2039:
2035:
2031:
2028:
2025:
2021:
2017:
2012:
2009:
1986:
1979:
1978:
1974:
1967:
1966:
1962:
1955:
1954:
1953:
1952:
1951:
1949:
1946:
1926:
1918:
1914:
1910:
1905:
1901:
1894:
1890:
1884:
1879:
1875:
1872:
1840:
1836:
1830:
1826:
1823:
1818:
1815:
1810:
1807:
1790:
1783:
1782:
1778:
1771:
1770:
1769:
1768:
1767:
1765:
1762:
1748:
1726:
1723:
1720:
1717:
1713:
1707:
1703:
1697:
1693:
1688:
1684:
1679:
1675:
1670:
1666:
1663:
1656:
1651:
1648:
1643:
1640:
1632:
1626:
1622:
1616:
1612:
1607:
1603:
1599:
1593:
1589:
1583:
1579:
1574:
1553:
1533:
1502:
1492:
1485:
1484:
1477:
1470:
1469:
1468:
1467:
1466:
1464:
1461:
1444:
1424:
1412:
1409:
1361:
1341:
1313:
1293:
1290:
1287:
1282:
1279:
1275:
1262:in 1873, with
1247:
1237:Euler's number
1224:
1200:
1185:Pierre Wantzel
1172:
1142:
1118:
1104:
1101:
1089:cubic equation
1038:
1014:
876:
843:
840:
837:
833:
829:
826:
823:
791:
788:
782:
779:
772:
769:
766:
763:
760:
757:
751:
748:
741:
716:
676:
673:
670:
657:Books of Kings
638:
635:
629:
626:
620:
617:
590:
587:
581:
578:
572:
569:
544:
541:
535:. In general,
502:
470:
446:
383:
382:
380:
379:
372:
365:
357:
354:
353:
352:
351:
343:
335:
330:
322:
321:
320:Related topics
317:
316:
315:
314:
309:
301:
300:
296:
295:
294:
293:
286:
278:
277:
273:
272:
271:
270:
265:
260:
255:
250:
248:William Shanks
245:
240:
235:
230:
225:
223:François Viète
220:
215:
210:
205:
200:
195:
190:
182:
181:
177:
176:
175:
174:
169:
164:
162:Approximations
159:
157:Less than 22/7
151:
150:
146:
145:
144:
143:
138:
130:
129:
125:
124:
123:
122:
117:
112:
104:
103:
99:
98:
78:
77:
68:
67:
15:
9:
6:
4:
3:
2:
6103:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6063:
6061:
6048:
6047:
6042:
6035:
6034:
6021:
6011:
6008:
6006:
6003:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5977:
5975:
5971:
5963:
5960:
5959:
5958:
5955:
5951:
5948:
5947:
5946:
5943:
5939:
5936:
5935:
5934:
5931:
5927:
5924:
5923:
5922:
5919:
5915:
5912:
5911:
5910:
5907:
5903:
5900:
5899:
5898:
5895:
5891:
5888:
5887:
5886:
5883:
5879:
5876:
5875:
5874:
5871:
5867:
5863:
5862:
5861:
5860:
5856:
5855:
5853:
5849:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5809:
5806:
5803:
5799:
5793:
5790:
5788:
5785:
5783:
5780:
5779:
5777:
5773:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5742:
5737:
5735:
5732:
5730:
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5701:
5699:
5695:
5689:
5686:
5685:
5683:
5681:
5677:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5653:
5651:
5650:Pons asinorum
5648:
5646:
5643:
5641:
5638:
5636:
5633:
5631:
5628:
5626:
5623:
5621:
5620:Hinge theorem
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5587:
5585:
5583:
5582:
5576:
5573:
5569:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5481:
5478:
5476:
5473:
5472:
5471:
5468:
5466:
5463:
5459:
5456:
5455:
5454:
5451:
5447:
5444:
5442:
5439:
5438:
5437:
5434:
5433:
5431:
5425:
5419:
5416:
5412:
5409:
5407:
5404:
5402:
5399:
5398:
5397:
5394:
5393:
5391:
5387:
5381:
5380:
5376:
5374:
5373:
5369:
5367:
5363:
5359:
5357:
5353:
5349:
5347:
5346:
5342:
5340:
5339:
5335:
5333:
5332:
5328:
5326:
5325:
5321:
5319:
5315:
5311:
5309:
5305:
5301:
5299:
5295:
5291:
5289:
5287:(Aristarchus)
5285:
5281:
5279:
5278:
5274:
5272:
5271:
5267:
5265:
5261:
5257:
5255:
5251:
5247:
5245:
5244:
5240:
5238:
5234:
5230:
5228:
5227:
5223:
5221:
5218:
5216:
5215:
5211:
5210:
5208:
5204:
5198:
5195:
5193:
5192:Zeno of Sidon
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4928:
4925:
4923:
4920:
4918:
4915:
4913:
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4824:
4822:
4820:
4816:
4812:
4808:
4801:
4796:
4794:
4789:
4787:
4782:
4781:
4778:
4770:
4766:
4762:
4758:
4754:
4750:
4745:
4741:
4740:
4735:
4730:
4726:
4722:
4717:
4713:
4709:
4705:
4700:
4696:
4695:
4690:
4685:
4684:
4672:
4665:
4657:
4653:
4649:
4645:
4644:
4636:
4629:
4625:
4623:9780877548034
4619:
4615:
4611:
4610:Bloom, Harold
4605:
4597:
4593:
4589:
4585:
4581:
4577:
4573:
4569:
4568:
4560:
4553:
4549:
4548:
4541:
4537:
4533:
4529:
4528:
4520:
4512:
4508:
4504:
4500:
4496:
4492:
4488:
4481:
4479:
4470:
4466:
4462:
4458:
4454:
4450:
4446:
4442:
4438:
4431:
4423:
4419:
4415:
4411:
4407:
4403:
4402:
4397:
4390:
4382:
4378:
4374:
4370:
4366:
4360:
4356:
4352:
4348:
4341:
4339:
4337:
4335:
4328:
4323:(2): 123–152.
4322:
4318:
4317:
4312:
4308:
4302:
4294:
4293:
4288:
4282:
4274:
4270:
4266:
4264:0-387-20571-3
4260:
4256:
4252:
4248:
4243:Reprinted in
4240:
4236:
4232:
4228:
4224:
4220:
4216:
4212:
4211:
4206:
4200:
4192:
4190:0-387-94673-X
4186:
4182:
4178:
4174:
4170:
4164:
4156:
4152:
4147:
4142:
4138:
4134:
4130:
4123:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4086:
4078:
4076:
4067:
4065:9781466887169
4061:
4057:
4056:
4051:
4045:
4029:
4028:
4021:
4014:. p. 96.
4013:
4012:
4007:
4001:
3985:
3981:
3977:
3973:
3969:
3965:
3961:
3960:
3955:
3948:
3940:
3936:
3932:
3925:
3916:
3915:Mathographics
3912:
3908:
3902:
3894:
3890:
3889:
3884:
3878:
3876:
3867:
3863:
3859:
3855:
3851:
3847:
3842:
3837:
3833:
3829:
3828:
3816:
3808:
3804:
3800:
3796:
3789:
3781:
3775:
3771:
3767:
3761:
3753:
3749:
3745:
3741:
3737:
3733:
3732:
3724:
3717:
3709:
3705:
3701:
3695:
3691:
3690:
3685:
3681:
3675:
3667:
3663:
3658:
3653:
3649:
3645:
3644:
3639:
3635:
3629:
3621:
3618:(in French).
3617:
3616:
3611:
3607:
3601:
3593:
3589:
3584:
3579:
3575:
3571:
3570:
3565:
3558:
3550:
3546:
3542:
3538:
3534:
3530:
3525:
3520:
3516:
3512:
3511:
3506:
3500:
3492:
3488:
3484:
3480:
3476:
3472:
3471:
3459:
3457:
3455:
3446:
3442:
3441:
3433:
3425:
3424:Ramanujan, S.
3419:
3417:
3415:
3413:
3405:
3402:Reprinted as
3399:
3397:0-387-96568-8
3393:
3389:
3385:
3379:
3377:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3339:
3327:
3325:
3316:
3312:
3308:
3304:
3300:
3297:(in German).
3296:
3295:
3290:
3286:
3285:Lindemann, F.
3280:
3278:
3269:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3240:
3228:
3220:
3217:(in French).
3216:
3212:
3208:
3202:
3194:
3190:
3186:
3180:
3176:
3172:
3168:
3161:
3155:
3149:
3145:
3141:
3135:
3127:
3123:
3119:
3115:
3114:
3106:
3098:
3096:9780199213122
3092:
3088:
3087:
3079:
3071:
3070:
3065:
3059:
3051:
3049:9780262013178
3045:
3041:
3040:
3032:
3024:
3020:
3016:
3014:0-8176-3148-8
3010:
3006:
3005:
3000:
2994:
2986:
2982:
2978:
2972:
2968:
2964:
2960:
2953:
2946:
2942:
2938:
2932:
2931:
2926:
2925:Heath, Thomas
2920:
2913:
2907:
2900:
2899:
2891:
2889:
2880:
2874:
2870:
2869:
2863:Reprinted in
2860:
2856:
2851:
2846:
2842:
2838:
2837:
2832:
2825:
2817:
2811:
2807:
2803:
2802:
2797:
2791:
2783:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2754:
2749:
2745:
2741:
2737:
2736:Bailey, D. H.
2731:
2729:
2712:
2705:
2701:
2691:
2688:
2685:
2682:
2676:
2673:
2670:
2667:
2666:
2660:
2658:
2657:
2652:
2648:
2647:
2642:
2638:
2637:Leopold Bloom
2633:
2631:
2627:
2626:Spanos (1978)
2623:
2622:Arnaut Daniel
2619:
2614:
2612:
2608:
2607:
2602:
2596:
2586:
2584:
2583:
2578:
2573:
2568:
2566:
2565:
2564:Vitruvian Man
2560:
2556:
2550:
2541:
2533:
2531:
2530:
2522:
2521:
2520:Vitruvian Man
2516:
2512:
2510:
2506:
2502:
2501:
2496:
2489:In literature
2486:
2484:
2480:
2475:
2472:
2456:
2442:
2438:
2436:
2435:
2430:
2426:
2417:
2415:
2414:Lewis Carroll
2411:
2406:
2392:
2384:
2380:
2376:
2372:
2368:
2367:Thomas Hobbes
2354:
2340:
2318:
2314:
2310:
2305:
2301:
2297:
2294:
2288:
2283:
2280:
2274:
2268:
2265:
2259:
2253:
2248:
2244:
2238:
2233:
2229:
2224:
2202:
2184:
2175:
2168:
2154:
2152:
2148:
2143:
2129:
2125:
2116:
2111:
2108:
2102:
2099:
2091:
2075:
2055:
2050:
2046:
2040:
2037:
2033:
2029:
2026:
2023:
2019:
2015:
2010:
2007:
1997:
1983:
1971:
1959:
1945:
1943:
1938:
1924:
1916:
1912:
1908:
1903:
1899:
1892:
1888:
1882:
1873:
1870:
1858:
1856:
1838:
1834:
1828:
1824:
1821:
1816:
1813:
1808:
1805:
1787:
1775:
1761:
1746:
1724:
1721:
1718:
1715:
1705:
1701:
1695:
1691:
1682:
1677:
1673:
1668:
1664:
1661:
1654:
1649:
1646:
1641:
1638:
1624:
1620:
1614:
1610:
1601:
1591:
1587:
1581:
1577:
1551:
1531:
1523:
1500:
1489:
1480:
1474:
1460:
1458:
1442:
1422:
1408:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1373:
1359:
1339:
1331:
1327:
1311:
1291:
1288:
1285:
1280:
1277:
1273:
1265:
1261:
1245:
1238:
1222:
1214:
1198:
1190:
1186:
1170:
1162:
1158:
1140:
1116:
1103:Impossibility
1100:
1098:
1094:
1090:
1086:
1082:
1077:
1075:
1071:
1066:
1064:
1056:
1052:
1036:
1028:
1012:
1004:
1000:
999:James Gregory
996:
992:
988:
984:
980:
975:
972:
968:
963:
959:
955:
951:
947:
942:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
896:
888:
883:
875:
873:
868:
864:
859:
857:
841:
838:
835:
831:
827:
824:
821:
813:
809:
805:
789:
786:
780:
777:
770:
767:
764:
761:
758:
755:
749:
746:
739:
731:
714:
704:
703:
702:Shulba Sutras
698:
697:
692:
674:
671:
668:
659:
658:
654:
653:Old Testament
636:
633:
627:
624:
618:
615:
606:
588:
585:
579:
576:
570:
567:
558:
554:
540:
538:
534:
530:
526:
521:
519:
514:
500:
492:
488:
484:
468:
460:
444:
436:
432:
427:
425:
421:
417:
413:
409:
405:
401:
397:
393:
389:
378:
373:
371:
366:
364:
359:
358:
356:
355:
350:
344:
342:
338:Six nines in
336:
334:
333:Basel problem
331:
329:
326:
325:
324:
323:
319:
318:
313:
310:
308:
305:
304:
303:
302:
298:
297:
292:
291:
287:
285:
282:
281:
280:
279:
275:
274:
269:
266:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
238:William Jones
236:
234:
231:
229:
228:Seki Takakazu
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
185:
184:
183:
179:
178:
173:
170:
168:
165:
163:
160:
158:
155:
154:
153:
152:
148:
147:
142:
141:Transcendence
139:
137:
136:Irrationality
134:
133:
132:
131:
127:
126:
121:
118:
116:
115:Circumference
113:
111:
108:
107:
106:
105:
101:
100:
97:
80:
79:
76:
70:
69:
65:
61:
60:
54:
45:
41:
37:
30:
26:
22:
6037:
6024:
5866:Thomas Heath
5857:
5740:
5724:Law of sines
5580:
5512:Golden ratio
5410:
5377:
5370:
5361:
5355:(Theodosius)
5351:
5343:
5336:
5329:
5322:
5313:
5303:
5297:(Hipparchus)
5293:
5283:
5275:
5268:
5259:
5249:
5241:
5236:(Apollonius)
5232:
5224:
5212:
5187:Zeno of Elea
4947:Eratosthenes
4937:Dionysodorus
4768:
4752:
4737:
4724:
4707:
4694:cut-the-knot
4692:
4670:
4664:
4647:
4641:
4635:
4627:
4613:
4604:
4571:
4565:
4559:
4545:
4534:(2): 52–56.
4531:
4525:
4519:
4494:
4490:
4486:
4444:
4440:
4436:
4430:
4405:
4399:
4395:
4389:
4346:
4320:
4314:
4301:
4291:
4281:
4246:
4217:(2): 69–72.
4214:
4208:
4199:
4172:
4163:
4136:
4132:
4122:
4089:
4083:
4054:
4044:
4032:. Retrieved
4026:
4020:
4009:
4000:
3988:. Retrieved
3984:the original
3963:
3957:
3947:
3938:
3934:
3924:
3914:
3901:
3887:
3834:(4): 40–45.
3831:
3825:
3815:
3798:
3788:
3769:
3760:
3738:(2): 31–36.
3735:
3729:
3716:
3688:
3674:
3647:
3641:
3628:
3619:
3613:
3600:
3573:
3567:
3557:
3524:math/0408159
3514:
3508:
3499:
3477:(2): 67–98.
3474:
3468:
3444:
3438:
3403:
3387:
3342:
3336:
3298:
3292:
3243:
3237:
3227:
3218:
3214:
3201:
3166:
3160:
3147:
3143:
3134:
3117:
3111:
3105:
3085:
3078:
3068:
3064:Cotes, Roger
3058:
3038:
3031:
3003:
2993:
2958:
2952:
2929:
2919:
2912:Knorr (1986)
2906:
2897:
2867:
2840:
2834:
2824:
2800:
2796:Plofker, Kim
2790:
2760:(1): 50–57.
2757:
2751:
2715:. Retrieved
2704:
2654:
2644:
2634:
2615:
2606:Princess Ida
2604:
2603:comic opera
2598:
2588:
2580:
2569:
2562:
2552:
2544:
2535:
2527:
2525:
2518:
2505:Aristophanes
2498:
2495:metaphorical
2492:
2476:
2447:
2434:The Athenæum
2432:
2428:
2422:
2407:
2382:
2364:
2194:
2147:Robert Dixon
2144:
2090:golden ratio
1996:E. W. Hobson
1993:
1939:
1859:
1797:
1519:
1414:
1400:
1374:
1106:
1078:
1067:
1002:
978:
976:
961:
943:
900:
860:
700:
694:
655:
546:
524:
522:
515:
428:
387:
386:
327:
288:
233:Takebe Kenko
172:Memorization
81:
40:
5933:mathematics
5741:Arithmetica
5338:Ostomachion
5307:(Autolycus)
5226:Arithmetica
5002:Hippocrates
4932:Dinostratus
4917:Dicaearchus
4847:Aristarchus
4753:Convergence
4725:Convergence
4712:Brady Haran
4708:Numberphile
3990:14 November
3606:Wantzel, L.
2945:pp. 220–235
2941:pp. 183–200
2937:pp. 172–174
2748:Plouffe, S.
2651:Thomas Mann
2641:James Joyce
2483:Paul Halmos
2371:John Wallis
1479:Kochański's
1258:, shown by
946:integration
812:Zu Chongzhi
481:is not the
461:. That is,
258:John Wrench
243:John Machin
198:Zu Chongzhi
6060:Categories
5985:Babylonian
5885:arithmetic
5851:History of
5680:Apollonius
5365:(Menelaus)
5324:On Spirals
5243:Catoptrics
5182:Xenocrates
5177:Thymaridas
5162:Theodosius
5147:Theaetetus
5127:Simplicius
5117:Pythagoras
5102:Posidonius
5087:Philonides
5047:Nicomachus
5042:Metrodorus
5032:Menaechmus
4987:Hipparchus
4977:Heliodorus
4927:Diophantus
4912:Democritus
4892:Chrysippus
4862:Archimedes
4857:Apollonius
4827:Anaxagoras
4819:(timeline)
4769:Mathologer
4489: ".
4447:: 95–125.
3622:: 366–372.
3447:: 350–372.
2901:, Book II.
2697:References
954:quadrature
903:Anaxagoras
893:(found by
730:Archimedes
537:quadrature
487:polynomial
299:In culture
284:Chronology
188:Archimedes
128:Properties
5446:Inscribed
5206:Treatises
5197:Zenodorus
5157:Theodorus
5132:Sosigenes
5077:Philolaus
5062:Oenopides
5057:Nicoteles
5052:Nicomedes
5012:Hypsicles
4907:Ctesibius
4897:Cleomedes
4882:Callippus
4867:Autolycus
4852:Aristotle
4832:Anthemius
4596:162823092
4511:194056860
4487:Imago dei
4469:155844205
4381:234128826
4239:122137198
3980:171077338
3866:123623596
3841:1111.1739
3752:120481094
3708:839010064
3367:119986449
3315:120469397
3193:132820288
2896:Euclid's
2643:'s novel
2555:Vitruvius
2500:The Birds
2457:π
2410:Victorian
2393:π
2341:π
2319:…
2203:π
2174:animation
2100:φ
2076:φ
2051:…
2030:φ
2016:⋅
1878:¯
1839:…
1809:≈
1806:π
1747:π
1719:π
1716:≈
1683:⋅
1662:≈
1647:−
1552:π
1532:π
1443:π
1423:π
1405:countably
1383:uses the
1360:π
1312:π
1289:−
1281:π
1223:π
1199:π
1171:π
1141:π
1117:π
1037:π
1013:π
991:hyperbola
971:Oldenburg
969:wrote to
939:Oenopides
839:≈
825:≈
822:π
787:≈
765:π
756:≈
715:π
672:≈
669:π
634:≈
619:≈
616:π
571:≈
568:π
523:The term
501:π
469:π
445:π
402:with the
203:Aryabhata
6010:Japanese
5995:Egyptian
5938:timeline
5926:timeline
5914:timeline
5909:geometry
5902:timeline
5897:calculus
5890:timeline
5878:timeline
5581:Elements
5427:Concepts
5389:Problems
5362:Spherics
5352:Spherics
5317:(Euclid)
5263:(Euclid)
5260:Elements
5253:(Euclid)
5214:Almagest
5122:Serenus
5097:Porphyry
5037:Menelaus
4992:Hippasus
4967:Eutocius
4942:Domninus
4837:Archytas
4656:25473619
4612:(1987).
4567:Speculum
4540:40682600
4461:27831895
4441:Traditio
4437:Paradiso
4325:—
4309:(1970).
4289:(1934).
4171:(1996).
4052:(2015).
4034:1 August
4008:(1872).
3935:Parabola
3909:(1987).
3885:(1913).
3768:(2008).
3636:(1918).
3608:(1837).
3541:30037438
3426:(1914).
3386:(1987).
3287:(1882).
3209:(1761).
3142:(1667).
3120:: 1–17.
3066:(1850).
3001:(1986).
2927:(1921).
2898:Elements
2798:(2009).
2782:14318695
2717:16 April
2684:Squircle
2663:See also
2630:O. Henry
2529:Paradise
2526:Dante's
2427:'s book
962:squaring
950:calculus
842:3.141593
689:Ancient
491:rational
392:geometry
94:26433...
62:Part of
5990:Chinese
5945:numbers
5873:algebra
5801:Related
5775:Centers
5571:Results
5441:Central
5112:Ptolemy
5107:Proclus
5072:Perseus
5027:Marinus
5007:Hypatia
4997:Hippias
4972:Geminus
4962:Eudoxus
4952:Eudemus
4922:Diocles
4588:2855144
4373:4272388
4273:2065455
4231:0784946
4155:1221681
4114:0037596
4106:3029832
3858:3029928
3807:1872936
3666:1560082
3592:2824428
3549:2125383
3491:2690037
3359:0774394
3268:1447977
3260:2974737
3023:0884893
2985:1800805
2914:, p. 25
2859:0875525
2774:1439159
2646:Ulysses
2618:sestina
2582:Dunciad
2088:is the
1155:were a
923:Eudemus
872:Proclus
867:polygon
808:Liu Hui
543:History
485:of any
457:) is a
424:circles
276:History
208:Madhava
193:Liu Hui
83:3.14159
6005:Indian
5782:Cyrene
5314:Optics
5233:Conics
5152:Theano
5142:Thales
5137:Sporus
5082:Philon
5067:Pappus
4957:Euclid
4887:Carpus
4877:Bryson
4654:
4620:
4594:
4586:
4538:
4509:
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2772:
2215:to be
2068:where
1324:is an
1049:is an
967:Newton
814:found
412:axioms
400:square
312:Pi Day
180:People
66:on the
27:, and
6000:Incan
5921:logic
5697:Other
5465:Chord
5458:Axiom
5436:Angle
5092:Plato
4982:Heron
4902:Conon
4652:JSTOR
4592:S2CID
4584:JSTOR
4536:JSTOR
4507:S2CID
4465:S2CID
4457:JSTOR
4418:JSTOR
4396:Birds
4377:S2CID
4235:S2CID
4102:JSTOR
3976:S2CID
3862:S2CID
3836:arXiv
3748:S2CID
3726:(PDF)
3537:JSTOR
3519:arXiv
3487:JSTOR
3435:(PDF)
3363:S2CID
3311:S2CID
3256:JSTOR
3189:S2CID
3146:[
2778:S2CID
2298:3.141
2041:3.141
1825:3.141
1665:3.141
1061:is a
952:, or
878:arcs.
802:. In
790:3.143
759:3.141
607:used
589:3.125
489:with
420:lines
149:Value
91:23846
88:89793
85:26535
5962:list
5250:Data
5022:Leon
4872:Bion
4618:ISBN
4359:ISBN
4259:ISBN
4185:ISBN
4060:ISBN
4036:2021
3992:2020
3895:–35.
3774:ISBN
3704:OCLC
3694:ISBN
3392:ISBN
3179:ISBN
3091:ISBN
3044:ISBN
3009:ISBN
2971:ISBN
2873:ISBN
2810:ISBN
2719:2012
2616:The
2408:The
2281:2143
1855:Milü
1739:for
1083:and
856:Milü
768:<
762:<
699:and
637:3.16
483:root
422:and
102:Uses
5864:by
5578:In
4576:doi
4550:by
4499:doi
4449:doi
4410:doi
4406:105
4398:".
4351:doi
4327:Pdf
4251:doi
4219:doi
4177:doi
4141:doi
4094:doi
3968:doi
3846:doi
3740:doi
3652:doi
3578:doi
3529:doi
3515:112
3479:doi
3467:".
3347:doi
3303:doi
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3244:104
3236:".
3171:doi
3122:doi
2963:doi
2845:doi
2762:doi
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2561:'s
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2315:582
2302:592
2047:640
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