2862:
735:
1403:
44:
1221:
186:
1382:
1014:
914:
1094:
1692:
679:
86:
580:
1226:
269:
769:
of a limit governs the number of terms of the expression needed to achieve a given number of digits of accuracy. In Viète's formula, the numbers of terms and digits are proportional to each other: the product of the first
1082:
1565:
932:
402:
293:, but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses and as a motivating example for the concept of
723:
gets arbitrarily large, these finite products have values that approach the value of Viète's formula arbitrarily closely. Viète did his work long before the concepts of limits and rigorous proofs of
824:
1231:
590:
1578:
585:
498:
1486:-gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a
1769:
1468:
191:
1216:{\displaystyle \pi =\lim _{k\to \infty }2^{k}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}}}} _{k{\text{ square roots}}},}
2214:
721:
701:
181:{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }
1025:
1503:
1377:{\displaystyle {\begin{aligned}\pi &=\lim _{k\to \infty }2^{k}{\sqrt {2-a_{k}}},\\a_{1}&=0,\\a_{k}&={\sqrt {2+a_{k-1}}}.\end{aligned}}}
358:
277:, who published it in 1593. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a
2560:
2296:
2244:
2181:
1899:
1867:
1826:
414:. As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation,
2040:
1414:, inscribed in a circle. The ratios between areas or perimeters of consecutive polygons in the sequence give the terms of Viète's formula.
2813:
2599:
2511:
2490:
2109:
1776:
1498:
2065:
324:, that has Viète's formula as a special case. Many similar formulas involving nested roots or infinite products are now known.
2032:
Maths for the
Mystified: An Exploration of the History of Mathematics and Its Relationship to Modern-day Science and Computing
2729:
347:
to (in principle) arbitrary accuracy had long been known. Viète's own method can be interpreted as a variation of an idea of
2355:
1712:
goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution
2329:
1394:
are now known, similar to Viète's in their use of either nested radicals or infinite products of trigonometric functions.
442:
1982:
Cullerne, J. P.; Goekjian, M. C. Dunn (December 2011). "Teaching wave propagation and the emergence of Viète's formula".
1009:{\displaystyle {\frac {2}{\pi }}=\cos {\frac {\pi }{4}}\cdot \cos {\frac {\pi }{8}}\cdot \cos {\frac {\pi }{16}}\cdots }
2234:
2898:
1490:(the diameter of the circle, counted twice) and a square, the ratio of perimeters of a square and an octagon, etc.
1482:
to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a
909:{\displaystyle {\frac {\sin x}{x}}=\cos {\frac {x}{2}}\cdot \cos {\frac {x}{4}}\cdot \cos {\frac {x}{8}}\cdots }
445:
2768:
484:
294:
2332:. Vol. 12. New York: John Wiley & Sons for the Mathematical Association of America. pp. 1–12.
2221:
1894:. Translated by Wilson, Stephen S. Providence, Rhode Island: American Mathematical Society. pp. 44–46.
483:, equal to the integral of products of the same functions, provides a motivating example for the concept of
2903:
1568:
1494:
341:
74:
2648:
1687:{\displaystyle \sin x=2^{n}\sin {\frac {x}{2^{n}}}\left(\prod _{i=1}^{n}\cos {\frac {x}{2^{i}}}\right).}
419:
282:
2861:
2597:
Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant".
727:
were developed in mathematics; the first proof that this limit exists was not given until the work of
1936:
674:{\displaystyle {\begin{aligned}a_{1}&={\sqrt {2}}\\a_{n}&={\sqrt {2+a_{n-1}}}.\end{aligned}}}
475:
in the limiting behavior of these speeds. Additionally, a derivation of this formula as a product of
274:
1738:
2696:
1441:
798:
766:
468:
355:
of a circle by the perimeter of a many-sided polygon, used by
Archimedes to find the approximation
278:
35:
2867:
2830:
2793:
2785:
2673:
2616:
2528:
2464:
2409:
2276:
2217:
2136:
2118:
1999:
1961:
1953:
1479:
1019:
724:
575:{\displaystyle \lim _{n\rightarrow \infty }\prod _{i=1}^{n}{\frac {a_{i}}{2}}={\frac {2}{\pi }},}
457:
313:
286:
2642:
Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for
1018:
Then, expressing each term of the product on the right as a function of earlier terms using the
734:
2486:
2292:
2240:
2230:
2177:
2036:
2030:
1895:
1863:
1822:
480:
406:
By publishing his method as a mathematical formula, Viète formulated the first instance of an
336:
to two French kings, and amateur mathematician. He published this formula in 1593 in his work
2444:
Translated into
English by Thomas W. Polaski. See final formula. The same formula is also in
2434:[On various methods for expressing the quadrature of a circle with verging numbers].
2163:
1887:
1818:
2822:
2777:
2738:
2657:
2608:
2569:
2520:
2401:
2367:
2284:
2265:
2169:
2128:
2074:
1991:
1945:
1402:
467:
Beyond its mathematical and historical significance, Viète's formula can be used to explain
423:
407:
333:
66:
2842:
2752:
2709:
2669:
2628:
2583:
2540:
2337:
2148:
2088:
1909:
1836:
2838:
2748:
2705:
2691:
2665:
2624:
2579:
2536:
2385:
2371:
2351:
2333:
2199:
2144:
2084:
1905:
1832:
1809:
1423:
1407:
728:
410:
known in mathematics, and the first example of an explicit formula for the exact value of
1732:
2450:
2388:(2007). "A simple geometric method of estimating the error in using Vieta's product for
2446:
2431:
2427:
818:
786:
706:
686:
321:
70:
43:
17:
2892:
2797:
2677:
2413:
2356:"Ueber die Convergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung"
2003:
1965:
1805:
814:
434:
352:
264:{\displaystyle {\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}.}
1995:
2882:
2694:(2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers".
2269:
1949:
1411:
1391:
317:
2724:
2107:
Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums".
1091:
that still involves nested square roots of two, but uses only one multiplication:
761:
terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.
456:
digits and 16 decimal digits in 1424. Not long after Viète published his formula,
2876:
2485:(1st ed.). Oxford, United Kingdom: Oxford University Press. pp. 57–58.
2453:[Various observations about angles proceeding in geometric progression].
2288:
2725:"Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products"
2226:
1475:
453:
58:
2857:
2661:
2574:
2555:
2405:
2173:
2079:
2060:
348:
2556:"Some closed-form evaluations of infinite products involving nested radicals"
2451:"Variae observationes circa angulos in progressione geometrica progredientes"
2743:
2360:
Historisch-litterarische
Abteilung der Zeitschrift für Mathematik und Physik
301:
2324:(1959). "Chapter 1: From Vieta to the notion of statistical independence".
300:
The formula can be derived as a telescoping product of either the areas or
2358:[On the convergence of a special product expansion due to Vieta].
2394:
International
Journal of Mathematical Education in Science and Technology
2321:
1855:
1087:
It is also possible to derive from Viète's formula a related formula for
476:
415:
2620:
1957:
2834:
2789:
2532:
2239:. Princeton, New Jersey: Princeton University Press. pp. 221–234.
2140:
1862:. Princeton, New Jersey: Princeton University Press. pp. 50, 140.
1471:
813:
Viète's formula may be obtained as a special case of a formula for the
305:
2483:
Euler's pioneering equation: the most beautiful theorem in mathematics
2224:(c. 1340 – 1425), but were not known in Europe until much later. See:
495:
Viète's formula may be rewritten and understood as a limit expression
2612:
2123:
1435:
309:
2826:
2781:
2524:
2132:
2326:
Statistical
Independence in Probability, Analysis and Number Theory
460:
used a method closely related to Viète's to calculate 35 digits of
2468:
2165:
An Atlas of
Functions: with Equator, the Atlas Function Calculator
1487:
1401:
733:
471:
in an infinite chain of springs and masses, and the appearance of
1077:{\displaystyle \cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}}
2432:"De variis modis circuli quadraturam numeris proxime exprimendi"
1419:
1560:{\displaystyle \sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}},}
785:
digits. This convergence rate compares very favorably with the
464:, which were published only after van Ceulen's death in 1610.
2811:
Rummler, Hansklaus (1993). "Squaring the circle with holes".
1817:(2nd ed.). Boulder, Colorado: The Golem Press. pp.
397:{\displaystyle {\frac {223}{71}}<\pi <{\frac {22}{7}}.}
1474:, the second term is the ratio of areas of an octagon and a
2231:"7.3.1 Mādhava on the circumference and arcs of the circle"
437:. However, this was not the most accurate approximation to
78:
2162:
Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010).
703:, the expression in the limit is a finite product, and as
34:. For formulas for symmetric functions of the roots, see
2283:. Berlin & Heidelberg: Springer. pp. 531–561.
2059:
Moreno, Samuel G.; García-Caballero, Esther M. (2013).
793:. Although Viète himself used his formula to calculate
426:
calls its appearance "the dawn of modern mathematics".
418:
highlights Viète's formula as marking the beginning of
2878:
Variorum de rebus mathematicis responsorum, liber VIII
469:
the different speeds of waves of different frequencies
338:
Variorum de rebus mathematicis responsorum, liber VIII
50:
Variorum de rebus mathematicis responsorum, liber VIII
2270:"The life of Pi: From Archimedes to ENIAC and beyond"
2202:
1741:
1581:
1506:
1444:
1229:
1097:
1028:
935:
827:
709:
689:
588:
501:
361:
194:
89:
2885:. The formula is on the second half of p. 30.
2208:
1763:
1686:
1559:
1462:
1376:
1215:
1076:
1008:
908:
801:version of his formula has been used to calculate
738:Comparison of the convergence of Viète's formula (
715:
695:
673:
574:
396:
263:
180:
2509:Servi, L. D. (2003). "Nested square roots of 2".
2436:Commentarii Academiae Scientiarum Petropolitanae
1245:
1105:
503:
332:François Viète (1540–1603) was a French lawyer,
2766:Allen, Edward J. (1985). "Continued radicals".
2196:Very similar infinite trigonometric series for
1934:to thousands of digits from Vieta's formula".
320:leads to a generalized formula, discovered by
1497:and Euler's formula. Repeatedly applying the
744:) and several historical infinite series for
8:
1418:Viète obtained his formula by comparing the
1925:
1923:
1921:
1919:
1886:Eymard, Pierre; Lafon, Jean Pierre (2004).
1470:, is the ratio of areas of a square and an
774:terms in the limit gives an expression for
48:
2471:. See the formula in numbered paragraph 3.
2260:
2258:
2256:
2102:
2100:
2098:
1977:
1975:
1800:
1798:
1796:
1794:
1792:
2742:
2573:
2201:
2122:
2078:
1752:
1740:
1668:
1659:
1647:
1636:
1619:
1610:
1598:
1580:
1544:
1528:
1505:
1452:
1445:
1443:
1353:
1341:
1328:
1301:
1282:
1270:
1264:
1248:
1230:
1228:
1203:
1199:
1178:
1164:
1156:
1148:
1140:
1131:
1124:
1108:
1096:
1048:
1035:
1027:
993:
974:
955:
936:
934:
893:
874:
855:
828:
826:
708:
688:
650:
638:
625:
610:
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559:
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533:
522:
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500:
381:
362:
360:
244:
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223:
212:
195:
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160:
152:
143:
127:
118:
103:
90:
88:
2504:
2502:
2463:Translated into English by Jordan Bell,
2016:
1881:
1879:
1493:Another derivation is possible based on
42:
1788:
789:, a later infinite product formula for
47:Viète's formula, as printed in Viète's
2316:
2314:
1850:
1848:
1846:
281:expression and marks the beginning of
2561:Rocky Mountain Journal of Mathematics
2054:
2052:
312:. Alternatively, repeated use of the
7:
1223:which can be rewritten compactly as
805:to hundreds of thousands of digits.
429:Using his formula, Viète calculated
289:and can be used for calculations of
30:This article is about a formula for
2035:. Leicester: Matador. p. 165.
2168:. New York: Springer. p. 15.
1255:
1115:
817:that has often been attributed to
797:only with nine-digit accuracy, an
778:that is accurate to approximately
757:is the approximation after taking
513:
224:
27:Infinite product converging to 2/π
25:
2814:The American Mathematical Monthly
2600:The American Mathematical Monthly
2512:The American Mathematical Monthly
2110:The American Mathematical Monthly
1438:. The first term in the product,
2860:
2281:From Alexandria, Through Baghdad
1777:List of trigonometric identities
1571:that, for all positive integers
1390:and other constants such as the
2066:Journal of Approximation Theory
1410:with numbers of sides equal to
2730:Pacific Journal of Mathematics
2723:Stolarsky, Kenneth B. (1980).
1950:10.1080/0025570X.2008.11953549
1888:"2.1 Viète's infinite product"
1764:{\displaystyle x=2^{n}\alpha }
1252:
1112:
821:, more than a century later:
510:
491:Interpretation and convergence
188:It can also be represented as
1:
2330:Carus Mathematical Monographs
2029:De Smith, Michael J. (2006).
1463:{\displaystyle {\sqrt {2}}/2}
77:of the mathematical constant
2289:10.1007/978-3-642-36736-6_24
340:. At this time, methods for
273:The formula is named after
2920:
441:known at the time, as the
29:
2662:10.1007/s11139-005-4852-z
2575:10.1216/RMJ-2012-42-2-751
2481:Wilson, Robin J. (2018).
2406:10.1080/00207390601002799
2174:10.1007/978-0-387-48807-3
2080:10.1016/j.jat.2013.06.006
1996:10.1088/0031-9120/47/1/87
1930:Kreminski, Rick (2008). "
1478:, etc. Thus, the product
2769:The Mathematical Gazette
2061:"On Viète-like formulas"
1735:, same identity taking
1495:trigonometric identities
485:statistical independence
295:statistical independence
2744:10.2140/pjm.1980.89.209
2222:Madhava of Sangamagrama
1084:gives Viète's formula.
929:in this formula yields
452:to an accuracy of nine
433:to an accuracy of nine
73:representing twice the
2554:Nyblom, M. A. (2012).
2210:
1860:Trigonometric Delights
1765:
1688:
1652:
1569:mathematical induction
1561:
1464:
1415:
1378:
1217:
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1010:
910:
762:
717:
697:
675:
576:
538:
398:
265:
228:
182:
54:
49:
18:Proof of Viète formula
2649:The Ramanujan Journal
2275:. In Sidoli, Nathan;
2211:
1766:
1689:
1632:
1562:
1465:
1434:sides inscribed in a
1405:
1379:
1218:
1079:
1011:
911:
737:
718:
698:
676:
577:
518:
443:Persian mathematician
420:mathematical analysis
399:
351:of approximating the
283:mathematical analysis
266:
208:
183:
46:
2266:Borwein, Jonathan M.
2236:Mathematics in India
2216:appeared earlier in
2209:{\displaystyle \pi }
2200:
1937:Mathematics Magazine
1739:
1579:
1567:leads to a proof by
1504:
1499:double-angle formula
1442:
1227:
1095:
1026:
933:
825:
707:
687:
586:
499:
359:
192:
87:
2697:Fibonacci Quarterly
2277:Van Brummelen, Glen
767:rate of convergence
683:For each choice of
2868:Mathematics portal
2455:Opuscula Analytica
2218:Indian mathematics
2206:
1771:on Viète's formula
1761:
1684:
1557:
1460:
1416:
1386:Many formulae for
1374:
1372:
1259:
1213:
1209:
1205: square roots
1197:
1119:
1074:
1020:half-angle formula
1006:
906:
763:
713:
693:
671:
669:
572:
517:
458:Ludolph van Ceulen
394:
314:half-angle formula
287:linear convergence
261:
178:
55:
2899:Infinite products
2298:978-3-642-36735-9
2246:978-0-691-12067-6
2220:, in the work of
2183:978-0-387-48807-3
1984:Physics Education
1901:978-0-8218-3246-2
1869:978-1-4008-4282-7
1828:978-0-88029-418-8
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716:{\displaystyle n}
696:{\displaystyle n}
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481:Rademacher system
389:
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173:
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109:
98:
65:is the following
16:(Redirected from
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2776:(450): 261–263.
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2692:Osler, Thomas J.
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1708:in the limit as
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1408:regular polygons
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408:infinite product
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334:privy councillor
308:converging to a
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128:
120:
119:
114:
105:
104:
99:
91:
81:
67:infinite product
52:
36:Vieta's formulas
33:
21:
2919:
2918:
2914:
2913:
2912:
2910:
2909:
2908:
2889:
2888:
2866:
2859:
2856:
2851:
2850:
2827:10.2307/2324662
2810:
2809:
2805:
2782:10.2307/3617569
2765:
2764:
2760:
2722:
2721:
2717:
2690:
2689:
2685:
2643:
2641:
2640:
2636:
2596:
2595:
2591:
2553:
2552:
2548:
2525:10.2307/3647881
2508:
2507:
2500:
2493:
2480:
2479:
2475:
2447:Euler, Leonhard
2445:
2428:Euler, Leonhard
2426:
2425:
2421:
2389:
2384:
2383:
2379:
2350:
2349:
2345:
2320:
2319:
2312:
2303:
2301:
2299:
2272:
2264:
2263:
2254:
2247:
2225:
2198:
2197:
2195:
2191:
2184:
2161:
2160:
2156:
2133:10.2307/2974641
2106:
2105:
2096:
2058:
2057:
2050:
2043:
2028:
2027:
2023:
2015:
2011:
1981:
1980:
1973:
1931:
1929:
1928:
1917:
1902:
1885:
1884:
1877:
1870:
1854:
1853:
1844:
1829:
1812:
1804:
1803:
1790:
1785:
1748:
1737:
1736:
1729:
1718:
1713:
1709:
1705:
1697:
1664:
1631:
1627:
1615:
1594:
1577:
1576:
1572:
1502:
1501:
1483:
1440:
1439:
1431:
1427:
1400:
1387:
1371:
1370:
1349:
1334:
1324:
1321:
1320:
1307:
1297:
1294:
1293:
1278:
1260:
1237:
1225:
1224:
1120:
1093:
1092:
1088:
1051:
1024:
1023:
931:
930:
924:
919:
830:
823:
822:
811:
802:
794:
790:
779:
775:
771:
758:
754:
749:
745:
739:
729:Ferdinand Rudio
705:
704:
685:
684:
668:
667:
646:
631:
621:
618:
617:
603:
593:
584:
583:
541:
497:
496:
493:
472:
461:
449:
448:had calculated
438:
430:
411:
357:
356:
343:
330:
290:
240:
190:
189:
85:
84:
79:
71:nested radicals
63:Viète's formula
39:
31:
28:
23:
22:
15:
12:
11:
5:
2917:
2915:
2907:
2906:
2901:
2891:
2890:
2887:
2886:
2872:
2871:
2855:
2854:External links
2852:
2849:
2848:
2821:(9): 858–860.
2803:
2758:
2737:(1): 209–227.
2715:
2704:(3): 202–204.
2683:
2656:(3): 305–324.
2634:
2607:(6): 510–520.
2589:
2568:(2): 751–758.
2546:
2519:(4): 326–330.
2498:
2491:
2473:
2419:
2400:(1): 136–142.
2377:
2343:
2310:
2297:
2252:
2245:
2205:
2189:
2182:
2154:
2117:(8): 716–724.
2094:
2048:
2041:
2021:
2009:
1971:
1944:(3): 201–207.
1915:
1900:
1875:
1868:
1842:
1827:
1806:Beckmann, Petr
1787:
1786:
1784:
1781:
1780:
1779:
1773:
1772:
1760:
1755:
1751:
1747:
1744:
1728:
1725:
1683:
1679:
1671:
1667:
1663:
1658:
1655:
1650:
1645:
1642:
1639:
1635:
1630:
1622:
1618:
1614:
1609:
1606:
1601:
1597:
1593:
1590:
1587:
1584:
1556:
1551:
1548:
1543:
1540:
1535:
1532:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1459:
1455:
1449:
1406:A sequence of
1399:
1396:
1369:
1362:
1359:
1356:
1352:
1348:
1345:
1340:
1337:
1335:
1331:
1327:
1323:
1322:
1319:
1316:
1313:
1310:
1308:
1304:
1300:
1296:
1295:
1292:
1285:
1281:
1277:
1274:
1267:
1263:
1257:
1254:
1251:
1247:
1243:
1240:
1238:
1236:
1233:
1232:
1212:
1202:
1196:
1182:
1177:
1174:
1171:
1168:
1163:
1160:
1155:
1152:
1147:
1144:
1139:
1136:
1127:
1123:
1117:
1114:
1111:
1107:
1103:
1100:
1070:
1066:
1063:
1060:
1057:
1054:
1047:
1042:
1039:
1034:
1031:
1005:
1000:
997:
992:
989:
986:
981:
978:
973:
970:
967:
962:
959:
954:
951:
948:
943:
940:
905:
900:
897:
892:
889:
886:
881:
878:
873:
870:
867:
862:
859:
854:
851:
848:
843:
839:
836:
833:
819:Leonhard Euler
810:
807:
787:Wallis product
752:
712:
692:
666:
659:
656:
653:
649:
645:
642:
637:
634:
632:
628:
624:
620:
619:
614:
609:
606:
604:
600:
596:
592:
591:
571:
566:
563:
558:
553:
548:
544:
536:
531:
528:
525:
521:
515:
512:
509:
505:
492:
489:
479:involving the
435:decimal digits
393:
388:
385:
380:
377:
374:
369:
366:
342:approximating
329:
326:
322:Leonhard Euler
275:François Viète
260:
253:
250:
247:
243:
239:
234:
231:
226:
221:
218:
215:
211:
207:
202:
199:
177:
172:
164:
159:
156:
151:
148:
142:
137:
131:
126:
123:
117:
112:
108:
102:
97:
94:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2916:
2905:
2904:Pi algorithms
2902:
2900:
2897:
2896:
2894:
2884:
2880:
2879:
2874:
2873:
2869:
2863:
2858:
2853:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2816:
2815:
2807:
2804:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2770:
2762:
2759:
2754:
2750:
2745:
2740:
2736:
2732:
2731:
2726:
2719:
2716:
2711:
2707:
2703:
2699:
2698:
2693:
2687:
2684:
2679:
2675:
2671:
2667:
2663:
2659:
2655:
2651:
2650:
2638:
2635:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2601:
2593:
2590:
2585:
2581:
2576:
2571:
2567:
2563:
2562:
2557:
2550:
2547:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2513:
2505:
2503:
2499:
2494:
2492:9780198794929
2488:
2484:
2477:
2474:
2470:
2466:
2460:
2456:
2452:
2448:
2441:
2437:
2433:
2429:
2423:
2420:
2415:
2411:
2407:
2403:
2399:
2395:
2387:
2381:
2378:
2373:
2369:
2365:
2362:(in German).
2361:
2357:
2353:
2347:
2344:
2339:
2335:
2331:
2327:
2323:
2317:
2315:
2311:
2300:
2294:
2290:
2286:
2282:
2278:
2271:
2267:
2261:
2259:
2257:
2253:
2248:
2242:
2238:
2237:
2232:
2228:
2223:
2219:
2203:
2193:
2190:
2185:
2179:
2175:
2171:
2167:
2166:
2158:
2155:
2150:
2146:
2142:
2138:
2134:
2130:
2125:
2120:
2116:
2112:
2111:
2103:
2101:
2099:
2095:
2090:
2086:
2081:
2076:
2072:
2068:
2067:
2062:
2055:
2053:
2049:
2044:
2038:
2034:
2033:
2025:
2022:
2019:, p. 67.
2018:
2017:Beckmann 1971
2013:
2010:
2005:
2001:
1997:
1993:
1989:
1985:
1978:
1976:
1972:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1938:
1926:
1924:
1922:
1920:
1916:
1911:
1907:
1903:
1897:
1893:
1892:The Number pi
1889:
1882:
1880:
1876:
1871:
1865:
1861:
1857:
1851:
1849:
1847:
1843:
1838:
1834:
1830:
1824:
1820:
1816:
1815:
1811:A History of
1807:
1801:
1799:
1797:
1795:
1793:
1789:
1782:
1778:
1775:
1774:
1758:
1753:
1749:
1745:
1742:
1734:
1731:
1730:
1726:
1724:
1716:
1701:
1694:
1681:
1677:
1669:
1665:
1661:
1656:
1653:
1648:
1643:
1640:
1637:
1633:
1628:
1620:
1616:
1612:
1607:
1604:
1599:
1595:
1591:
1588:
1585:
1582:
1570:
1554:
1549:
1546:
1541:
1538:
1533:
1530:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1500:
1496:
1491:
1489:
1481:
1477:
1473:
1457:
1453:
1447:
1437:
1425:
1421:
1413:
1412:powers of two
1409:
1404:
1397:
1395:
1393:
1384:
1367:
1360:
1357:
1354:
1350:
1346:
1343:
1338:
1336:
1329:
1325:
1317:
1314:
1311:
1309:
1302:
1298:
1290:
1283:
1279:
1275:
1272:
1265:
1261:
1249:
1241:
1239:
1234:
1210:
1200:
1194:
1180:
1175:
1172:
1169:
1166:
1161:
1158:
1153:
1150:
1145:
1142:
1137:
1134:
1125:
1121:
1109:
1101:
1098:
1085:
1068:
1064:
1061:
1058:
1055:
1052:
1045:
1040:
1037:
1032:
1029:
1021:
1016:
1003:
998:
995:
990:
987:
984:
979:
976:
971:
968:
965:
960:
957:
952:
949:
946:
941:
938:
922:
918:Substituting
916:
903:
898:
895:
890:
887:
884:
879:
876:
871:
868:
865:
860:
857:
852:
849:
846:
841:
837:
834:
831:
820:
816:
815:sinc function
808:
806:
800:
788:
783:
768:
755:
742:
736:
732:
730:
726:
710:
690:
681:
664:
657:
654:
651:
647:
643:
640:
635:
633:
626:
622:
612:
607:
605:
598:
594:
569:
564:
561:
556:
551:
546:
542:
534:
529:
526:
523:
519:
507:
490:
488:
486:
482:
478:
470:
465:
459:
455:
447:
444:
436:
427:
425:
421:
417:
409:
404:
391:
386:
383:
378:
375:
372:
367:
364:
354:
353:circumference
350:
346:
339:
335:
327:
325:
323:
319:
315:
311:
307:
303:
298:
296:
288:
284:
280:
276:
271:
258:
251:
248:
245:
241:
237:
232:
229:
219:
216:
213:
209:
205:
200:
197:
175:
170:
162:
157:
154:
149:
146:
140:
135:
129:
124:
121:
115:
110:
106:
100:
95:
92:
82:
76:
72:
68:
64:
60:
51:
45:
41:
37:
19:
2883:Google Books
2877:
2818:
2812:
2806:
2773:
2767:
2761:
2734:
2728:
2718:
2701:
2695:
2686:
2653:
2647:
2637:
2604:
2598:
2592:
2565:
2559:
2549:
2516:
2510:
2482:
2476:
2458:
2457:(in Latin).
2454:
2439:
2438:(in Latin).
2435:
2422:
2397:
2393:
2380:
2363:
2359:
2346:
2325:
2302:. Retrieved
2280:
2235:
2227:Plofker, Kim
2192:
2164:
2157:
2124:math/0411380
2114:
2108:
2070:
2064:
2031:
2024:
2012:
1990:(1): 87–91.
1987:
1983:
1941:
1935:
1891:
1859:
1810:
1733:Morrie's law
1714:
1699:
1695:
1492:
1417:
1392:golden ratio
1385:
1086:
1017:
920:
917:
812:
781:
764:
750:
740:
682:
494:
466:
428:
405:
337:
331:
328:Significance
318:trigonometry
299:
272:
62:
56:
40:
2366:: 139–140.
1476:hexadecagon
799:accelerated
725:convergence
454:sexagesimal
59:mathematics
2893:Categories
2881:(1593) on
2461:: 345–352.
2442:: 222–236.
2372:23.0263.02
2304:2024-08-20
2073:: 90–112.
1783:References
1480:telescopes
1398:Derivation
349:Archimedes
304:of nested
302:perimeters
75:reciprocal
2798:250441699
2678:123023282
2469:1009.1439
2414:120145020
2352:Rudio, F.
2322:Kac, Mark
2204:π
2004:122368450
1966:125362227
1856:Maor, Eli
1759:α
1696:The term
1657:
1634:∏
1608:
1586:
1542:
1526:
1511:
1358:−
1276:−
1256:∞
1253:→
1235:π
1195:⏟
1173:⋯
1138:−
1116:∞
1113:→
1099:π
1062:
1033:
1004:⋯
996:π
991:
985:⋅
977:π
972:
966:⋅
958:π
953:
942:π
904:⋯
891:
885:⋅
872:
866:⋅
853:
835:
731:in 1891.
655:−
565:π
520:∏
514:∞
511:→
477:integrals
376:π
285:. It has
238:π
233:
225:∞
210:∏
201:π
176:⋯
141:⋅
116:⋅
96:π
2875:Viète's
2621:27641976
2449:(1783).
2430:(1738).
2354:(1891).
2279:(eds.).
2268:(2014).
2229:(2009).
1958:27643107
1858:(2011).
1808:(1971).
1727:See also
1704:goes to
416:Eli Maor
306:polygons
2843:1247533
2835:2324662
2790:3617569
2753:0596932
2710:2437033
2670:2193382
2629:2231136
2584:2915517
2541:1984573
2533:3647881
2338:0110114
2149:1357488
2141:2974641
2089:3090772
1910:2036595
1837:0449960
1472:octagon
2841:
2833:
2796:
2788:
2751:
2708:
2676:
2668:
2627:
2619:
2582:
2539:
2531:
2489:
2412:
2370:
2336:
2295:
2243:
2180:
2147:
2139:
2087:
2039:
2002:
1964:
1956:
1908:
1898:
1866:
1835:
1825:
1698:2 sin(
1436:circle
582:where
310:circle
53:(1593)
2831:JSTOR
2794:S2CID
2786:JSTOR
2674:S2CID
2617:JSTOR
2529:JSTOR
2465:arXiv
2410:S2CID
2273:(PDF)
2137:JSTOR
2119:arXiv
2000:S2CID
1962:S2CID
1954:JSTOR
1819:94–95
1488:digon
1426:with
1420:areas
316:from
279:limit
2487:ISBN
2293:ISBN
2241:ISBN
2178:ISBN
2037:ISBN
1896:ISBN
1864:ISBN
1823:ISBN
1430:and
765:The
422:and
379:<
373:<
2823:doi
2819:100
2778:doi
2739:doi
2658:doi
2646:".
2609:doi
2605:113
2570:doi
2521:doi
2517:110
2402:doi
2392:".
2368:JFM
2285:doi
2170:doi
2129:doi
2115:102
2075:doi
2071:174
1992:doi
1946:doi
1702:/2)
1654:cos
1605:sin
1583:sin
1539:cos
1523:sin
1508:sin
1422:of
1246:lim
1106:lim
1059:cos
1030:cos
988:cos
969:cos
950:cos
888:cos
869:cos
850:cos
832:sin
780:0.6
504:lim
365:223
230:cos
69:of
57:In
2895::
2839:MR
2837:.
2829:.
2817:.
2792:.
2784:.
2774:69
2772:.
2749:MR
2747:.
2735:89
2733:.
2727:.
2706:MR
2702:45
2700:.
2672:.
2666:MR
2664:.
2654:10
2652:.
2625:MR
2623:.
2615:.
2603:.
2580:MR
2578:.
2566:42
2564:.
2558:.
2537:MR
2535:.
2527:.
2515:.
2501:^
2408:.
2398:38
2396:.
2364:36
2334:MR
2328:.
2313:^
2291:.
2255:^
2233:.
2176:.
2145:MR
2143:.
2135:.
2127:.
2113:.
2097:^
2085:MR
2083:.
2069:.
2063:.
2051:^
1998:.
1988:47
1986:.
1974:^
1960:.
1952:.
1942:81
1940:.
1918:^
1906:MR
1904:.
1890:.
1878:^
1845:^
1833:MR
1831:.
1821:.
1791:^
1723:.
1721:/2
1717:=
1575:,
1022::
999:16
927:/2
923:=
748:.
487:.
384:22
368:71
297:.
83::
61:,
2845:.
2825::
2800:.
2780::
2755:.
2741::
2712:.
2680:.
2660::
2644:π
2631:.
2611::
2586:.
2572::
2543:.
2523::
2495:.
2467::
2459:1
2440:9
2416:.
2404::
2390:π
2374:.
2340:.
2307:.
2287::
2249:.
2186:.
2172::
2151:.
2131::
2121::
2091:.
2077::
2045:.
2006:.
1994::
1968:.
1948::
1932:π
1912:.
1872:.
1839:.
1813:π
1754:n
1750:2
1746:=
1743:x
1719:π
1715:x
1710:n
1706:x
1700:x
1682:.
1678:)
1670:i
1666:2
1662:x
1649:n
1644:1
1641:=
1638:i
1629:(
1621:n
1617:2
1613:x
1600:n
1596:2
1592:=
1589:x
1573:n
1555:,
1550:2
1547:x
1534:2
1531:x
1520:2
1517:=
1514:x
1484:2
1458:2
1454:/
1448:2
1432:2
1428:2
1388:π
1368:.
1361:1
1355:k
1351:a
1347:+
1344:2
1339:=
1330:k
1326:a
1318:,
1315:0
1312:=
1303:1
1299:a
1291:,
1284:k
1280:a
1273:2
1266:k
1262:2
1250:k
1242:=
1211:,
1201:k
1181:2
1176:+
1170:+
1167:2
1162:+
1159:2
1154:+
1151:2
1146:+
1143:2
1135:2
1126:k
1122:2
1110:k
1102:=
1089:π
1069:2
1065:x
1056:+
1053:1
1046:=
1041:2
1038:x
980:8
961:4
947:=
939:2
925:π
921:x
899:8
896:x
880:4
877:x
861:2
858:x
847:=
842:x
838:x
803:π
795:π
791:π
782:n
776:π
772:n
759:n
753:n
751:S
746:π
741:×
711:n
691:n
665:.
658:1
652:n
648:a
644:+
641:2
636:=
627:n
623:a
613:2
608:=
599:1
595:a
570:,
562:2
557:=
552:2
547:i
543:a
535:n
530:1
527:=
524:i
508:n
473:π
462:π
450:π
439:π
431:π
412:π
392:.
387:7
344:π
291:π
259:.
252:1
249:+
246:n
242:2
220:1
217:=
214:n
206:=
198:2
171:2
163:2
158:+
155:2
150:+
147:2
136:2
130:2
125:+
122:2
111:2
107:2
101:=
93:2
80:π
38:.
32:π
20:)
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