Pi (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Pi" in English language version.

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Euler, Leonhard (1727). "Tentamen explicationis phaenomenorum aeris" (PDF). Commentarii Academiae Scientiarum Imperialis Petropolitana (in Latin). 2: 351. E007. Archived (PDF) from the original on 1 April 2016. Retrieved 15 October 2017. Sumatur pro ratione radii ad peripheriem, $I : π$ English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine: " $π$ is taken for the ratio of the radius to the periphery [note that in this work, Euler's $π$ is double our $π$ .]"
Euler, Leonhard (1736). "Ch. 3 Prop. 34 Cor. 1". Mechanica sive motus scientia analytice exposita. (cum tabulis) (in Latin). Vol. 1. Academiae scientiarum Petropoli. p. 113. E015. Denotet $1 : π$ rationem diametri ad peripheriam English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine : "Let $1 : π$ denote the ratio of the diameter to the circumference"

americanscientist.org

Hayes, Brian (September 2014). "Pencil, Paper, and Pi". American Scientist. Vol. 102, no. 5. p. 342. doi:10.1511/2014.110.342. Retrieved 22 January 2022.

ams.org

mathscinet.ams.org

Mollin, R. A. (1999). "Continued fraction gems". Nieuw Archief voor Wiskunde. 17 (3): 383–405. MR 1743850.
Borwein, Jonathan M. (2014). "The life of $π$ : from Archimedes to ENIAC and beyond". In Sidoli, Nathan; Van Brummelen, Glen (eds.). From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren. Heidelberg: Springer. pp. 531–561. doi:10.1007/978-3-642-36736-6_24. ISBN 978-3-642-36735-9. MR 3203895.
Gibbons, Jeremy (2006). "Unbounded spigot algorithms for the digits of pi" (PDF). The American Mathematical Monthly. 113 (4): 318–328. doi:10.2307/27641917. JSTOR 27641917. MR 2211758.
Martini, Horst; Montejano, Luis; Oliveros, Déborah (2019). Bodies of Constant Width: An Introduction to Convex Geometry with Applications. Birkhäuser. doi:10.1007/978-3-030-03868-7. ISBN 978-3-030-03866-3. MR 3930585. S2CID 127264210.

See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, § 5.3.3, pp. 111–112.
Howe, Roger (1980). "On the role of the Heisenberg group in harmonic analysis". Bulletin of the American Mathematical Society. 3 (2): 821–844. doi:10.1090/S0273-0979-1980-14825-9. MR 0578375.
Tate, John T. (1950). "Fourier analysis in number fields, and Hecke's zeta-functions". In Cassels, J. W. S.; Fröhlich, A. (eds.). Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Thompson, Washington, DC. pp. 305–347. ISBN 978-0-9502734-2-6. MR 0217026.

archive.org

Jones, William (1706). Synopsis Palmariorum Matheseos. London: J. Wale. pp. 243, 263. p. 263: There are various other ways of finding the Lengths, or Areas of particular Curve Lines or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to Circumference as 1 to
${\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=$
$3.14159, & c. = π$ . This Series (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. John Machin; and by means thereof, Van Ceulen's Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
Reprinted in Smith, David Eugene (1929). "William Jones: The First Use of $π$ for the Circle Ratio". A Source Book in Mathematics. McGraw–Hill. pp. 346–347.
Remmert 2012, p. 148.
Weierstrass, Karl (1841). "Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt" [Representation of an analytical function of a complex variable, whose absolute value lies between two given limits]. Mathematische Werke (in German). Vol. 1. Berlin: Mayer & Müller (published 1894). pp. 51–66. Remmert, Reinhold (2012). "Ch. 5 What is π?". In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.). Numbers. Springer. ISBN 978-1-4612-1005-4.
Baltzer, Richard (1870). Die Elemente der Mathematik [The Elements of Mathematics] (in German). Hirzel. p. 195. Archived from the original on 14 September 2016.
Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. p. 183. ISBN 978-0-07-054235-8.
Posamentier & Lehmann 2004, p. 25 Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
Schlager, Neil; Lauer, Josh (2001). Science and Its Times: Understanding the Social Significance of Scientific Discovery. Gale Group. ISBN 978-0-7876-3933-4. Archived from the original on 13 December 2019. Retrieved 19 December 2019., p. 185.
Boyer & Merzbach 1991, p. 168. Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (2 ed.). Wiley. ISBN 978-0-471-54397-8.
Boyer & Merzbach 1991, p. 202 Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (2 ed.). Wiley. ISBN 978-0-471-54397-8.
Sandifer, Ed (2009). "Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118.
Euler, Leonhard (1755). "§2.2.30". Institutiones Calculi Differentialis (in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318. E 212.

Euler, Leonhard (1798) [written 1779]. "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae". Nova Acta Academiae Scientiarum Petropolitinae. 11: 133–149, 167–168. E 705.

Chien-Lih, Hwang (2004). "88.38 Some Observations on the Method of Arctangents for the Calculation of $π$ ". Mathematical Gazette. 88 (512): 270–278. doi:10.1017/S0025557200175060. S2CID 123532808.

Chien-Lih, Hwang (2005). "89.67 An elementary derivation of Euler's series for the arctangent function". Mathematical Gazette. 89 (516): 469–470. doi:10.1017/S0025557200178404. S2CID 123395287.
Posamentier & Lehmann 2004, p. 284 Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
Lindemann, F. (1882). "Über die Ludolph'sche Zahl". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. 2: 679–682.
Oughtred, William (1648). Clavis Mathematicæ [The key to mathematics] (in Latin). London: Thomas Harper. p. 69. (English translation: Oughtred, William (1694). Key of the Mathematics. J. Salusbury.)
Barrow, Isaac (1860). "Lecture XXIV". In Whewell, William (ed.). The mathematical works of Isaac Barrow (in Latin). Harvard University. Cambridge University press. p. 381.
Gregorius, David (1695). "Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae" (PDF). Philosophical Transactions (in Latin). 19 (231): 637–652. Bibcode:1695RSPT...19..637G. doi:10.1098/rstl.1695.0114. JSTOR 102382.
Arndt & Haenel 2006, pp. 39–40
Posamentier & Lehmann 2004, p. 105 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
Arndt & Haenel 2006, pp. 43
Posamentier & Lehmann 2004, pp. 105–108 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
For instance, Pickover calls π "the most famous mathematical constant of all time", and Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the Givenchy π perfume, Pi (film), and Pi Day as examples. See: Pickover, Clifford A. (1995). Keys to Infinity. Wiley & Sons. p. 59. ISBN 978-0-471-11857-2. Peterson, Ivars (2002). Mathematical Treks: From Surreal Numbers to Magic Circles. MAA spectrum. Mathematical Association of America. p. 17. ISBN 978-0-88385-537-9. Archived from the original on 29 November 2016.
Posamentier & Lehmann 2004, p. 118
Arndt & Haenel 2006, p. 50 Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 211–212
Posamentier & Lehmann 2004, pp. 36–37
Hallerberg, Arthur (May 1977). "Indiana's squared circle". Mathematics Magazine. 50 (3): 136–140. doi:10.2307/2689499. JSTOR 2689499. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.

arxiv.org

Plouffe, Simon (2022). "A formula for the $n$ th decimal digit or binary of $π$ and powers of $π$ ". arXiv:2201.12601 [math.NT].
L. Esposito; C. Nitsch; C. Trombetti (2011). "Best constants in Poincaré inequalities for convex domains". arXiv:1110.2960 [math.AP].
Benjamin Nill; Andreas Paffenholz (2014). "On the equality case in Erhart's volume conjecture". Advances in Geometry. 14 (4): 579–586. arXiv:1205.1270. doi:10.1515/advgeom-2014-0001. ISSN 1615-7168. S2CID 119125713.
T. Friedmann; C.R. Hagen (2015). "Quantum mechanical derivation of the Wallis formula for pi". Journal of Mathematical Physics. 56 (11): 112101. arXiv:1510.07813. Bibcode:2015JMP....56k2101F. doi:10.1063/1.4930800. S2CID 119315853.

bbc.co.uk

Palmer, Jason (16 September 2010). "Pi record smashed as team finds two-quadrillionth digit". BBC News. Archived from the original on 17 March 2011. Retrieved 26 March 2011.

bnf.fr

gallica.bnf.fr

Euler, Leonhard (1922). Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio (in Latin). Lipsae: B.G. Teubneri. pp. 133–134. E101. Archived from the original on 16 October 2017. Retrieved 15 October 2017.

books.google.com

Andrews, Askey & Roy 1999, p. 59. Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). Special Functions. Cambridge: University Press. ISBN 978-0-521-78988-2.
Arndt & Haenel 2006, p. 17. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Oughtred, William (1652). Theorematum in libris Archimedis de sphaera et cylindro declarario (in Latin). Excudebat L. Lichfield, Veneunt apud T. Robinson. $δ . π$ :: semidiameter. semiperipheria
Arndt & Haenel 2006, p. 8. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Remmert 2012, p. 129. Remmert, Reinhold (2012). "Ch. 5 What is π?". In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.). Numbers. Springer. ISBN 978-1-4612-1005-4.
Remmert 2012, p. 148.
Weierstrass, Karl (1841). "Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt" [Representation of an analytical function of a complex variable, whose absolute value lies between two given limits]. Mathematische Werke (in German). Vol. 1. Berlin: Mayer & Müller (published 1894). pp. 51–66. Remmert, Reinhold (2012). "Ch. 5 What is π?". In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.). Numbers. Springer. ISBN 978-1-4612-1005-4.
Arndt & Haenel 2006, p. 5. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 22–23. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 22, 28–30. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 3. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 6. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 33. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 240. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 242. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Kennedy, E.S. (1978). "Abu-r-Raihan al-Biruni, 973–1048". Journal for the History of Astronomy. 9: 65. Bibcode:1978JHA.....9...65K. doi:10.1177/002182867800900106. S2CID 126383231. Ptolemy used a three-sexagesimal-digit approximation, and Jamshīd al-Kāshī expanded this to nine digits; see Aaboe, Asger (1964). Episodes from the Early History of Mathematics. New Mathematical Library. Vol. 13. New York: Random House. p. 125. ISBN 978-0-88385-613-0. Archived from the original on 29 November 2016.
Andrews, Askey & Roy 1999, p. 14. Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). Special Functions. Cambridge: University Press. ISBN 978-0-521-78988-2.
Arndt & Haenel 2006, p. 167. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. pp. 67–77, 165–166. ISBN 978-0-88920-324-2. Archived from the original on 29 November 2016. Retrieved 5 June 2013.
Plofker, Kim (2009). Mathematics in India. Princeton University Press. p. 27. ISBN 978-0691120676.
Arndt & Haenel 2006, p. 170. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 175, 205. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 171. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 176. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 15–16, 175, 184–186, 205. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 176–177. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 177. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 178. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 179. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 180. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 182. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 182–183. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 183. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 185–191 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 185–186. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Princeton University Press. p. 264. ISBN 978-0-691-13526-7.
Arndt & Haenel 2006, p. 187. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Vieta, Franciscus (1593). Variorum de rebus mathematicis responsorum. Vol. VIII.
Arndt & Haenel 2006, p. 188. Newton quoted by Arndt. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 189. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 192–193. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 72–74 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 192–196, 205. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 194–196 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Borwein, J.M.; Borwein, P.B. (1988). "Ramanujan and Pi". Scientific American. 256 (2): 112–117. Bibcode:1988SciAm.258b.112B. doi:10.1038/scientificamerican0288-112.
Arndt & Haenel 2006, pp. 15–17, 70–72, 104, 156, 192–197, 201–202 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 69–72. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, Formula 16.10, p. 223. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 196. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Oughtred, William (1648). Clavis Mathematicæ [The key to mathematics] (in Latin). London: Thomas Harper. p. 69. (English translation: Oughtred, William (1694). Key of the Mathematics. J. Salusbury.)
Arndt & Haenel 2006, p. 166. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Cajori, Florian (2007). A History of Mathematical Notations: Vol. II. Cosimo, Inc. pp. 8–13. ISBN 978-1-60206-714-1. the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented $3.14159...$ by $δ : π$ , as did Oughtred more than a century earlier
Smith, David E. (1958). History of Mathematics. Courier Corporation. p. 312. ISBN 978-0-486-20430-7.
Arndt & Haenel 2006, p. 165: A facsimile of Jones' text is in Berggren, Borwein & Borwein 1997, pp. 108–109. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (1997). Pi: a Source Book. Springer-Verlag. ISBN 978-0-387-20571-7.
Segner, Joannes Andreas (1756). Cursus Mathematicus (in Latin). Halae Magdeburgicae. p. 282. Archived from the original on 15 October 2017. Retrieved 15 October 2017.
Euler, Leonhard (1747). Henry, Charles (ed.). Lettres inédites d'Euler à d'Alembert. Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche (in French). Vol. 19 (published 1886). p. 139. E858. Car, soit π la circonference d'un cercle, dout le rayon est $= 1$ English translation in Cajori, Florian (1913). "History of the Exponential and Logarithmic Concepts". The American Mathematical Monthly. 20 (3): 75–84. doi:10.2307/2973441. JSTOR 2973441. Letting $π$ be the circumference (!) of a circle of unit radius
Euler, Leonhard (1736). "Ch. 3 Prop. 34 Cor. 1". Mechanica sive motus scientia analytice exposita. (cum tabulis) (in Latin). Vol. 1. Academiae scientiarum Petropoli. p. 113. E015. Denotet $1 : π$ rationem diametri ad peripheriam English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine : "Let $1 : π$ denote the ratio of the diameter to the circumference"
Segner, Johann Andreas von (1761). Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm (in Latin). Renger. p. 374. Si autem $π$ notet peripheriam circuli, cuius diameter eſt $2$
Arndt & Haenel 2006, p. 205. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 197. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 15–17. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 131. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 132, 140. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 87. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 111 (5 times), pp. 113–114 (4 times). For details of algorithms, see Borwein, Jonathan; Borwein, Peter (1987). Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity. Wiley. ISBN 978-0-471-31515-5. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 17–19 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 18. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 103–104 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 104 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 104, 206 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 110–111 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Bailey, David H.; Borwein, Jonathan M. (2016). "15.2 Computational records". Pi: The Next Generation, A Sourcebook on the Recent History of Pi and Its Computation. Springer International Publishing. p. 469. doi:10.1007/978-3-319-32377-0. ISBN 978-3-319-32375-6.
Arndt & Haenel 2006, p. 39 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 39–40
Posamentier & Lehmann 2004, p. 105 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
Arndt & Haenel 2006, pp. 43
Posamentier & Lehmann 2004, pp. 105–108 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
Arndt & Haenel 2006, pp. 77–84. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 77. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 117, 126–128. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 20
Bellards formula in: Bellard, Fabrice. "A new formula to compute the n^th binary digit of pi". Archived from the original on 12 September 2007. Retrieved 27 October 2007. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, pp. 41–43 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 43 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory (2005 ed.). Mineola, NY: Dover Publications. ISBN 978-0-486-44568-7. LCCN 2005053026. OCLC 61200849.
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Danesi, Marcel (January 2021). "Chapter 4: Pi in Popular Culture". Pi ( $π$ ) in Nature, Art, and Culture. Brill. p. 97. doi:10.1163/9789004433397. ISBN 9789004433373. S2CID 224869535.
For instance, Pickover calls π "the most famous mathematical constant of all time", and Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the Givenchy π perfume, Pi (film), and Pi Day as examples. See: Pickover, Clifford A. (1995). Keys to Infinity. Wiley & Sons. p. 59. ISBN 978-0-471-11857-2. Peterson, Ivars (2002). Mathematical Treks: From Surreal Numbers to Magic Circles. MAA spectrum. Mathematical Association of America. p. 17. ISBN 978-0-88385-537-9. Archived from the original on 29 November 2016.
Posamentier & Lehmann 2004, p. 118
Arndt & Haenel 2006, p. 50 Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Arndt & Haenel 2006, p. 14 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
Petroski, Henry (2011). Title An Engineer's Alphabet: Gleanings from the Softer Side of a Profession. Cambridge University Press. p. 47. ISBN 978-1-139-50530-7.
Freiberger, Marianne; Thomas, Rachel (2015). "Tau – the new $π$ ". Numericon: A Journey through the Hidden Lives of Numbers. Quercus. p. 159. ISBN 978-1-62365-411-5.
Arndt & Haenel 2006, pp. 211–212
Posamentier & Lehmann 2004, pp. 36–37
Hallerberg, Arthur (May 1977). "Indiana's squared circle". Mathematics Magazine. 50 (3): 136–140. doi:10.2307/2689499. JSTOR 2689499. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka. Posamentier, Alfred S.; Lehmann, Ingmar (2004). $π$ : A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.

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Grienbergerus, Christophorus (1630). Elementa Trigonometrica (PDF) (in Latin). Archived from the original (PDF) on 1 February 2014. His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < $π$ < 3.14159 26535 89793 23846 26433 83279 50288 4199.
Roy, Ranjan (1990). "The Discovery of the Series Formula for $π$ by Leibniz, Gregory and Nilakantha" (PDF). Mathematics Magazine. 63 (5): 291–306. doi:10.1080/0025570X.1990.11977541. Archived from the original (PDF) on 14 March 2023. Retrieved 21 February 2023.
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Euler, Leonhard (1727). "Tentamen explicationis phaenomenorum aeris" (PDF). Commentarii Academiae Scientiarum Imperialis Petropolitana (in Latin). 2: 351. E007. Archived (PDF) from the original on 1 April 2016. Retrieved 15 October 2017. Sumatur pro ratione radii ad peripheriem, $I : π$ English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine: " $π$ is taken for the ratio of the radius to the periphery [note that in this work, Euler's $π$ is double our $π$ .]"
Euler, Leonhard (1736). "Ch. 3 Prop. 34 Cor. 1". Mechanica sive motus scientia analytice exposita. (cum tabulis) (in Latin). Vol. 1. Academiae scientiarum Petropoli. p. 113. E015. Denotet $1 : π$ rationem diametri ad peripheriam English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine : "Let $1 : π$ denote the ratio of the diameter to the circumference"
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Arndt & Haenel 2006, p. 20
Bellards formula in: Bellard, Fabrice. "A new formula to compute the n^th binary digit of pi". Archived from the original on 12 September 2007. Retrieved 27 October 2007. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka.
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For instance, Pickover calls π "the most famous mathematical constant of all time", and Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the Givenchy π perfume, Pi (film), and Pi Day as examples. See: Pickover, Clifford A. (1995). Keys to Infinity. Wiley & Sons. p. 59. ISBN 978-0-471-11857-2. Peterson, Ivars (2002). Mathematical Treks: From Surreal Numbers to Magic Circles. MAA spectrum. Mathematical Association of America. p. 17. ISBN 978-0-88385-537-9. Archived from the original on 29 November 2016.
Gill, Andy (4 November 2005). "Review of Aerial". The Independent. Archived from the original on 15 October 2006. the almost autistic satisfaction of the obsessive-compulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits long)
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Knuth, Donald (3 October 1990). "The Future of TeX and Metafont" (PDF). TeX Mag. 5 (1): 145. Archived (PDF) from the original on 13 April 2016. Retrieved 17 February 2017.

wolfram.com

mathworld.wolfram.com

Weisstein, Eric W. "Lindemann-Weierstrass Theorem". mathworld.wolfram.com. Retrieved 26 October 2024.
Weisstein, Eric W. "Circle". mathworld.wolfram.com. Retrieved 22 January 2025. A=1/2(2 $π$ r)r= $π$ r^2
Weisstein, Eric W. "Circumference". mathworld.wolfram.com. Retrieved 22 January 2025. C=2 $π$ r
Weisstein, Eric W. "Ellipse". mathworld.wolfram.com. Retrieved 22 January 2025. A=... $π$ ab.

worldcat.org

search.worldcat.org

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worldscientific.com

Brezinski, C. (2009). "Some pioneers of extrapolation methods". In Bultheel, Adhemar; Cools, Ronald (eds.). The Birth of Numerical Analysis. World Scientific. pp. 1–22. doi:10.1142/9789812836267_0001. ISBN 978-981-283-625-0.