Machin-like formula (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Machin-like formula" in English language version.

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archive.org

Jones, William (1706). Synopsis Palmariorum Matheseos. London: J. Wale. pp. 243, 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.
Beckmann, Petr (1971). A History Of Pi. USA: The Golem Press. p. 102. ISBN 0-88029-418-3.
Størmer, Carl (1897). "Sur l'application de la théorie des nombres entiers complexes a la solution en nombres rationnels ${\textstyle x_{1}\ x_{2}\ldots x_{n}}$ ${\textstyle c_{1}\ c_{2}\ldots c_{n}}$ ${\textstyle k}$ de l'équation: ${\textstyle c_{1}\operatorname {arc\ tg} x_{1}+{}}$ ${\textstyle c_{2}\operatorname {arc\ tg} x_{2}+\cdots +}$ ${\textstyle c_{n}\operatorname {arc\ tg} x_{n}={}}$ ${\textstyle k{\tfrac {\pi }{4}}}$ ". Archiv for Mathematik og Naturvidenskab. 19 (3): 1–95.
Størmer, Carl (1896). "Solution complète en nombres entiers $m$ , $n$ , $x$ , $y$ et $k$ de l'équation ${\textstyle m\operatorname {arc\ tg} {\tfrac {1}{x}}+n\operatorname {arc\ tg} {\tfrac {1}{y}}=k{\tfrac {\pi }{4}}.}$ ". Mathematisk-naturvidenskabelig Klasse. Skrifter udgivne af Videnskabsselskabet i Christiania. 1895 (11): 1–21.
Størmer, Carl (1899). "Solution complète en nombres entiers de l'équation ${\textstyle m\operatorname {arc\,tang} {\frac {1}{x}}+n\operatorname {arc\,tang} {\frac {1}{y}}=k{\frac {\pi }{4}}}$ " [Complete solution in whole numbers of the equation ...]. Bulletin de la Société Mathématique de France (in French). 27: 160–170. doi:10.24033/bsmf.603.
Euler, Leonhard (1744) [written 1737]. "De variis modis circuli quadraturam numeris proxime exprimendi". Commentarii academiae scientiarum Petropolitanae. 9: 222–236. E 74.
Letter from Jakob Hermann to Gottfried Leibniz, 21 August 1706. Published in Gerhardt, C.I., ed. (1859). "XXII. Hermann an Leibniz.". Leibnizens mathematische Schriften. Vol. 4. H.W. Schmidt. pp. 302–304.

arxiv.org

https://arxiv.org/pdf/2108.07718.pdf (2021)

doi.org

Lehmer, Derrick Henry (1938). "On Arccotangent Relations for π". American Mathematical Monthly. 45 (10): 657–664. doi:10.2307/2302434. JSTOR 2302434.
Wetherfield, Michael (2016). "The Enhancement of Machin's Formula by Todd's Process". The Mathematical Gazette. 80 (488): 333–344. doi:10.2307/3619567. JSTOR 3619567. S2CID 126173230.
Størmer, Carl (1899). "Solution complète en nombres entiers de l'équation ${\textstyle m\operatorname {arc\,tang} {\frac {1}{x}}+n\operatorname {arc\,tang} {\frac {1}{y}}=k{\frac {\pi }{4}}}$ " [Complete solution in whole numbers of the equation ...]. Bulletin de la Société Mathématique de France (in French). 27: 160–170. doi:10.24033/bsmf.603.
Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for $π$ ". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896.

jjj.de

Jörg Uwe Arndt: "Matters Computational" section 32.5.2, page 637.

jstor.org

Lehmer, Derrick Henry (1938). "On Arccotangent Relations for π". American Mathematical Monthly. 45 (10): 657–664. doi:10.2307/2302434. JSTOR 2302434.
Wetherfield, Michael (2016). "The Enhancement of Machin's Formula by Todd's Process". The Mathematical Gazette. 80 (488): 333–344. doi:10.2307/3619567. JSTOR 3619567. S2CID 126173230.
Chien-Lih, Hwang. "More Machin-Type Identities". The Mathematical Gazette. 81 (490). JSTOR 3618793.
Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for $π$ ". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896.

pacific.edu

scholarlycommons.pacific.edu

Euler, Leonhard (1744) [written 1737]. "De variis modis circuli quadraturam numeris proxime exprimendi". Commentarii academiae scientiarum Petropolitanae. 9: 222–236. E 74.

semanticscholar.org

api.semanticscholar.org

Wetherfield, Michael (2016). "The Enhancement of Machin's Formula by Todd's Process". The Mathematical Gazette. 80 (488): 333–344. doi:10.2307/3619567. JSTOR 3619567. S2CID 126173230.

youtube.com

The biggest hand calculation in a century! [Pi Day 2024]. Retrieved 2024-04-02 – via www.youtube.com.