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Morphing Lord Brouncker's continued fraction for π into the product of Wallis

Published online by Cambridge University Press:  23 January 2015

Thomas J. Osler
Thomas J. Osler*
Affiliation:
Mathematics Department, Rowan University, Glassboro, NJ 08028 USAe-mail:osler@rowan.edu
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Extract

Three of the oldest and most celebrated formulae for π are:

The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenient notation

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 532 , March 2011 , pp. 17 - 22
Copyright
Copyright © The Mathematical Association 2011

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References

1. Vieta, F., Variorum de Rebus Mathematicis Reponsorum Liber VII, (1593) in Opera Mathematica (reprinted), Georg Olms Verlag Hildesheim, New York (1970) pp. 398-400 and 436–446.Google Scholar
2. Wallis, John The Arithmetic of Infinitesimals (translated from Latin by Stedall, Jacqueme A.) Springer Verlag, New York.Google Scholar
3. Stedall, Jacqueline A., Catching Proteus: The collaborations of Wallis and Brouncker: I. Squaring the Circle, Notes and Records of the Royal Society of London, 54 (September 2000) pp. 293–316.Google Scholar
4. Osler, T. J., The united Vieta's and Wallis's products for π, Amer. Math. Monthly 106 (1999) pp. 774–776.Google Scholar
5. Lange, L J., An elegant continued fraction for π, Amer. Math. Monthly, 106 (1999), pp. 456–458.Google Scholar
6. Perron, O., Die Lehre von den Kettenbruchen, Band II, Teubner, Stuttgart (1957). (There is a Chelsea edition of thio book in which our equation (9) appears on p. 255 and not p. 35.)Google Scholar
7. Berndt, B. C., Ramanujan's Notebooks Part II, Springer-Verlag, New York (1989).Google Scholar
8. Euler, L, De fractionibus continuis Wallisii (On the continued fractions of Wallis), originally published in Memoires de l'academie des sciences de St.-Petersbourg 5 (1815) pp. 24–44. Also see Opera omnia: Series 1, Volume 16 pp. 178 - 199. On the web at the Euler Archive http://www.math.dartmouthxdu/~euler/ Google Scholar
9. Dutka, J., Wallis's product, Brouncker's continued fraction and Leibniz's series, Arch. History Exact Sciences 26 (1982) pp. 115–126.Google Scholar
10. Khrushchev, S., A recovery of Brouncker's proof for the quadrature continued fraction, Publicacions matematiques 50 (2006) pp. 3–42.Google Scholar