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Zbl 0665.26007
Almkvist, Gert; Berndt, Bruce C.
Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$ , and the Ladies Diary. (English)
[J] Am. Math. Mon. 95, No.7, 585-608 (1988). ISSN 0002-9890

This is an amusing and educational guided tour of the Lagrange-Gauss medium arithmetico-geometricum and its neighbourhood: elliptic integrals, approximations of the perimeter of the ellipse, of $\pi$, and of elementary functions. (The referee would have liked to see also a report on the connection to the exact discussion of the pendulum with large swings via elliptic integrals, but you cannot have it all.) \par Historic arguments in proofs are given with gaps pointed out but usually not filled in. An essential tool, whose source was little known, is a transformation going back to the English mathematician {\it John Landen} (1719-1790). While he introduced it in the Philos. Trans. R. Soc. London (61, 298-309 (1771) and 65, 283-289 (1775)), it is pointed out that he published mostly in the journal Ladies Diary. In our day and age it may raise interest and approval that this ladies' journal, with a circulation of several thousands, published, in addition to ``remarkable events'' and ``enigmas'', mainly solutions of mathematical problems. That about 96 \% of the conributors were men, may be less welcome information.
[J.Aczél]

MSC 2000:: *26D15 Inequalities for sums, series and integrals of real functions
40A25 Approximation to limiting values
33E05 Elliptic functions and integrals
33B10 Elementary functions
01A50 Mathematics in the 18th century
01A55 Mathematics in the 19th century

Keywords: arithmetic-geometric mean; convergence; Landen's transformation; inequality; elliptic integrals; approximations of the perimeter of the ellipse

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