Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, \(\pi\) , and the Ladies Diary. (English) Zbl 0665.26007

Almkvist, Gert; Berndt, Bruce C.

Am. Math. Mon. 95, No. 7, 585-608 (1988).

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.)
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 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.

Reviewer: J.Aczél

MSC:

26D15	Inequalities for sums, series and integrals
40A25	Approximation to limiting values (summation of series, etc.)
33E05	Elliptic functions and integrals
33B10	Exponential and trigonometric functions
01A50	History of mathematics in the 18th century
01A55	History of mathematics in the 19th century

Keywords:

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