zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π , and the Ladies Diary. (English) Zbl 0665.26007

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 π, 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
26D15Inequalities for sums, series and integrals of real functions
40A25Approximation to limiting values (summation of series, etc.)
33E05Elliptic functions and integrals
33B10Exponential and trigonometric functions
01A50Mathematics in the 18th century
01A55Mathematics in the 19th century