1. In the beginning there was Gauss

In the year 1985, one of us had the luxury of attending a graduate course on Elliptic Functions given by Henry McKean at the Courant Institute. Among the many beautiful results he described in his unique style, there was a calculation of Gauss: take two positive real numbers a and b, with a > b, and form a new pair by replacing a with the arithmetic mean . and b with the geometric mean. Then iterate:

starting with a = a and b0 = b. Gauss [1799] was interested in the initial conditions a = 1 and . The iteration generates a sequence of algebraic numbers which rapidly become impossible to describe explicitly; for instance is a root of the polynomial The numerical behavior is surprising; a6 and b6 agree to 87 digits. It is simple to check that

and then he recognized the reciprocal of this number as a numerical approximation to the elliptic integral

It is unclear to the authors how Gauss recognized this number: he simply knew it. (Stirling's tables may have been a help; [Borwein and Bailey 2003] contains a reproduction of the original notes and comments.) He was particularly interested in the evaluation of this definite integral as it provides the length of a lemniscate. In his diary Gauss remarked, ‘This will surely open up a whole new field of analysis’ [Cox 1984; Borwein and Borwein 1987].