Department of Mathematics
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I have the result, but I do not yet know how to get it.
—Carl Friedrich Gauss

One of the central problems in the classical geometry of the plane curves is the determination of the length of an arc of a given curve. As we all know very well, the case of the circle had been solved in antiquity. Considerable progress was made on "rectifying the curves" during the 16th and 17th centuries through the efforts of Wallis, Neil, Fagnano, Lagrange and many others, and one was able to determine the lengths of various curves, among them the cycloid, the parabola and the semi-cubical parabola, but not, for example, the ellipse or the lemniscate. These two curves had to wait for the emergence of the differential and integral calculus. In these two cases, the arc length turns out to be an elliptic integral, which cannot be expressed in terms of the well-known functions (read "elementary functions") of calculus. Gauss, while he was still a teenager, invented the concept of arithmetic-geometric mean (a binary law of composition) so that he can give explicit formulas for the length of the perimeter of these curves. In this talk we will review some of the elementary properties of AGM and describe the connection between AGM and the length of the lemniscate. For an ancient topic like the AGM, there are bound to be many excellent references. Here are a couple of modern references.

References:

1. Cox, David A. The arithmetic-geometric mean of Gauss, L' Enseign. Math. (2) 30 (1984), 275--330.
2. Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, CMS series, Toronto, 1987.