Arc Length and Elliptic Functions
The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae). The period lattices are of a very special form, being proportional to the Gaussian integers. For this reason the case of elliptic functions with complex multiplication by the square root of minus one is called the lemniscatic case in some sources.
Read more about this topic: Lemniscate Of Bernoulli
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