Some Milestones of Lemniscatomy

This little article is an attempt to kindle the reader’s interest in some modern research in the domain of complex multiplication by sketching a few episodes from the history of the very first example of this theory, the lemniscatic elliptic integral and the corresponding elliptic functions. Even though treating such an example offers elementary proofs of some key properties, our exposition is not mathematically self-contained. All we hope is that it will motivate the reader to look up some of the further literature which we indicate. At the same time, we have tried to draw attention to the differences between the various periods of the history of lemniscatomy, and to a few historical details which may not be generally known. 1. Prelude in antiquity.1 The story from which we are about to tell a few instructive episodes does not really begin, but is vaguely suggested to begin, in the dawn of times, or rather in the twilight of Greek mathematics, where the sources which are extant today do not quite permit us to know what was said and thought, and at what time. The leading Neo-Platonist of his time, Proclus from Lykia (on today’s maps, this i