I always had this question, but was unable to get a definitive answer to it.
There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it possible to reconstruct the theory of elliptic functions from purely geometric considerations? If this is possible, then to what extent?
Geometric means geometric constructions to prove main formulas but without use of functions of complex variables.
Note that I don't imply that this geometric theory would be practical. From modern standpoint, this construction probably will be considered as a waste of time. I'm interested in the theoretical possibility of such a development. Also I'm interested in understanding the intuition behind such a development, why it is possible?