Gestalt switches in Poincaré׳s prize paper: An inspiration for, but not an instance of, chaos

I analyse in detail the construction of asymptotic surfaces in Sections 16-19 of Poincaré (1890), also known as the prize paper. There are two prime reasons for doing so. Firstly, this part of the prize paper contains an interesting argumentative strategy, which I call Poincaré׳s gestalt switch. Secondly, it has been claimed that the prize paper contains one of the first descriptions of chaotic motion. I will argue that the latter claim is false, although both the gestalt switches and the graphical representation which Poincaré (1890) chose for the asymptotic surfaces might well have provided the inspiration for later works in chaos theory.