Some insights from total collapse

We discuss the Sundman-Weierstrass theorem of total collapse in its historical context. This remarkable and relatively simple result, a type of stability criterion, is at the crossroads of some interesting developments in the gravitation Newtonian N-body problem. We use it as motivation to explore the connections to such important concepts as integrability, singularities and tyipicality in order to gain some insight on the transition from a predominantly quantitative to a novel qualitative approach to dynamical problems that took place at the end of the 19th century.