TITLE: Who discovered the winding number?
QUESTION [6 upvotes]: I would guess it was well known by the time of Cauchy. But are there earlier references to it?

REPLY [13 votes]: Grünbaum and Shephard  suggest that the winding numbers (for closed polygons) have been discussed in the literature at least since 1769. 
See
A.L.F.  Meister,  Generalia de genesi figurarum  planarum et  inde pendentibus earum ajfectionibus,  Novi Comm. Soc.  Reg.  Scient.  Gotting.  1 (1769/70),  pp.  144-180.