I would guess it was well known by the time of Cauchy. But are there earlier references to it?



Who discovered the winding number?
[Grünbaum and Shephard](http://www.ams.org/journals/tran/1990-322-01/S0002-9947-1990-1024774-2/S0002-9947-1990-1024774-2.pdf)  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.


