I've been learning about Riemann surfaces and the Riemann-Roch theorem, and I've been able to experience some of its great power. However, I'm curious as to what the history behind divisors on curves is, namely:
Why are they called divisors?
What inspired their creation? (Is there a motivating example?)
The formalism makes complete sense, and the results surely justify the theory, but how did anyone think, "Oh let's look at the free abelian group on the points of this surface and hope something good happens"? To me, they just seem like useful book-keeping objects to record local information about meromorphic functions, but is there some kind of geometric or intuitive way to think about these objects as more than just formalities that work?
Thank you