Let $(a, b)_v$ denote the Hilbert symbol on the completion $K_v$ of a global field $K$ at a place $v$. The **Hilbert reciprocity law** $\prod_v (a, b)_v = 1$ is a strict generalization of quadratic reciprocity, to which it reduces in the case $K = \mathbb{Q}, a = p, b = q$. Hilbert had this to say about his law:

The reciprocity law... reminds [sic] the Cauchy integral theorem, according to which the integral of a function over a path enclosing all of its singularities always yields the value $0$. One of the known proofs of the ordinary quadratic reciprocity law suggests an intrinsic connection between this number-theoretic law and Cauchy's fundamental function-theoretic theorem.

(I am working off of a translation here.) Does anyone have any idea what proof Hilbert could be referring to?