Gauss's (unpublished and largely unknown) proof of the quartic reciprocity law
probably used lattice point arguments. The details were supplied by several authors
at the end of the 19th century (for references, see e.g. Hill's article below).

A modern approach using geometric ideas similar to those above was provided in several articles by R. Hill, such as <a href="http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118775537&page=record">this one</a>.