I'm referring to this proof. The key formula ("Eisenstein's Lemma") is
$$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where $u=2,4,\ldots,p-1$}$$
The sum in the exponent is easily seen to count the number of lattice points in this rectangle that are below the diagonal and having even $x$-coordinate,
where $p$ and $q$ are our primes in question, and via some clever flipping, quadratic reciprocity pops out.
I seem to recall in my first summer at PROMYS, some counselors tried to work out a version for quartic reciprocity, using (if I remember correctly) a similarly-defined lattice in $\mathbb{Z}[i]\times\mathbb{Z}[i]$. However, I don't remember the details, or if they were successful. I'd be interested to see if a "low-tech" proof using lattice point counting can work for cubic or quartic reciprocity laws (or even more generally, but I suspect that would be overly optimistic).
Best Answer
Gauss's (unpublished and largely unknown) proof of the quartic reciprocity law probably used lattice point arguments.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 this one.
Edit (2015). For reconstructions of Gauss's ideas see the recently published book Gauss's reciprocity laws in number theory (in German).