  We give a heuristic argument supporting conjectures of Bhargava on the
asymptotics of the number of $S_n$-number fields having bounded discriminant.
We then make our arguments rigorous in the case $n=3$ giving a new elementary
proof of the Davenport-Heilbronn theorem. Our basic method is to count elements
of small height in $S_n$-fields while carefully keeping track of the index of
the monogenic ring that they generate.
