  We give simple proofs of the Davenport--Heilbronn theorems, which provide the
main terms in the asymptotics for the number of cubic fields having bounded
discriminant and for the number of 3-torsion elements in the class groups of
quadratic fields having bounded discriminant. We also establish second main
terms for these theorems, thus proving a conjecture of Roberts. Our arguments
provide natural interpretations for the various constants appearing in these
theorems in terms of local masses of cubic rings.
