  We develop a partial trace formula which circumvents some technical
difficulties in computing the Selberg trace formula for the quotient
$SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$. As applications, we establish the
Weyl asymptotic law for the discrete Laplace spectrum and prove that almost all
of its cusp forms are tempered at infinity. The technique shows there are
non-lifted cusp forms on $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$ as well
as non-self-dual ones. A self-contained description of our proof for
$SL_2({\Z})\backslash \U$ is included to convey the main new ideas. Heavy use
is made of truncation and the Maass-Selberg relations.
