  An exploratory study of chiral four-nucleon interactions in nuclear and
neutron matter is performed. The leading-order terms arising from pion-exchange
in combination with the chiral $4\pi$-vertex and the chiral NN$3\pi$-vertex are
found to be very small. Their attractive contribution to the energy per
particle stays below $0.6\,$MeV in magnitude for densities up to $\rho
=0.4\,$fm$^{-3}$. We consider also the four-nucleon interaction induced by
pion-exchange and twofold $\Delta$-isobar excitation of nucleons. For most of
the closed four-loop diagrams the occurring integrals over four Fermi spheres
can either be solved analytically or reduced to easily manageable one- or
two-parameter integrals. After summing the individually large contributions
from 3-ring, 2-ring and 1-ring diagrams of alternating signs, one obtains at
nuclear matter saturation density $\rho_0=0.16\,$fm$^{-3}$ a moderate
contribution of $2.35\,$MeV to the energy per particle. The curve $\bar
E(\rho)$ rises rapidly with density, approximately with the third power of
$\rho$. In pure neutron matter the analogous chiral four-body interactions
lead, at the same density $\rho_n$, to a repulsive contribution that is about
half as strong. The present calculation indicates that long-range multi-nucleon
forces, in particular those provided by the strongly coupled $\pi N
\Delta$-system with its small mass-gap of $293\,$MeV, can still play an
appreciable role for the equation of state of nuclear and neutron matter.
