  Using PCA data from the Rossi X-Ray Timing Explorer (RXTE), we track the
spectral states of the neutron star transient system Aql X-1 through a complete
outburst cycle. We find a hard-to-soft state transition during the very early,
rising phase of the outburst and show that there is a hysteresis effect such
that the transition back to the hard state occurs at a luminosity $\sim $ 5
times {\it lower} than the hard-to-soft transition. This hysteresis effect
rules out the propeller mechanism as the sole cause of state transitions in Aql
X-1. Assuming the propeller mechanism only operates at a luminosity equal to or
below that of the observed soft-to-hard transition requires that the magnetic
field of Aql X-1 be less than $7\times 10^7$ Gauss, the lowest neutron star
field known to date. To compare the state transition behavior of Aql X-1 with
that found in transient black hole systems, we use RXTE ASM data to compute
hardness-intensity diagrams for four black hole candidate transients where the
ASM data should also give us state information throughout much of the outburst
cycles. In all four systems, we find evidence for a hard-to-soft state
transition during the rising outburst phase and for the source staying in a
soft state down to much lower luminosities during the declining phase, i.e., a
hysteresis effect. This similarity suggests a common origin for state
transitions in low magnetic field neutron star and black hole systems, and the
hysteresis effect rules out the ``Strong ADAF Principle'' for determining the
state of an accretion disc. We discuss the general implications of these
observations for current models of state transitions. We note the contrast to
previous observations of the non-transient systems Cygnus X-1 and X-3 which do
not show a hysteresis effect.
