  We construct, for any finite dimension $n$, a new hidden measurement model
for quantum mechanics based on representing quantum transition probabilities by
the volume of regions in projective Hilbert space. For $n=2$ our model is
equivalent to the Aerts sphere model and serves as a generalization of it for
dimensions $n \geq 3$. We also show how to construct a hidden variables scheme
based on hidden measurements and we discuss how joint distributions arise in
our hidden variables scheme and their relationship with the results of Fine.
