  A novel geometric formalism for statistical estimation is applied here to the
canonical distribution of classical statistical mechanics. In this scheme
thermodynamic states, or equivalently, statistical mechanical states, can be
characterised concisely in terms of the geometry of a submanifold ${\cal M}$ of
the unit sphere ${\cal S}$ in a real Hilbert space ${\cal H}$. The measurement
of a thermodynamic variable then corresponds to the reduction of a state vector
in ${\cal H}$ to an eigenstate, where the transition probability is the
Boltzmann weight. We derive a set of uncertainty relations for conjugate
thermodynamic variables in the equilibrium thermodynamic states. These follow
as a consequence of a striking thermodynamic analogue of the Anandan-Aharonov
relations in quantum mechanics. As a result we are able to provide a resolution
to the controversy surrounding the status of `temperature fluctuations' in the
canonical ensemble. By consideration of the curvature of the thermodynamic
trajectory in its state space we are then able to derive a series of higher
order variance bounds, which we calculate explicitly to second order.
