  The classical Heisenberg model on the trillium and distorted windmill
lattices exhibits a degenerate ground state within large-$N$ theory, where the
degenerate wavevectors form a surface and line, in 3-dimensional space,
respectively. We name such states partially ordered to represent the existence
of long-range order along the direction normal to these degenerate manifolds.
We investigate the effects of thermal fluctuations using Monte Carlo (MC)
methods, and find a first order transition to a magnetically ordered state for
both cases. We further show that the ordering on the distorted windmill lattice
is due to order by disorder, while the ground state of the trillium lattice is
unique. Despite these different routes to the realization of low temperature
ordered phases, the static structure factors obtained by large-$N$ theory and
MC simulations for each lattice show quantitative agreement in the cooperative
paramagnetic regime at finite temperatures. This suggests that a remnant of the
characteristic angle-dependent spin correlations of partial order remains above
the transition temperatures for both lattices. The possible relevance of these
results to $\beta$-Mn, CeIrSi, and MnSi is discussed.
