Cometariums are not navigation devices, but we present them here because of their relation to predicting the paths of objects in the sky.
A cometarium shows the long, parabolic path that a comet takes through space. Many (including the one shown below) had keys that could be turned, moving an arm that pushed the comet along its path. The arm is sometimes geared to approximate the velocity of the comet, which is faster at its closest approach to the sun and slower at great distances.
As they were expensive scientific curios rather than useful tools, relatively few cometariums were constructed. Most working models are probably now in museums.
The first cometarium was constructed in 1732 by J.T. Desaguliers of the UK, though it was not called that at the time. (He intended to model the movement of Mercury at the time.) They became more common around 1758, with the return of Halley's comet.
- The mechanics and origin of cometaria, an article in the Journal of Astronomical History and Heritage.
- On Ptolemy's equant, Kepler's second law and the non-existent 'empty focus' cometarium (PDF download, 2 MB), an article from the Journal of the Royal Astronomical Society of Canada. Contains some nice gear diagrams for a cometarium.