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Peter Monta,

Excellent summary. That is about the size of it. :) 

I recommend two experiments to see the impact of collimation error in its most basic form. First, make a fixed-angle sextant...

A fixed-angle sextant is nothing more than two small mirrors affixed perpendicular, with some angle between them, to a board or even a lump of stiff clay. The idea is to remove everything from a sextant except the mirrors. There are no shades, no index arm, no micrometer, no scope, not even a graduated arc. Just two mirrors sticking up on a little board. Now when you look into this pair of mirrors, you see two superimposed views, just as in a proper sextant. And when the views coincide perfectly, the "instrument" measures one fixed angle, and quite exactly, too. The angle can be determined by a variety of methods including the simple process of a practical test of altitude from a known location. Make yourself a few dozen of these fixed-angle sextants, each properly calibrated, and you would have no need for an expensive professional sextant.

But is a fixed-angle "sextant" accurate? Does it even deserve the name "sextant"? In this primitive proto-sextant, we can see the impact of various errors of orientation isolated and abstracted from all other issues. How do yaw, pitch, and roll affect the instrument's ability to measure an angle properly? We can get to grips with the matter experimentally, by observing angles. But this is also a great case for geometric analysis in mathematical vacuo.

Once we understand the proto-sextant --this "root-sextant", consisting of nothing more than two mirrors mounted on a board, we can begin to generalize. How do real sextants manage these errors from "wobbling about"? If you yaw, is there an error? If you pitch, is there an error (generally no, and this is the whole point of double-reflection)? And if you roll the sextant, is there an error? And when you contemplate that latter question, you have to start thinking about the axes about which you are making these rotations. About which axis should you roll the instrument when swinging the arc, so-called? And going further, are there rules which apply for small-angle rotations of a degree or two that do not apply for larger rotations? Probably yes. Does that matter in practical terms? And does order count (always problematic with rotations!) if I yaw then pitch or pitch and yaw after?

For a second experiment in collimation error, make or find yourself a Bris sextant. This is also an example of a fixed angle sextant with the reflective surfaces glued or attached to each other directly without even a backing board which allows for multiple fixed-angle reflections. These are genuine "instruments of reflection" with the potential accuracy that follows. But how do you use one? How do we swing the arc? How do we manage collimation? Just what the heck is going on with the process of observation with a Bris sextant, and how do we compare that with a traditional sextant? And how does it compare with a pure fixed-angle sextant?

Just some things to think about...

Frank Reed