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Frank Reed, you wrote:  "how does this thing work??"

Frank
I’ve been wondering that, but you always ask these questions at my bedtime, so here are just a few first impressions. 
1.  The sextant shape is a red herring.  It’s really a ‘triangle solver’.
2.  The actuate arm probably swivels.  Probably about the index mirror end and takes the index mirror with it.
3.  The screw threaded shaft up the middle of the drum pushes the actuate arm up against the compression spring in the radius arm moving the double reflected image compared to the direct image in the eyepiece.
4.  If tangent rules, moving the drum along the height scale alters the length of the ‘base’ (adjacent side) of the triangle, the movement up or down of the screw threaded shaft being the ‘perpendicular’ (opposite side).
5.  The next step is for one of our mathematicians come up with the maths, which hopefully is a simple tangent problem and not some nth order differential equation.
6.  Incidentally, did you note that in your photograph the left-hand end of the actuate arm has fallen out of its slot. 
Analogue Rules OK.  DaveP
PS.  written before I noted Murray had also replied.