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At Google Books I came across an 1844 book published by instrument
makers Troughton and Simms. Thanks to its chapter on the Troughton
reflecting circle, I now understand how that device works, although at
first the principle of the "reverse reading" wasn't clear. Finally I
realized it is equivalent to inverting a marine sextant, directing the
telescope in the same direction as before (say, to the horizon), and
moving the index arm off the arc to sight the body, which is now at a
negative altitude from the point of view of the instrument. Of course in
this orientation there's a range of angles where the horizon glass
blocks the index mirror's view. The same thing occurs with the
reflecting circle, according to Troughton's explanation.

The chapter closes with some good advice: "Should these hints about the
adjustments set any over-handy gentleman on tormenting his instrument,
it will not be what was intended by them; they were added, that, in case
of accident, those who are so unfortunate, might be enabled thereby to
put their own instrument in order."

http://books.google.com/books?id=FyQJAAAAIAAJ

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