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David Pike wrote: "Has anyone any idea of the light paths and why anyone might be interested in measuring 180°?"

You could use the sextant (or quintant) to measure the actual dip of the horizon. Lets use the sun as an example: For this you take one "normal" measurement (lets say the lower limb) h1. Then you turn 180 degrees (sun in your back) and take a second shot now over your head (your zenit) of the sun with the now "upper" limb (the same limb as before), you get a second "height" h2 greater 90 degrees. With equation dip = (h1 + h2 - 180 degrees) / 2 you can calculate the actual dip at your position.

If you use a standard sextant up to 120 degrees, the body must be at a height of minimum 60 degrees, to get h2 correctly. (you need for the second shot 90 degrees up to the zenit and then (90 - 60) = 30 degrees to the body, thats 120 degrees max.)  If you have a quintant with an arc up to 160 degrees, the body would need to be only at minimum height of only 20 degrees.

Maybe this method could also be used if your horizon below the body is not visible/usable, but the horizon in your back is visible. I am not sure about errors involved by such "overhead" measurements of heights; i have never done it myself yet.

Best reagrds,

Uwe