Marine sextant

What happens

Light from the star or the Sun's lower limb strikes the index mirror, which turns with the index arm, then the half-silvered horizon mirror, and enters the telescope beside the direct view of the horizon. Both reflections are computed as reflections, d′ = d − 2(d·m)·m; together they turn the ray by twice the angle between the mirrors, whatever its incidence, so the arc is divided in double degrees: the rack gives whole degrees and the drum minutes. The sea horizon lies below the true horizontal by the dip, 1.76′·sqrt(h), and a star appears higher than it is by the refraction (Bennett).

For a star at 35° seen from 10 m the dip is 5.57′ and the refraction 1.42′: the image meets the horizon at a reading of 35.1164°, and the reduced sight gives 35° back. An index error of +2′ moves the reading to 35.1497°; left uncorrected, the sight is 9.0′ too high. The Sun at 20°: its lower limb comes down at 19.8710°. A frame tilted about the line of sight needs a larger reading: 1.4′ more at 2°, 8.9′ at 5° and 36.2′ at 10°. On a rolling deck the image swings in an arc and touches the horizon at its lowest point, where the frame is plumb.

Model: plane mirrors and exact vector reflections; the star dead ahead; no prismatic or centring error. An observer swinging the arc keeps a tenth of the roll in the frame.

Try this

Bring the image down with the arm and the drum until it touches the horizon, then take a sight. Let the ship roll and swing the arc: set the reading where the image just touches the horizon at the bottom of its path.