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I'm just a bubble sextant owner, but based on what I've read it seems
that the normal method for measuring a marine sextant's index error is
to bring the direct and reflected images of an object (horizon etc.)
into coincidence. There is also the method of splitting a celestial body
and bringing opposite limbs into contact. Trying to think outside the
box, I looked at another approach, utilizing the eye's ability to
accurately bisect the gap between two closely spaced lines.

The method I propose uses a test target of several parallel, equally
spaced horizontal lines. The lines should appear closely spaced but
clearly resolved at *double* the distance at which the target will be
used. (The reason for that will become clear.) The white spaces between
lines should be should be considerably wider than the lines, perhaps
three times wider.

Set up the target at a measured distance and sight it through the
sextant. Bring the reflected and direct images into coincidence. Due to
the parallax between the two lines of sight, the sextant will be set to
a slight negative angle ("off the arc"). From the vertical distance
between the telescope and index mirror, and the distance to the target,
you can compute the parallax. Add parallax to the sextant reading to
obtain index error.

That's a preliminary value. The point of coincidence is hard to judge
when attempting to put black lines on top of each other. It's easier to
bisect a narrow white space with a black line. That's the reason for the
multiple lines on the target. A slight movement of the micrometer knob
will split the images so the black lines of one will bisect the spaces
of the other, thus doubling the number of visible lines and halving
their spacing.

The additional small angle to attain this condition can be calculated
from the line spacing and distance to the target. Additional bisections
occur at multiples of that angle, allowing an average index error to be
calculated using more than one point on the worm. (Doing a full
revolution would require a large target, though: five feet at 100 yards.)

Many variants of this method are possible. For example, the simplest
test target would be a pair of lines plus a single line to bisect the
space between them. If the pair and the single are spaced to match the
vertical offset between the direct and reflected lines of sight at the
sextant, parallax is eliminated.

To check parallax, make another measurement at half the original
distance. It would be best to construct another target so the sight
picture looks the same at the shorter distance. The results from this
second test will reveal any residual parallax, since IE is constant with
distance while parallax varies in inverse proportion.

Unfortunately I don't have a marine sextant to try any of this.

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