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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Perpendicularity again. was: Adjusting Central Mirror
From: Herbert Prinz
Date: 2004 Oct 12, 10:01 -0400

```Alexandre Eremenko wrote:

> [...] you can move your eye (or the sextant) so that
> the point, where the direct and reflected images of the arc meet,
> moves along the right vertical edge of the index mirror.
> You cannot get a match of these arcs at every point of this
> vertical edge. Either they match at the lower point of the edge of the
> mirror,
> or in the upper point of this edge  or somewhere in the middle.
> But never everywhere.

>
> This does not depend on the particular sextant.
> Every sextant with sufficiently large mirrors will
> behave like this (and this can be shown mathematically).
>
> So WHERE exactly should you place your eye with respect to the frame
> for the proper test?
>
> No manual says this precisely.

Alexandre,

I cannot convince myself that every sextant suffers from this problem. Furthermore, I do not
understand how the size of the mirror would be responsible.

You are describing a parallactic effect which can only be caused by a different distance of
the individual cylinders from the front edge of the index mirror. As long as the pivot of
the index arm (= centre of the arch) is in the plane of the mirror and one cylinder
coincides with the image of the other, the front edge of the mirror is equidistant from both
cylinders and you can therefore look at them at any vertical angle. If, on the other hand
the mirror surface is off centre, you will get the effect that you describe. In this case,
you must look at the cylinders in a plane parallel to the limb.

Now, how does the size of the mirror come into play? It may amplify an effect that is
already there, mainly because it allows wider vertical viewing angles, and to a lesser
extent because the edge gets closer to the limb. But a large mirror by itself does not
create the effect.

The pragmatic approach to solving this problem is to ask the question: Is there a good
geometrical reason why one would want to use any other plane for the inspection than that in
which the light travels during normal operation, i.e. the axis of the telescope and the
centre of the mirrors? And if there isn't any reason, well, there you go. Jim Thompson was
right on spot when he asked "Isn't that what twin cylinders are for?  I assume that by
lining up their tops as they sit on the index arm, the eye is forced into the correct
angle." If they are one inch tall (as he confirmed in the mean time), their tops will be
exactly in the correct plane when placed on the limb of an Astra III. And so will be your
eye when you see their tops in the centre of the vertical edge of the index mirror.

In an earlier message, Alexandre said:

>
> The same simple geometry shows WHAT this test really checks.
> It checks the perpendicularity of the index mirror to the
> plane passing through the upper edges of both visors AND YOUR EYE.

I don't think so. The test checks the perpendicularity of the mirror to the LINE passing
through the upper edges of both visors. This is a stronger condition.

Somewhat less intuitive is the situation where you inspect the limb instead of the tops of
vanes. To annihilate any potential effect of parallax, you would have to position your eye
in the plane of the limb. As you noticed yourself, this is strictly impossible because the
index mirror is slightly raised, sitting on top of the index arm. So you try the best you
can. But then, every _good_ manual will tell you that this method may be less accurate.

I checked two of my sextants. The Astra, which has a front silvered mirror that is pushed
out ca 5 mm clearly produces the effect, while the good old Plath with a back silvered
mirror which seems pretty well centre aligned displays no noticeable parallax or possibly a
tiny one in the opposite direction.

Let me add to your book list one more Russian title which is one of the best post war
textbooks on nautical astronomy (far superior to what Dutton, "Bowditch" or Mueller/Krauss
have to offer on the subject):

B. Krasavtsev, B. Khlyustin, Nautical Astronomy, MIR Publishers, Moscow 1970. This is a
revised translation of a 1960 Russian edition.

The authors do not address the specific phenomenon we are discussing here. But they do
recommend to view the vanes from a distance of 30 to 40 cm past the inner edge of the index
mirror, they do say to align the tops, and a diagram clearly shows these tops at the centre
of the vertical edge of the mirror.

The authors give an extensive mathematical analysis of the instrument errors that might be
of interest to some. I could copy a few pages.

Herbert Prinz

```
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