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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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