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    The repeating circle.
    From: George Huxtable
    Date: 2005 Jan 1, 21:51 +0000

    The repeating circle
    I'm preparing a posting for the list, as promised, about Mayer's invention
    of the first repeating circle, including a translation (not by me) of his
    paper from the original Latin.
    In doing so, I have been reading Chauvenet's account (in "Spherical and
    Practical Astronomy", vol 2, pages 119 onward) of the workings of such a
    circle (perhaps Borda's), in which the method of use differs somewhat from
    that proposed by Mayer.
    And I don't understand how the method that Chauvenet describes could
    possibly work, over the whole range of possible angles. That's rather a
    surprise to me, as Chauvenet is normally so very reliable. So perhaps it's
    me that is getting it wrong. Which is why I am asking for help, from anyone
    else who has access to a copy of Chauvenet. I would scan and post a copy of
    the relevant pages (119 to 123) myself, but am painfully aware how such
    images produced by my ancient Mac are incompatible with some PC's, so it
    would be kind if another list-member with access to Chauvenet would make
    those pages more widely available.
    The problem shows up in comparing Chauvenet's fig. 28 and 29, which show
    the two alternate positions of the instrument, used alternately in
    measuring the angle AB subtended between two sky objects A and B.
    I see no difficulty with Chauvenet's fig. 29, in which the geometry of the
    telescope T, the two mirrors m and M, are very familiar, just like an
    observation with a normal sextant. In this case the object B being viewed
    in the index mirror is to the right (i.e. clockwise on the diagram) of the
    direction in which the telescope is directly viewing the other object A. No
    problems here. We can now forget about fig. 29.
    The problem occurs with fig. 28, which shows the object being viewed in the
    index mirror (which is now A) to the left (i.e. anticlockwise) of the
    direction in which the telescope is viewing the other object (this time,
    Take a careful look at that diagram 28. It shows that light arriving at the
    index mirror M from A now has to cross the view-line of the telescope Tm,
    something that never ever happens with a sextant. As shown in the diagram,
    the telescope tube itself doesn't get in the way of that light-ray, because
    the telescope T has been pulled far back, much further than it would be in
    a sextant, to leave a good gap between T and m. But the instrument has to
    be able to cope with a wide range of angles subtended between A and B, all
    angles from 0 degrees to (say) 120 degrees. Let's say the angle AB to be
    measured happens to be about half of what Chauvenet has shown on his
    diagram. Then how could light from A (in that new direction) reach the
    index mirror M, without colliding with the other mirror m on the way in?
    That's the problem that I can't resolve. It seems to me that mirror m must
    inevitably obscure the index mirror's view over quite a swathe of
    directions. If the required angle AB fell into that range, the instrument
    would be unusable, wouldn't it?.
    In addition to describing the use of the circle as above, Chauvenet in art.
    109 considers another way of using the instrument. Instead of exchanging
    the views of the two objects between telescope-direct and index-mirror, the
    telescope is always used directly for observations of one object (A, say)
    and the index mirror always used for the other, the necessary switching
    between the two modes taking place by inverting the instrument and holding
    it by the other hand. That's all very well in itself, but used that way, it
    appears to me to be just as susceptible as before to the same obscuration,
    over a similar range of angles.
    Unfortunately, such repeating circles are precious museum objects so I
    don't expect ever to get a chance to try one out to see if the difficulty I
    am foreseeing actually presents a problem in real-life.
    I would be interested to learn if any listmember agrees with my analysis
    above, or better still, can show that I am imagining difficulties or can
    point to some trick for circumventing them.
    Mayer's own proposed method avoided such crossings-over between the two
    directions. He always used a configuration as shown in fig. 29, and
    interleaved between several such observations a zeroing observation in
    which the mirrors were "set parallel" in a similar manner to an index-error
    check on a sextant. So Mayer's method avoided any need to view directions
    to the left of the view-line of the telescope as in fig. 28, those
    directions in which the problem occurs. In addition, this allowed Mayer to
    bring his telescope close up to the horizon mirror. However, Mayer's
    procedure required twice as many observations, altogether, for a similar
    accuracy of result.
    More on Mayer soon.
    A happy New Year to all.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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