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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,
B).

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.

George.

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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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