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Dear Fred,

1. Now I think I understand what Hadley writes
about wires. He worries about parallelism of
the line of sight to the plane of the quadrant.
The lack of parallelism can arise from
a) collimation error and
b) if the objects touch away from the middle line
of the field of view.
Hadley assumes that collimation is perfect and
the distance of each wire from the central line
(the line parallel to the plane of the arc, and passing
through the optical axis) is 1 and 3/8 degrees.
Then, if the objects touch on one of the wires,
the error in minutes is
-2 tan(A/2),
where A is the measured angle.

Chauvenet, on p. 113 of volume II gives the precise formula,
as well as the following approximate one:
error=-i^2 sin(1') tan(A/2),
where the error is in minutes, and i is the distance of a wire
from the center line in minutes.

In my sextant, this distance i (half of the distance between
the wires) is approx 1 degree,
something
in between of what Headley recommends and the 30' distance
"of the best sextant telescopes" according to Chauvenet.

Now both Hadley and Chauvenet recommend to have the
objects touching as close as possible to the central line,
and it is clear that the wires help to do this.
(The total field of view of my telescope is about 6 degrees).

But Headley goes even further. He suggests making several
(more than two) parallel wires, over the field of view,
and than applying a special correction, depending on
the place where the objects touch.

From my experience there is at least one reason why one would want
to touch the objects away from the central line. The reason
is the following. I found that under certain conditions
my horizon glass
reflects very
well by its FRONT (unsilvered surface). So that its transparent
part of it works like a "full view mirror". I find this feature
very useful, especially in taking the distances.
This requires a touch near the LEFT wire, rather than in
the middle. Now I am waiting for good weather to test
Hadley's formula in this situation:-)

2. Inspecting the formula, we see that there is indeed a big
problem with measuring distances close to 180 degrees,
when the tan(A/2) blows up.

This is the mathematical explanation of why

a) the quadrant with back sight feature cannot be
used as a dipmeter, and

b) why everyone experienced the problems with finding
the index correction for back sights with Wollaston method.
(See my previous messages on this subject. I can conclude
from his paper that Wollaston did not actually experiment
with a quadrant, just speculating on the basis of
"thought experiments":-)

Same difficulties should occur with "periscope method"
by Blith as described by George, and with George's
self-made periscope attachment.

This also explains why the other method of finding the index
correction for back sights described by George is superior
to the Wollaston method.
I recall: this other methodi involves
a measurement of an angle close to 90 degrees
(which can be taken by BOTH fore sight and back sight).
First you determine the index correction for the fore sight.
Then you measure some angle close to 90 degrees by a fore sight.
And then you measure the same angle by the back sight,
to determine the back sight index correction.

It seems clear that this method can hardly be practiced
in sea. (Indeed, the only appropriate objects would
be two stars at approx 90 degrees distance, the Moon
is not appropriate because the angle should not change while
you switch from the fore to back observation). I think
this is the
reason why back sights were
abandonned by the mariners. I can add that all sorts of
collimation errors (including the non-parallelism described
above) will interfere with the index correction performed
with this "90-degree method".

It seems extremelly interesting to me, how the dipmeter
designers overcame this difficulty. Unfortunately the picture
in the Shufeldt paper (as it was transmitted by fax) does
not permit me to determine the details of its construction.
and Shufeldt
himself does not say anything about this.

Alex.

   

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