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From: Alexandre E Eremenko (no email)
Date: Mon Oct 31 2005 - 20:18:30 EST
Dear John,
My math estimate from the observationbs is that
the two angles between the adjacent glass panes are
about 5.2d and 6.5d.
(The rest of this message is pure math.
I apologise to the list members who don't like it.
Just skip it then).
Here is the theory of the Bris sextant. The ray from the Sun to your eye
experiences on its way an even number of reflections.
Two reflections are equivalent to a rotation
by twice the angle between the two reflection planes.
(This is a theorem we teach in our linear algebra courses;
the ordinary sextant
is based exactly on this theorem: the ray from the Sun experiences
exactly two
reflections, one in the Index mirror and another in the Horison mirror).
Now we have 3 mirrors. Say 1, 2, 3. Let the angle between 1 and 2 be A/2,
and the angle between 2 and 3 be B/2. (The mirrors are enumerated in
the natural order, from the Sun to your eye, for example.
Consider first the three bright Suns.
They correspond to the rays which experience 2 reflections.
The possible combinations of 2 reflections are
(21), (32) and (31).
The angles of deflection are A, B and A+B.
Now consider the dim Suns. They correspond to the rays
experiencing 4 reflections.
There are 8 combinations of 4 reflections possible,
(2121), (2132), (2131), (3232), (3231), (3121), (3132), (3131).
Some of them will produce the same angles. (These Suns
will look slightly brighter).
The possible angles of deflectio of the ray are only 5:
2A, 2B, 2A+B, 2B+A and 2A+2B.
Now, assuming without loss of generality, that A<B<2A,
we order all angles in the order of magnitude:
A, B, 2A, A+B, 2B, 2A+B, A+2B, 2A+2B.
Eight angles, exactly as I see in my Bris sextant.
Now compare this with my measurement,
say for the Lower limb, which is more complete:
rounding to the tenths of a degree they are
10.5, 13.0, 21.0, 23.5, 26.0, 34.0, 36.0, 46.0.
The best fit seems to be A=10.5d and B=13.0d,
from which we deduce the angles between the panes: 5.2d and 6.5d.
Alex.
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