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George replied"
> Bill comes across as a careful and intelligent observer. His recent
> observations are troubling him, and deserve our careful scrutiny. But
> he needs to be aware that in working to fractional minutes in that
> way, he is getting near to the limits of what's possible in terms of
> the instrument he is using, and his eyes.

Thank you, and limits understood.  I consider, in static conditions, +/-
0.1' precision to be about my limit on a good day.  The Astra is rated at
+/- 20" accuracy along the arc.  Hence my apparent lack of concern over
matching 4SD (or interpolated 4SD) exactly with two observations.
>
> First, we can dispose of one discordant result as due to faulty
> arithmetic. Where he refers to "Later tests done with sun Hc nominally
> 32d, left eye" and gets  (in minutes of arc): Vertical average on the
> arc 31.625, and vertical average off the arc 30.405, I make the
> resulting index error to be 0.6' on the arc, not as Bill derives
> 0.11'. Which would make it completely concordant with his first
> vertical set.

Your math is correct.  My input was not.  A blunder on my part.
On the arc is correct at 31.625.  Off the arc was a typo.  Reading average
55d 28!595 off.  60 - 28!595 = 31.405 (not 30.405 as stated).  So 0.11' IE
was correct.
>
> It would be interesting to know what magnification telescope is in
> use.

Stock Astra 3.5X star scope.
>
> The fact that different results are obtained with differing sextant
> orientations leads to a strong suspicion of flexing under gravity
> somewhere. Not very likely to be in the frame itself, more likely to
> be in the fixing of the mirrors in their mountings, or in the fixing
> of the mountings themselves with respect to arm and frame. If Bill
> pushes the mirror-glasses slightly, at the corners (with a gentle soft
> tool) is there the slightest sign of any motion? Clamping the handle
> in a firm vice, can any shift be detected in a reflected image when
> that is tried?

That is a possibility, but I make it a practice to never touch a mirror
unless absolutely necessary for cleaning.  They will in fact move slightly
in many cases.

> I'm aware that Bill has discounted gravity-flexure because of
> differences in behaviour seen by left or right eye, but he has not
> spelled out those differences for us in detail, so I am disinclined to
> accept those assurances at face value. More information about that
> please, better specified. One test for flexure would be to make the
> same measurement with the sextant vertical (in its normal
> orientation); then inverted.

This I have already done in the vertical versus horizontal tests posted if I
understand you correctly. All published results were from the left eye only,
with the exception of noting the reversal of trend with the right eye. I am
sorry for any confusion I may have caused. Horizontal tests were done in two
stages, one set with handle up and one with handle down.  No noticeable
difference between the two sets, so they were lumped together in the data
posted.

Regarding rigorous right-eye testing, at this juncture I have not done that.
I determined early on that my left eye was more precise.  I made a dozen or
so observations with the right eye to establish a baseline this time, but
the results were poor--as usual. Despite being able to focus to the point
that I could see a sunspot clearly with the left eye, I could not bring it
into focus (or even see it) with the right eye after repeated attempts.
Standard deviations of the observations I did make with the right eye
vertically and horizontally were 0.25' and 0.3' respectively.  As a layman,
I could not justify comparing the left eye numbers with 0.1'SD to the right
eye numbers

What I can propose regarding gravity on the frame or mirrors is that I would
expect changes from vertical to horizontal to trend in the same direction
for both eyes or any  observer.  That was clearly not so.  Achieve tangency
(observed) with the left eye and sextant vertical, then rotate the sextant
to horizontal position, handle up then down, and the images clearly
separate.  Achieve tangency (observed) with the right eye and sextant
vertical, then rotate the sextant to horizontal position, handle up then
down, and the images clearly overlap.
>
> Now for the odd effects that Bill reports in his vision, in which
> circles are distorted in different ways by different eyes. How does he
> tell this? If he looks with just one eye, is a circle, to him, clearly
> an uncircle? By about what sort of percentage? Is it only when looking
> with both eyes that he becomes aware of the discrepancy? And about how
> much discrepancy, as a percentage of the diameter, is he aware of?

Good questions.  I am most aware of it at night, looking at lighted globes
around the property, although it is noticeable in the daylight as well.
Without my eyeglasses it is markedly an uncircle, perhaps 15% +/- 5%
distortion per eye.  With glasses, not noticeable unless one is looking for
it, but there. Not noticeable in the scope without glasses.  Of course focus
is off with the naked eye and globes, so I get the added effect of soft
edges.  I have run tests with the scope and glasses, but I cannot support
with any confidence a change in observed values.
>
> I am speaking here as a layman, but one with some experience of
> optical defects. The human eye-system is clever; it projects an image
> on to the retina, the pixels of which map to the brain as a perceived
> image. Part of learning, from infancy, is correlating that image with
> real life. If circles had always been ellipses on his retina, due to
> some astigmatic defect in his optics, the brain would have mapped that
> ellipse as actually corresponding to a circle. So if Bill in now
> conscious of circles reading-out in his head as ellipses, it could be
> because of a recent change, in optics or retina, that his brain hasn't
> adapted to yet. Either that, or perhaps he wears specs with an
> astigmatic correction, that his brain has adapted to, so he sees
> distortion when he takes them off; as magnification in one plane, or
> another.

Excellent points, but it does raise some questions for me based on my
observations.  I (most of us) are used to using two eyes, so they brain may
be mapping to correct for both eyes simultaneously,  I submit for
consideration some observations.

With both eyes open (no glasses) the lighted globe, while somewhat out of
focus, appears to me as a circle.  Somehow the brain has correct the
problem. One might argue that the average of the two just happened to be a
circle.  Of course when speaking of averaging, that ignores eye dominance.

In comparing scopes critically for two weeks, (one on each eye at some
times) I noted the slight color differences between the two images.  Ah, I
thought, different lens coatings.  To my surprise, when I swapped out the
scopes and eyes, the tint did not follow the scope.  It was may eyes' color
perception that was different.

In the test for scope magnification (proposed by you it I recall) the scope
was on one eye, both eyes were open, a for a brief period of time one could
compare fence posts with both eyes to determine magnification.  But the
brain quickly adjusted and removed one image from perception.  As it does
with color temperature.

Thinking back to psychology/visual perception I recall the experiment of
putting inverting goggles on a subject.  It took days or weeks (and
headaches) before the brain flipped the image so the images were again
upright despite the inverting goggles.

So it seems the brain can rapidly (seconds) adjust for input, or be very
stubborn about giving up on the tried and true.  One question here is when
one goes from using two eyes to one eye, how quickly will the brain remap
the information?

FYI, my eyes are about 20/60, but I only wear glasses for driving (maybe 2
15minutes sessions per day, when attending sporting events, or when coastal
piloting. "No glasses" is pretty much the norm for my optical system.

>
> But it shouldn't matter much to Bill's sextant problem, as I see it.
> Bill is not using his eye-telescope combination to measure anything,
> such as the height of a distant building, when using his sextant. He
> is simply using it as a null-sensor, to determine coincidence between
> the two views, of the top of the building in one view, and the bottom
> of it in the other. It's the sextant, not his eye, that measures that
> height. That's why you can switch to a telescope with a different
> magnification, or even no telescope at all, and get the same result.

I am still wrestling with that concept.  One line of reasoning is that if my
eye is distorting the image in the scope, then it appears taller than
reality, so my angle is too great.  The other line of thought followed your
explanation.  Given our track record, I am inclined to take your explanation
on faith, or at least give it 100:1 odds. 
>
> Now, about irradiation. There are special problems when using the Sun.
> A lot of dark shade is needed to make it tolerable. Often, the darkest
> shade for the horizon mirror is less black that that for the horizon
> mirror, because the reflected horizon is much less bright than the Sun
> is. Is that the case here? Or, with a full-view mirror, the two images
> may not be equally bright, or may be differently coloured. Is that the
> case here? If the two Sun images that Bill is comparing are not the
> same brightness, for any reason, then the effect of irradiation
> enters. This is a defect in every human eye, which causes brighter
> objects to appear to have their boundary, with adjacent dimmer
> objects, extended, so that they look slightly bigger. It differs from
> one person to another, and for all I know, may well differ from one
> eye to another. Is that part of the difference that Bill sees when he
> shifts from one eye to the other?

I have the traditional split horizon mirror/glass, and three of the shades
for both mirrors appear to be identical, so (ignoring a very slight possible
theoretical difference between mirror and glass sides) the images appear to
be equally bright.  This variable seems to dovetail with the building-height
question above.  Which matters?  What the scope and a "perfect" eye would
see (with no irradiation effects) or what my optical system sees?  In both
cases my system would see an image larger than life (along the vertical axis
with the preferred left eye).
>
> The effects of irradiation could be removed by using for index
> checking the horizon or a distant building, without shades. Perhaps,
> for the Sun, a single shade, cobbled-in (securely !) just in front of
> the telescope objective, instead of the normal shades, would equalise
> the brightness (and also remove shade errors).

This seems to be the logical first step.  Is the offset between
natural-horizon IE and sun IE connected to the shades?  If so, try to
isolate which set of shades and to what magnitude.  If not, tackle the eye
distortion and irradiation variables.

Which brings up a question.  Before it was determined that personal
differences, scope and filter differences etc. made in impossible to
accurately calculate and adjust for irradiation, what were the publish
magnitude(s) for irradiation?
>
> In reply to a question from Fred-
>
>> Third, have you tried changing the position of tangency to try to at
>> least get the SDs to agree?
>
> Bill replied-
>
> "Tangent" is by definition "tangent."  It is not, nearly touching.  It
> is
> not overlapping.  It is tangent... "
>
> Well, in cases where irradiation can have an effect, tangency is not
> so clear-cut a matter as Bill seems to think it is.

I both agree and disagree. It seems to depend entirely on one's frame of
reference.  If what the observer sees is the determining factor (and this
seems to be the way the sextant game is played in round #1) then what he/she
sees as tangent is tangent.  If the eye or irradiation distorts the image,
it could in reality be tangent, but the observer sees it as separated or
overlapped in error.  From a purely scientific standpoint the second
scenario is correct, as a part of the measuring system (the eye) introduces
a systematic error.  At the risk of comparing apples to oranges, we must
adjust the observed altitude of a body to its position with no air for our
calculations. How to do that for my eye is the end goal of the exercise.
Quantify the systematic error(s) and determine how best to correct for the
error (without eye surgery ;-)

As an example, let's assume my horizon IE is 0 and is both subjectively and
objectively correct (no darned spheres in play;-).

Then I calculate IE with the sun, with my (left) eye and irradiation
problems.  4SD is 63'(theoretical for simplicity.) I perceive and measure
on-the-arc tangency by separating the images until tangent (32') after they
are in reality separated, so the on-the-arc figure is too high. I perceive
and measure off-the-arc tangency (59d 29' = 31') before they are in fact
tangent, so the off-the-arc figure is too low.  In a perfect world, if IE
was in fact 0, the on and off measurements should have been 31.5' and 31.5'
respectively.

If that discrepancy cannot be traced to the shades between the no-shade
linear horizon alignment and the sun tangency with shades, what's left?
Irradiation and eye distortion, or problems with the sextant 1/2d from the 0
point.  That latter topic has been discussed and various excellent test
methods suggested when Alex raised the issue.
>
> I am puzzled about the changes in index error and side error that are
> reported when adjusting the focus of the telescope eyepiece. Are these
> observed effects clear-cut and reproducible? It's just that a point,
> seen directly, should be exactly coincident with its reflection, when
> observed through two mirrors which are exactly parallel in both
> planes, no matter what the direction of observation is. So I don't see
> how such changes to the telescope can change those errors. I wouldn't
> be so foolish as to claim that these effects can't happen, in the face
> of claims by two respected observers. But I ask, seriously, whether
> that evidence is strong, reproducible, and watertight. And if it is,
> can anyone suggest how it comes about?

I would say yes.

Method 1: Same star. Null out side error with one star image slightly below
the other.  With one person and one eye, make a slight change in focus.  You
can see two images become misaligned, introducing side error.  The fault
with this method is the size of the images are changing as the focus is
changed, and the range of focus change is limited before the images are
unusable.

Method 2:  (This is for folks that have devices to correct their vision, one
person, one eye.)  Same star. Null out side error with one star image
slightly below the other with focus set for naked eye.  Now put on your
glasses (or contacts) and refocus.  Side alignment will shift.

Method 3: Two people (Alex and I have experienced this with my Astra when
observing together and exchanging sextants). Same star. Null out side error
with naked eye with one star image slightly below the other.  Switch
observers and let the new observer refocus for their uncorrected vision. The
new observer will notice side error.

Alex and I have both experience side error nulled out with a star, then when
we looked at the sun the next day there was side error.  In my testing I
went back that night and the side error was gone.  I do not put a lot of
stock in that as proof.  Yes, the scope was refocused between night and day
observations, but there are other variables such as temperature, color of
body, and shades in place.

NOTE:  This assumes refocus is more than a hair.

What is more difficult to notice, unless you are looking for it, is a change
is vertical spacing as well.

In my mental model, the changes in the relationship is similar to Polaris
moving around the pole.  A some point you may see a larger amount of shift
along the X axis, but little along the Y axis.  In other cases you may see
the opposite.  In some cases, almost equal X and Y shift.

I may be ducking the question, that being what if the two images of one body
were perfectly superimposed?  I'll have to give that more thought.

In the case of two images of one body, or two bodies, the cause is in the
scope optics.  The lenses may not be perfectly ground.  Even if they are
perfect, the issue of lens collimation (are the lenses mounted perfectly in
the barrel) may come into play.  A likely culprit is the focusing mechanism
itself, as pointed out by Red who also experiences side-alignment changes
with his C&P.  Any eccentricity in the focusing mechanism or lens(es) not
parallel to the permanently mounted lenses could throw it off.

Bill





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