Chris Lusby Taylor wrote:
Bill Maddux gives a very logical reasoning to the subject of how
accurately one can read a sundial, assuming the reading is of the centre
of a symmetrical shadow line with fuzzy edges.
I am inclined to agree that judging the centre of such a shadow must be
easier than judging the position of the shadow cast by an edge. I believe
the optimum width of gnomon is such that the umbra is of zero width. Even
then, the figure of 1/60 width seems surprisingly good to me.
The list is not a suitable place for full details or the diagrams, graphs,
etc. needed to explore this thoroughly. When I spoke of the edge gradient
of a shadow, it was with the thought that the light gradients mentioned
apply to both the external boundaries of a shadow image and two the
internal bounderies of an aperture's image.
As for the symmetrical cut, try this little experiment: Cut a paper
rectangle with a width of at least 100 mm, or draw a pair of parallel lines
on a larger sheet at least 10 cm appart. Set this up at comfortable arm's
length, and try to judge the center of the space between the lines or
paper's edges. Mark the judged center with a fine pencil point, then
locate the center by carefull measurement with a millimeter scale, and
compare to determine your error. (Or for repeated trials, use a
wall-mounted small slate, or chalk board, make the judgement from a
distance, and fixate on its location as you approach and make a mark.
Erase and repeat this, and record and assess results by simple statistics.
I think you will be surprised. Recruit naive observers, and see how well
they do. (Also, compare the judgement of a horizontal interval vs a
vertical one. I think you may find higher precision in one orientation than
in the other.)
Note that when centering a symmetrical object on a symmetrical target, a
given error is doubled in its effect on symmetry since added distance on
one side is subtracted from the other. If the centering involves placing a
somewhat broadened fuzzy image relative to a double-edged defining
target-interval, a comparison of the asymmetry of the edge-spacings may aid
the process. I.e., if the difference in target width and image width is a
small interval compared to the width of the image, decentering by a quarter
of that small interval makes a three quarters discrepancy at one side while
diminishing that at the other to one quarter. Thus their ratio is a very
visible: three to one, instead of the one to one when centered. Also
realize that psycho-physical resolution is better when distinguishing two
closely spaced, extended, linear objects, than for two spots at the same
separation. (This has to do with neural signal processing. and integration
of adjacent receptor element responses.) If you use a circular spot, I
suggest it not be too small, and that you use a target with concentric
rings. You can get moire type effects at the arc edges that are very
sensitive for alignment. (These are not diffraction effects, since the
aperture acts as in a pinhole camera and forms an image of the sun's full
half-degree wide disk.) You can learn a lot by playing with full-scale
mockups in real sunlight under a variety of conditions. I recommend it
over simulations, whether with artificial sources or virtual on a
computer. It is a fun aspect of our dialing hobby.
==
You also wrote:
3. When a sextant is used to find the sun's altitude in a known place, it
is a sundial. You adjust a sextant to make the lower 'limb' of the sun
appear to touch the horizon. This can be done with great accuracy. Even my
plastic sextant is calibrated to 0.2' of arc, so I am sure the bestbrass
sextants are better than that. You add on the radius of the sun, and
correct for refraction in the atmosphere, to get the true altitude of the
centre. This is going to beat any spot or shadow at times of day when the
sun's altitude is changing fast.
As for sextant measurement: I don't think we should consider this naked
eye, not even if the sight is via a simple peep-hole, for to me the two
plane-surface mirrors and required optical filters are man-made optical
elements, even if no image magnification is involved. (Note that
depending on the size of the eye-piece aperture and distance of the eye
behind it,, you can easily get resolution limitation due to diffraction
image size increase at reduced effective 1/F number, and so not have the
nominal eye's 1 minute of arc best-resolution when observing contact of
the arcuate limb with a visible horizon. Such uncertainty is included in
full in the corrected altitude's final value.)
I would , however, suggest that if you use your sextant for this purpose,
set up a shallow container of water, or better oil. (Oil: motor lubricant,
or my preference vegetable cooking, has a higher refractive index than
water, hence stronger reflection at any given angle, and oils have better
viscous damping
