Dear Frank, Try as I might, I can find no published information on where people judge the edge of a shadow to be. It would certainly be an interesting experiment. As John Davis observed, the human eye is very non-linear. Experience with photography seems to suggest that, like the ear, it is logarithmic. I imagine one's judgement of a shadow would to some extent depend on how clear the sky was and might well vary from person to person. For instance, someone well experienced with the phenomenon might have trained themselves to judge the 50% point more accurately. My own informal experiments with a shadow sharpener suggest that I tend to judge the edge at about 10% of full illumination, which corresponds to about 70% of the 64 seconds passed, or 45 seconds, but it may depend on a lot of factors including the colour and optical qualities of the surface - white matt paper v polished brass, for instance.
One point you make that I don't understand is that you expect different results going from dark-to-light than light-to-dark. Why should that be? Shadows move so slowly that I'd imagine one would err towards the darker side consistently regardless of whether the dark area is shrinking or growing. Regards Chris ----- Original Message ----- From: "Frank King" <[EMAIL PROTECTED]> To: "Chris Lusby Taylor" <[EMAIL PROTECTED]> Cc: <sundial@uni-koeln.de>; <[EMAIL PROTECTED]> Sent: Sunday, February 17, 2008 6:22 PM Subject: Re: Monumental Sundial; 14 missing seconds > Dear Chris, > > I am a bit behind with my reading and I have only > just read your comments, and corrected comments, > on the umbra discussion. > > Subject to your corrections, I concur with almost > all you say but I feel a little amplification of > one of your follow-up remarks is needed. You > say... > > > If the gnomon is ... a conventional wedge-shaped > > gnomon with two style edges, then you can adjust > > the hour lines, by a little under a minute. > > I am very nearly happy so far! > > > This adjustment is definitive, in the sense that > > it is irrespective of the time of day, time of year, > > size of sundial, type of sundial (horizontal or > > vertical) or latitude. > > Yes, I am still happy [subject to your qualification > later that "the sun moves faster across the sky at > the equinoxes than at the solstices"]. > > My need for amplification is confined to the > 50 seconds figure here... > > > It is an angular change (about 50 seconds of time), > > not a linear change. > > It is indeed an angular change; it's the 50 seconds > figure which I feel needs a bit of amplification... > > At the equinoxes the sun obligingly trundles along > the celestial equator (well close to it) at a rate > of 1 degree every 4 minutes of time or 1 arc-minute > every 4 seconds of time. > > Consider a bug, sitting on the 2pm hour-line (say), > looking at the sun (via special eye protection) as it > approaches the business edge of a fat wedge gnomon. > > There will be a period lasting just over two minutes > (to be justified below) of significant interest: > > 1. Since sunrise the bug has been able to see the > entire solar disc and has been basking in full > sunlight. Then, about 13:59 sun time, the sun > makes first contact with the edge of the gnomon > and, soon afterwards, the bug notices a drop in > light. > > 2. The solar disc steadily slips behind the edge > and, about 14:01 sun time, we have second contact. > Thereafter, the sun is wholly behind the edge and > the bug is in maximum shadow. > > Taking the angular diameter of the sun to be 32' the > time taken from first contact to last contact will > be 128 seconds. This, measured in time, is the full > width of the umbra. > > The time from 14:00 to second contact is 64 seconds > and I pondered how to reduce this to 50 seconds... > > [Aside: of course I agree that you divide by the > cosine of the declination if you are not at an > equinox and the angular diameter varies a little > from 32'. These, as you say, are minor matters > though the first INCREASES the 64 seconds.] > > You later say, and again I concur, that "most people > seem to judge the edge very close to the dark side." > > The key words here are "very close". > > To reduce the true 64 seconds to the apparent > 50 seconds you seem, implicitly, to be saying > that "very close" translates into 14 seconds of > time, or 3.5 arc-minutes, about 10% of the solar > diameter. > > This feels about right but I should love to see > the results of some properly set-up experiments. > > I can imagine that the results would be different > for going from shadow to light (morning times) > from going from light to shadow (afternoon times). > > Do you have some mathematical justification that > has escaped me for the missing 14 seconds or is > this just a sensible estimate? > > Best wishes > > Frank > --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
