Dear Geoff, You get full marks and go to the top of the class!!
Your analysis almost exactly parallels what I had in mind and your description is delightfully eloquent! SOLUTION TO THE PUZZLE The S-shaped curves in your attachments are very convincing. It seems not to be well known that temporary-hour curves DO converge on a point. Perhaps this is because the point corresponds to a declination equal to the co-latitude. Your key sentence (against which an examiner would write a huge tick!) is: > The thing to note about these curves is that > they are symmetrical about the eqinoctial > point for any pair of +/- declinations... Just so, and all because the amount of extra daylight we get at a given positive declination exactly matches the amount of extra darkness we get at the negative equivalent of that declination. You continue... > any great circle drawn through an eqinoctial point > and the corresponding summer solstitial point must > pass through the winter solstitial point as well. ... and given that any great circle... > will project into a straight line on any plane > surface... ... you duly solve the puzzle! You are right about the symmetry of the corresponding afternoon time so... > ... any corresponding pair of hour lines will if > extended meet on the meridian. Yes indeed and I simply choose the 3h-9h pair. > I hope this reasoning is sound... Yes, absolutely perfect! THE COROLLARY You don't explicitly attend to the corollary but that can be taken as obvious. Given that a gnomonic projection onto a plane guarantees that great circles project into straight lines, the orientation is irrelevant... Accordingly, my declining reclining dial still shows the 3h and 9h straight-line approximations intersecting on the (now non-vertical) 6h line. YOUR PARTING THOUGHT > One thought occurs to me when looking at the > s-curves. The line joining the solstitial points > gives a zero mean error over the year but it > should be possible to find a better overall fit > by joining for example +/-15 degrees by a straight > line. What do you think? It depends what your goal is. A naive best fit is not necessarily what you want... Over the course of a year, the solar declination spends a long time hovering around the solstitial extremities but fair whistles across the equinoxes! If you want to keep to a minimum, the mean absolute error in indicated time over a half year (solstice to solstice), then you have to integrate over time and not declination. This would bias your pairing towards +/- 23.5 degrees (though, I suspect, not quite the whole way). MY PARTING THOUGHT My very limited understanding of sundial history suggests that the vast majority of sundials prior to 1400 or 1500 were intended to indicate temporary hours and most of these did a pretty poor job! It is a pity that, now we could do this job rather well, the market for our efforts is close to null! Maybe we should teach about Terce, Sext and None in schools!! Best wishes Frank [I am just off to Italy so there will be a welcome period of silence for a few days!] --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
