Dear Frank et al, After a second night sleeping on your nice puzzle I realised that I DID make a small goof in one of my assertions and no one has picked me up on it!!
In the formula: tan(dec) = [-]sin(d)/sqrt(t1^2 - 2.t1.t2.cos(d) + t2^2) I asserted (correctly) that the argument of the square root function couldn't be negative. Dave Bell kindly confirmed this. I (also correctly) asserted that the argument could be zero. I then (incorrectly) asserted that when the argument is zero, the associated declination would be 90 degrees. My (false) reasoning was that with zero as the value of the square root, the right-hand side would have a zero denominator and hence an infinite value overall. In fact, for the argument of the square root function to be zero we require... Either t1 = t2 and d = 0 or t1 = -t2 and d = 180 In both cases the numerator, sin(d), is also zero and we have nought-over-nought which is well known to give bad vibrations to mathematicians! These two sets of requirements correspond to: Either two places at the same latitude and with zero separation of longitude (meaning that the two places are coincident). or two places of opposite latitude and 180 degrees of longitude apart (meaning the two places are diametrically opposite on the Earth's surface). In both sets of circumstances there is no unique great circle through the pairs of points and, in consequence, no single associated solar declination. The second case is interesting in demonstrating something fairly obvious but perhaps not widely known: If, standing in a particular place, you note the instant of sunset (or sunrise) you can ponder the thought that someone at the other end of the Earth's diameter from you is simultaneously experiencing sunrise (or sunset). This is independent of where you are or the time of year. One of the many nice features of this puzzle is that there is no need to know the time of sunset (or sunrise) and, in consequence, you can lay any understanding of hour-angles on one side. Hey, isn't that just what you wanted to know! Best wishes Frank --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial