I've heard that dialists traditionally disregard atmospheric-refraction, when calculating sunrise an sunset times. That allows the use of spherical-trigonometry's tangent-formula, instead of the altitude-formula, a co-ordinate-transformation.
But the orrery derivation of the altitude-formula seems just as easy as the derivation of spherical-trigonometry's tangent-formula. In fact, the orrery-derivations of the alt and az formulas seem, to me, easier. ...even though those formulas are larger than the tangent-formula. The tangent formula, being briefer, involves less arithmetic, but the orrery derivation of the alt and az formulas seem more naturally and easily explained. ------------------------ By the way, though I'd explain declination-line construction by the altitude-method, there might be advantage in calculating it by the trig-at-the-dial method. For one thing, the alt & az formulas can have subtraction, which can cause a loss of significant digits (which would only rarely matter, with today's many-digits machines). Also, if you want the measurement to be straightforward, instead of looking for the point on the hour line that's the right distance from the sub-nodus point, which isn't on the hour-lline, then you'd need to calculate the solar altitude and azimuth both. That, and the conversion to rectangular co-ordinates, and then a little work with those co-ordinates, probably amounts to a bit more arithmetic than the trig-at-the-dial method. 48 Tu Novembeer 19th 1524 UTC Michael Ossipoff
--------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
