Dear Frans, Roger, Karlheinz, Gianni et al, Braunschweig is fascinating and I have a new design to offer. See:
http://www.cl.cam.ac.uk/users/fhk1/BraunschweigV.pdf and I have an associated puzzle for anyone who enjoys spherical geometry. See below! First, some history... THE SIMPLE CASE Frans, citing Zinner, said: > The usual medieval sundial was vertical and > semi-circular, with 12 (equally spaced) > temporal hour lines. A perpendicular gnomon > was inserted into the center, where all lines > intersect. This kind of sundial has an orthogonal gnomon and a set of 13 lines (I include 0 and 12) at 15-degree intervals. Even on a direct south-facing wall this will not indicate temporary hours very accurately. Gianni has quantified the errors. AN IMPROVEMENT An obvious improvement is to vary the spacing so that the hour lines are correct at the equinoxes. The 1334 Braunschweig dial seems to be an example. Roger supplied the angles from the brochure "Die Sonnenuhren am Braunschweiger Dom". Gianni has another paper confirming that the lines on this dial closely follow this spacing (though he uses the date 1350). Roger is a little harsh: > The early dial at Braunschweig, perhaps 1334, > is similar to the early mass dials... These > represent a crude understanding of time... Though he acknowledges: > There is a moderate improvement at Braunschweig > as the angles ... represent the timelines at the > equinox... THE IDEAL Mathematically exact temporary hour lines are not straight so you cannot use a gnomon (a style which casts a *line* shadow). You have to use a nodus (which casts a *point* shadow). Running from the summer solstice point to the winter solstice point, the line for a given temporary hour forms a narrow S-shape. If you draw the straight line from the summer solstice point to the winter solstice point you divide the S as in a $-sign. Here is the... PUZZLE Show that the straight line from the summer solstice point to the winter solstice point passes through the equinoctial point but through no other point on the S-shape. A DIFFERENT IMPROVEMENT Noting that summer, equinoctial and winter points are collinear, another obvious step is to use the straight lines through these triplets as approximations to the true temporary hour lines. Unfortunately these straight lines do not intersect at a common point but we can proceed as follows: 1. Take a vertical direct south-facing wall. 2. Set up a nodus as just described. 3. Mark the points for the temporary hours along the winter and summer hyperbolas and along the equinoctial line. 4. Note that, for a given hour, the three points in a triplet are collinear. 5. Take a particular pair of straight lines: the line through the 3h triplet and the line through the 9h triplet. 6. Note that these two lines intersect on the meridian line. Call the point of intersection point G. 7. Construct a gnomon from the nodus to point G. At Braunschweig this gnomon would dip downwards 8.56 degrees. 8. Draw a fan of straight lines from G through each of the equinoctial hour points. 9. Label these lines 1 to 11. 10. The result is what you see in: http://www.cl.cam.ac.uk/users/fhk1/BraunschweigV.pdf The diagram includes a hollow circle showing the perpendicular point from the nodus to the dial. ERROR ANALYSIS The diagram shows the correct positions of the hour points at the different temporary hours at the three seasons. The greatest errors are at 1h and at 11h at the winter solstice. Here are the true temporary hour times at which the shadow falls on the hour lines labelled 6 to 11: Summer Equinoxes Winter 6.000 6.000 6.000 7.023 7.000 6.949 8.029 8.000 7.936 9.000 9.000 9.000 no sun 10.000 10.181 no sun 11.000 11.515 These hour lines are exactly right at the equinoxes and approximately right at any other time of year. They are, by design, correct at 3h, 6h and 9h at the solstices and at the equinoxes and very nearly correct at other declinations (because the S-shapes are so narrow). Note that 3h, 6h and 9h were important times. These are the times that a Praetor (or appointed deputy) would ring a bell or something. The shadow is over half an hour late at 11h in winter but otherwise the errors are small. Of course, half a winter temporary hour is not much more than 20 ordinary minutes so it isn't so bad as it seems. The advantage of using a gnomon over using a nodus is, as usual, that the shadow of the gnomon continues to indicate the time when the shadow of the nodus is off the dial. QUESTIONS This seems such an obvious design that there must be examples of it in existence. Where is there such an example and by what name is this variant known? FURTHER DEVELOPMENT Note that there is no requirement for the wall to be direct south-facing or even for it to be vertical. See: http://www.cl.cam.ac.uk/users/fhk1/BraunschweigVd.pdf This variant is also for the latitude of Braunschweig but for a wall declining 20 degrees East and reclining 10 degrees backwards). I again show the perpendicular point for the nodus and I have included a horizon line too. The diagram is itself supposed to be reclining so the horizon line is, correctly, above the perpendicular point. The design procedure still works because the 3h and 9h lines still intersect on the meridian line (the 6h line) even though that line is not vertical on a reclining dial... COROLLARY TO THE PUZZLE Show that the intersection of the straight lines through the 3h and 9h triplets falls on the meridian line even when the dial is declining and reclining. Many congratulations if you have reached this point. Even more congratulations if you have solved the puzzle and its corollary. Time to stop. Best wishes Frank --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
