On Sat, 7 Jun 2003, John Carmichael wrote: > I know you wrote the list asking for answers, not questions. But I don't > understand the whole basis of the Britannica instructions. > > They say: > > A horizontal dial designed for Chicago's latitude radiates as a > > 42 deg ellipse. > > I've never heard of using an ellipse to construct a horizontal dial. Neither > Mayall or Waugh use this "ellipse method". And I've never heard of an > ellipse being described in terms of "degrees". What in the world is a 42 > deg ellipse? > > I'd love to see the drawings. Bet others would too. (I went to the > Britannica.com but couldn't view the article because I'm not a member, and > it wasn't in my printed Britannica).
I don't think I had ever seen this construction either, but it seemed logical, and not at all difficult, for a geometric method. The article was not in Britannica itself, but in a magazine article referenced from the on-line database: Tech Directions Dec 2002, Vol. 62 Issue 5, p 17 "It's High Time to Make Sundials!" by Mark Schwendau I would have liked to see the images too, but they weren't in the copy in the database. As Chris pointed out, maybe Schwendau used the co-latitide, and possibly, that's even what he should have used! I'll have to think about how this would work in the polar and equatorial limits... At any rate, the basic method is like this: Draw a circle, with radius 1.0. Draw another circle at the same center, with radius 1.3426 (supposedly = 1/sin(lat) ). From the center, draw radial lines at 15 degree intervals. These correspond to hour lines for a polar dial, with 1200 at the top. The rest of the construction determines the new lines, as projected on the tilted plane at the dial's latitude. Draw a horizontal line through the intersection of the 1100 and 1300 hour lines with the outer circle. Draw a vertical from the intersection of the 1100 line with the inner circle, extending it to intersect the horizontal line you just drew. That intersection is the new direction for 1100. Draw a final hour line from the dial center through it, and proceed to do the same for all other points you want on the dial. You will see that the set of points define an ellipse, which Schwendau describes as "a 42 degree ellipse". From a drafting perspective (no pun intended!), that makes some sense, if you think of looking at a circle on a tilted plane. Depending upon how that defining angle is measured, it means a 90 degree ellipse is either a circle (seen normal to the surface) or collapses to a line (seen in the plane of the circle.) This has to go back to the initial question, of latitude vs. co-latitude. At first glance, I think the author's words were correct, but his numbers wrong: 1/sin(lat) is undefined at latutude=0, where a horizontal dial fails. At latitude=90, the inner and outer circles are identical (1/1), where a horizontal dial is simply an equatorial dial. (The dial plate is parallel to the equator.) Maybe later I can put up a drawing on my Web cache of sundial odds and ends. ( http://www.AdvanceAssociates.com/Sundials ) Dave 37.28N 121.97W -
