I haven't seen the original article, but you seem to be right. Chicago is 41 degrees 53' north. The ellipse should indeed have axes in the ratio 1:sin(41deg 53'). The Britannica seems to have taken the co-latitude instead. That would be 48.1 degrees. Perhaps the author was British and more familiar with latitudes in the fifties.
Chris Lusby Taylor 51.4N 1.3W ----- Original Message ----- From: "Dave Bell" <[EMAIL PROTECTED]> To: <[email protected]> Sent: Saturday, June 07, 2003 2:04 AM Subject: Dial calculation mystery > I was showing one of my daughters some of the research resources on-line > at our local county library system. Pulled up Britannica, and entered > Sundial (of course.) There were a number of useful articles, and I > followed through to an article on laying-out a dial. The author chose a > geometric layout method, to make a horizontal dial for Chicago, IL. So > far, so good. I followed the details, and played with it a bit, before > realizing there was what looked like a glaring error. Or is it me?? > > ======================================================== > Gnomon Construction > > Sundial construction depends upon principles of constructive, or gnomonic, > geometry. A horizontal dial designed for Chicago's latitude radiates as a > 42 deg ellipse. > > For this example we use a 42 deg ellipse to determine the hour lines' > radiation on a horizontal dial at 42 deg latitude. To construct such an > ellipse, we need concern ourselves only with the relative lengths of the > ellipse's major and minor axes. Since we require only the radials and not > the ellipse's perimeter, exact size is unnecessary. > > One way to constructively deduce these lengths is by drawing a gnomonic > triangle for the selected latitude (Fig. 2). Construct right triangle ABC > such that < ABC equals the desired ellipse for a given latitude, in this > case, 42 deg. > > Where (X) represents the dial face ellipse's minor axis, (X) is used to > determine the length of a 42 deg ellipse's major axis. Those with a > trigonometry background and a scientific calculator can also use the > formula (1/sin 42) to derive the exact proportions. > > > Dial Plate Construction > > The gnomonic triangle holds all the information you need to draw a sundial > plate and its hour lines by the elliptical coordinate method. See Fig. 3. > > 1. Using the data from the previous gnomon example, draw two concentric > circles such that the inner circle has a radius of 1 and the outer circle > has a radius 1.3426. > =========================================================== > > Since I don't live at Chicago's latitude, I naturally went to compute the > ratio I would need here, in not-so-northern California. That's when I > discovered that 1/sin(42) is more like 1.49+, and asin(1/1.3426) = 48.1. > > Did the author just screw up, or am I missing something? > > Dave > 37.28N 121.97W > > - > -
