When I said that the vertical hour-lines should be drawn at distance, to the left, from the middle vertical line, that is proportional to the cosine of the hour-angle...
I should say *equal to* the cosine of the hour-angle, instead of proportional to it. ...where the length of the first horizontal line, from the right edge to the point where the vertical line is drawn, is one unit. Michael Ossipoff On Sat, May 13, 2017 at 7:03 PM, Michael Ossipoff <email9648...@gmail.com> wrote: > Fred-- > > Thanks for your answer. I'll look for Fuller's article. > > One or twice, I verified for myself, by analytic geometry, that the > Universal Capuchin Dial agrees with the formula that relates altitude, > time, declination and latitude. > > But that wasn't satisfying. Verifying a construction isn't the same as > finding one. Without knowing in advance what the construction and use > instructions are, I don't know of a way to design such a dial. > > ...or how the medieval astronomers and dialists arrived at it. > > But there's an exasperatingly tantalizing approach that gets partway. > ...based on the formula for time in terms of altitude, latitude and > declination: > > cos h = (sin alt - sin lat sin dec)/(cos dec cos lat) > > Dividing each term of the numerator by the denominator: > > cos h = sin alt/(cos dec cos lat) - tan lat tan dec > > If, in the drawing of the dial, the sun is toward the right, and you tip > the device upward on the right side to point it at the sun, then the > plum-line swings to the left, and the distance that the plum-bob moves to > the left is the length of the thread (L) times sin alt. > > So that seems to account for the sin alt, at least tentatively. > > Constructing the dial, if you draw a horizontal line in from a point on > the right-hand, side a distance L equal to the length of that thread, then > draw a vertical line there, and then, from that side-point, draw lines > angled upward by various amounts of latitude, then each line will meet the > vertical line a distance of L tan lat, up from the first (horizonal) line. > > So the distance from the horizontal line, up the vertical line to a > particular latitude-mark is L tan lat. > > At each latitude-mark, make a horizontal line. > > From the bottom of that vertical line, where it meets the horizontal line, > draw lines angled to the right from the vertical line by various amounts of > declination. Draw them up through all the horizontal lines. > > Because a latitude-line is L tan lat above the original bottom horizontal > line, then the distance to the right of the vertical line, at which one of > the declination-lines meets that latitude-line is L tan lat tan dec. > > That's where we fix the upper end of the plumb-line. Then, when we tip the > instrument up on the right, to point at the sun, and the plumb-bob swings, > its distance to the left of the middle will be: sin alt - tan lat tan dec. > > That's starting to look like the formula. > > Maybe it would be simpler to just say that L is equal to 1. > > But we want sin alt/(cos lat cos dec). > > The instructions for using the Universal Capuchin dial talk about > adjusting the distance of the bead from the top of the string before using > the dial, and that's got to be how you change sin alt to sin alt/(cos lat > cos dec). > > I guess I could study how that's done, by reading the construction and use > instructions again. > > I guess you'd want to make the plumb-line's length equal to sec lat sec > dec instead of 1. ...and there must be some way to achieve that by > adjusting the bead by some constructed figure, as described in the > use-instructions. > > But it isn't obvious to me how that would be done--especially if that > bead-adjustment is to be done after fixing the top-end of the plumb-line in > position. > > Maybe it would be easier if the bead-adjustment is done before fixing the > top end of the plumb-line, so that you know where you'll be measuring from. > I don't know. > > And then there's the matter of cos h. > > Just looking at afternoon... > > Because positive h is measured to the right from the > meridian--afternoon---and because, the later the afternoon hour, the lower > the sun is--then, in the afternoon, it seems to make sense for a larger > bead-swing to the left to represent an earlier hour...an hour angle with a > larger cosine. > > I guess, for afternoon, the vertical hour lines are positioned to the left > of middle by distance proportional to the cosine of the hour-angle. > > ------------- > > So, this isn't an explanation, but just a possible suggestion of the start > of an explanation. > > Maybe it can become an explanation. > > But I still have no idea how an orthographic projection leads to the > construction of the Universal Capuchin dial. > > (If a Capuchin dial isn't universal, it loses a big advantage over the > Shepard's dial, or the related Roman Flat altitude dial.) > > Michael Ossipoff > > > > > > > On Sat, May 13, 2017 at 3:47 PM, Fred Sawyer <fwsaw...@gmail.com> wrote: > >> Take a look at A.W. Fuller's article Universal Rectilinear Dials in the >> 1957 Mathematical Gazette. He says: >> >> "I have repeatedly tried to evolve an explanation of some way in which >> dials of this kind may have been invented. Only recently have I been >> satisfied with my results." >> >> The rest of the article is dedicated to developing his idea. >> >> Note that it's only speculation - he can't point to any actual historical >> proof. That's the problem with this whole endeavor; there is no known >> early proof for this form of dial - either in universal or specific form. >> (It seems that the universal form probably came first.) >> >> It was published in 1474 by Regiomontanus without proof. He does not >> claim it as his own invention and in fact refers to an earlier unidentified >> writer. There has been speculation that he got it from Islamic scholars - >> but nothing has been found in Islamic research that would qualify as a >> precursor. The dial is somewhat similar to the navicula that may have >> originated in England - but that dial is only an approximation to correct >> time. >> >> In discussing this history, Delambre says: >> >> "All the authors who have spoken of the universal analemma, such as >> Munster, Oronce Fine, several others and even Clavius, who demonstrates all >> at great length, contented themselves with giving the description of it >> without descending, as Ozanam says, to the level of demonstration." >> >> "At this one need not be surprised, seeing that it rests on very hidden >> principles of a very profound theory, such that it seems that it was >> reserved to [Claude Dechalles] to be able to penetrate the obscurity." >> >> So Dechalles gave what was evidently the first proof in 1674 - 200 years >> after Regiomontanus' publication. But as Delambre further notes: >> >> Dechalles’ proof … is long, painful and indirect, … without shedding the >> least light on the way by which one could be led to [the dial’s] origin. >> >> So - pick whichever proof makes sense for you. >> >> Fred Sawyer >> >> >> >> >
--------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
