When I said that there isn't an obvious way to measure to make the plumb-line length equal to sec lat sec dec, I meant that there' s no obvious way to achieve that *with one measurement*.
I was looking for a way to do it with one measurement, because that's how the use-instructions say to do it. In fact, not only is it evidently done with one measurement, but that one measurement has the upper end of the plumb-line already fixed to the point from which it's going to be used, at the intersection of the appropriate latitude-line and declination-line. That's fortuitous, that it can be done like that, with one measurement, and using only one positioning of the top end of the plumb-line. But of course it's easier, (to find) and there's an obviously and naturally-motivated way to do it, with *two* measurements, before fixing the top-end of the plumb-line at the point where it will be used. The line from that right-edge point (from which the first horizontal is drawn) to the point where the appropriate latitude-line intersects the vertical has a length of sec lat. So, before fixing the top end of the plumb-line where it will be used from, at the intersection of the appropriate lat and dec lines, just place the top end of the plumb line at one end of that line mentioned in the paragraph before this one, and slide the bead to the other end of that line. ...to get a length of thread equal to sec lat. Then, have a set of declination marks at the right edge, just like the ones that are actually on a Regiomontanus dial, except that the lines from the intersection of the first horizontal and vertical lines, to the declination (date) marks at the right-margins are shown. Oh, but have that system of lines drawn a bit larger, so that the origin of the declination-lines to the right margin is a bit farther to the left from the intersection of the first horizontal and the first vertical. ...but still on a leftward extension of the first horizontal. That's so that there will be room for the 2nd measurement, the measurement that follows. And have closely spaced vertical lines through those diagonal declination-lines to the right margin. So now lay the thread-length that you've measured above, along the first horizontal, with one end at the origin of the declination-lines to the margin. Note how far the thread reaches, among the closely-spaced vertical lines through those margin declination-lines. Now measure, from the origin of the margin declination-lines along the appropriate margin declination-line, to that one of the closely-spaced vertical lines that the thread reached in the previous paragraph. With the left end of the thread at the origin of the margin declination-lines, slide the bead along the thread to that vertical line. That will give a thread length, from end to bead, of sec lat sec dec. ...achieved in the easy (to find) way, by two measurements, before fixing the thread (plumb-line) end to the point from which it will be used. I wanted to mention that way of achieving that end-to-bead thread-length, to show that it can be easily done, and doesn't depend on the fortuitous way that's possible and used by the actual Regiomontanus dial, whereby only one thread-length measurement is needed, and the only positioning of the thread-end is at the point from which it will be used. Having said that, I suppose it would be natural for someone to look for a fortuitous way that has the advantages mentioned in the paragraph before this one. And I suppose it would be natural to start the trial-and-error search from the thread-end position where the thread will eventually be used, to have the advantage of only one thread-end positioning. One would write formulas for the distance of that point to various other points, with those distances expressed in terms of sec lat and sec dec (because sec lat sec dec is the sought thread-length). And I suppose it would be natural to start that trial-and-error search by calculating the distance from there to the right-margin end of the first horizontal, and points on the right margin...because that's still an empty part of the dial card. And, if you started with that, you'd find the fortuitous method that the actual Regiomontanus dial uses, to achieve the desired end-to-bead thread-length. (But, if that didn't do it, of course you might next try other distances. And if you didn't find a one-measurement way to do it (and can't say that you'd expect to), then of course you could just use the naturally and obviously motivated 2-measurement method that I described above). The distance calculations needed, to look for that fortuitous, easier-to-do (but not to find) one-measurement method are relatively big calculations with longer equations with more terms. ---------------------- By the way, I earlier mentioned that I'd verified for myself, by analytic geometry, that the Regiomontanus dial agrees with the formula that relates time, altitude, declination and latitude. That involved big (maybe page-filling, it seems to me) equations with lots of terms. When a proposition is proved in that way, that proof shows that the proposition is true, but it doesn't satisfyingly show why it's true, what makes it true. The naturally and obviously motivated construction that I've described here is much better in that regard. The only part that gets elaborately-calculated is the finding of that fortuitous, easy to do (but not easy to find) way to get the right thread-length with only one measurement, when the thread-end is already positioned for use. But, as I mentioned, the desired end-to-bead thread-length can be easily achieved by the obviously and naturally-motivated two-measurement method that I described above. Michael Ossipoff On Sat, May 13, 2017 at 9:23 PM, Michael Ossipoff <email9648...@gmail.com> wrote: > 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 >>> >>> >>> >>> >> >
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