Michael, I seem to recall that sec^2(x)=1+tan^2(x)
Therefore sec^2(lat).sec^2(dec)=(1+tan^2(lat)).(1+tan^2(dec)) =1+tan^2(lat)+tan^2(dec)+tan^2(lat).tan^2(dec) =(1+tan dec tan lat)^2 + (tan dec - tan lat)^2 I guess that this relationship, which is just a variant of sin^2+cos^2=1, should have been known to the dial designer. Geoff On 15 May 2017 at 16:32, Michael Ossipoff <email9648...@gmail.com> wrote: > Thanks for the Regiomontanus slide. > > Then the original designer of that dial must have just checked out the > result of that way of setting the bead, by doing the calculation to find > out if > squrt((1+tan dec tan lat)^2 + (tan dec - tan lat)^2)) = sec lat sec dec, > as a trial-and error trial that? > > Or, I don't know, is that a trigonometric fact that would be already known > to someone who is really experienced in trig? > > ----------------------- > > What's the purpose of the lower latitude scale, on the dial shown in that > slide? > > ------------------------ > > When I described my folded-cardboard portable equatorial-dial, I > mis-stated the declination arrangement: > > Actually, the sliding paper tab (made by making two slits in the bottom of > the tab, and fitting that onto an edge of the cardboard) is positioned via > date-markngs along that edge. The declination reading, and therefore the > azimuth, is correct when the shadow of a certain edge of the tab, > perpendicular to the cardboard edge on which it slides, just reaches the > hour-scale on the surface that's serving as a quarter of an Disk-Equatorial > dial. > > Actually, that dial was intended as an emergency backup at sea, where > there would always be available a horizon by which to vertically orient the > dial. > > The use of a plumb-bob for that purpose was my idea, because, on land > there often or usually isn't a visible horizon, due to houses, trees, etc. > Maybe, in really flat land, even without an ocean horizon, even a > land-horizon could be helpful, but such a horizon isn't usually visible in > most places on land. > > But then, with the plumb-line, it's necessary to keep the vertical surface > parallel to the pendulum-string, and keep the pendulum-string along the > right degree-mark, while making sure that the declination-reading is right, > when reading the time. > > ...Four things to keep track of at the same time. ...maybe making that > the most difficult-to-use portable dial. > > With the Equinoctical Ring-Dial, the vertical orientation, about both > horizontal axes, is automatically achieved by gravity, so only time and > declination need be read. > > And, with a pre-adjustable altitude-dial, only the sun-alignment shadow > and the time need to be read. > > With my compass tablet-dials, one mainly only had to watch the compass and > the time-reading. Of course it was necessary to hold the dial horizontal, > without a spirit-level, but that didn't keep them from being accurate. > > Michael Ossipoff > > > > On Sun, May 14, 2017 at 5:18 PM, Fred Sawyer <fwsaw...@gmail.com> wrote: > >> Michael, >> >> See the attached slide from my talk. All the various dials work with a >> string of this length. They vary simply in where the suspension point is >> placed. The pros and cons of the various suspension points were part of my >> presentation. >> >> Fred Sawyer >> >> >> On Sun, May 14, 2017 at 4:40 PM, Michael Ossipoff <email9648...@gmail.com >> > wrote: >> >>> 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 >>>>>> >>>>>> >>>>>> >>>>>> >>>>> >>>> >>> >> > > --------------------------------------------------- > https://lists.uni-koeln.de/mailman/listinfo/sundial > > >
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