Wow. What can I say. Your approach makes more sense in every way, than the way that I'd been trying to find how the bead-setting procedure could have been arrived at.
I'd wanted to start with various pairs of points, and then find out if any of them are separated by a distance of sec lat sec dec. But of course (now it's obvious) it makes a lot more sense to start with sec lat sec dec, and find out if it can be made into a distance. ...which is of course how you approached the problem. If we expect the distance on the dial to be a diagonal distance, then it will be the sum of two squares, all in a square-root sign. Most likely it will be a diagonal distance, which means the it will be the sum of two squares, all in a square-root sign. Of course it _needn't_ be a diagonal distance. It could be all horizontal or all vertical. But, a lot of distances already on the dial are expressed as tangents, and more could be. So, converting the sec to tan makes sense, for a start. The familiar identity that relates sec and tan involves their squares. That suggests a diagonal distance, adding some confirmation to the initial impression that a diagonal distance might be more likely. So (tan lat + 1) and (tan dec + 1) are multiplied together, resulting in four terms, each of which is a square (...though of course the 1 needn't snecessrily have been gotten by squaring--except that it wasn't gotten by multiplying other numbers. So maybe it should be considered a square). The fact that there are four squares suggests that the two squared expressions are both binomials. ...and that the squares' middle terms cancel eachother out. (Of course maybe the inventor didn't have a way to be sure that sec lat sec dec can be written as a distance on the dial at all. But, if not, he evidently checked out the possibility.) So, if the four squares are the squares of the terms of two binomials, with their middle terms canceling out, there are 3 ways in which the two binomials could be assembled from the square roots of those four squares. In a way, it doesn't matter which way it's done, as long as it results in a distance. But, for that diagonal distance, of course it's necessary that the two squared binomials reapresnt distances in mutually perpendicular directions. Well there's an obvious distance there, among the square-roots of those terms: tan lat tan dec. It's horizontal, and is the distance of the string-hang-point forward (sunward) from the middle vertical. And if the 1 is added to it, that's the horizontal distance of the string-hang-point from the rear edge of the dial-card. Of the square-roots of the other two terms, tan lat is the vertical distance of the string-hang point above the main horizontal, the first horizontal. So that works--a horizontal distance and a vertical distance, which are needed for a diagonal distance. And of course naturally (tan lat - tan dec) would be a vertical distance from the string-hang-point, to the upper-end of a line has been drawn across half the dial-card's 2-unit width, one end on the first horizontal, with the line angled up by the declination-angle. Since the horizontal distance suggested was from the string-hang-point to the rear edge of the dial, and because the string-hang point is tan lat above the first horizontal, then that suggests that the measurement should be from the string-hang point, to a point that is tan dec above the first horizontal, on the rear margin of the dial. ...leading to the Regiomontanus's way of setting the bead. And, in that way, that beat-setting method is naturally arrived at. So thanks for pointing out that natural approach, making choices than make more sense than the approach I was considering. Michael Ossipoff On Mon, May 15, 2017 at 2:54 PM, Geoff Thurston <thurs...@hornbeams.com> wrote: > 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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