Of course, because only the four squared-terms are present, the two binomials have to be chosen so that, when they're both squared, their resulting middle terms cancel eachother out. (tan lat tan dec + 1) and (tan lat - tan dec) meet that requirement.
Michael Ossipoff On Mon, May 15, 2017 at 9:25 PM, Michael Ossipoff <email9648...@gmail.com> wrote: > 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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