When I mentioned (at least hypothetically) a mechanical method and an empirical method for constructing a Reclining-Declining Dial, they were intended as ways to assure someone that s/he knows of a way that such a dial can be constructed. (I feel that it's unsatisfying to own, use, or visit a sundial (or use a map-projection) whose construction-explanation you haven't heard.
...though of course the empirical method that I described is perfectly do-able without difficulty. But there's another empirical method that's a lot easier to build than the mechanical method, and a lot faster than the empirical method that I described. It combines attributes of both of them: On a flat plywood base, on the ground, build and fassten a Disk-Equatorial or a Band-Equatorial dial. Under that dial's gnomon, make and fasten a block or plate having a surface that's oriented to the base, and to the gnomon-north of the Equatorial, in the same manner that you want your Reclining-Declining Dial to have with the horizontal and north. Directly under the Equatorial's gnomon (as determined by a plumb-line or T-Square) contact a stick with the base. By Plumb-line or T-square, ensure that the entire stick is directly under the Equatorial's gnomon. By measuring the stick's length and it's high-end's height, you can make its angle to the base equal to that of the Equtorial's gnomon. So, when the stick is fastened in that orientation, it's parallel to the gnomon of the Equatorial. (...further checkable by vertical ruler-measurements between both ends of the stick and the Equatorial.) Now, out in the sunlight, rotate the base until the equatorial's gnomon-shadow point to one of its hour (or half-hour) lines. Then mark a line on the block or plate representing the Reclining-Declining surface. That's the same hour-line, for for the Reclining-Declining Dial. Do that for every one of the Equatorial's hour-lines. If necessary, of course, the base could be tipped upward at one end, with that end resting on a boulder, tree-stump, box, etc. ...or leaned on a house, or laid on an inclined-surface. The orientation of the base doesn't matter, as long as it's in the sunlight, and it's oriented so that the Equatorial Dial's gnomon-shadow is on an hour-line. Then, of course copy the hour-lines on your Reclining-Declining model to the actual Reclining-Declining Dial that you want to build. Michael Ossipoff On Wed, Jul 18, 2018 at 2:10 PM, Michael Ossipoff <email9648...@gmail.com> wrote: > Yes, the central-gnomon Equatorial dials, with their gnomon parallel to > the Earth's axis, and their circular measuring-scale parallel to plane of > the equator is educational, because its measurement of Solar Time is > completely direct. > > And yes, you're quite right: The construction of the Horizontal Dial is > easily described in terms of a central-gnomon equatorial (...say, a > Disk-Equatorial). Stand a disk-equatorial with its disk resting on the > ground and its gnomon (by choosing the right length for it) parallel to the > Earth's axis. Draw an east-west line through the point where the disk > contacts the ground. > > Extend the Disk-Equatorial's radial hour-lines (as "rays") to where they > meet that east-west line on the ground. > > Obviously, when the gnomon's shadow is along a radial hour-line of the > Disk-Equatorial, it will also go to the point where that ray meets the > ground. So draw a line from the ground-contacting end of the gnomon to that > point on the east-west line. > > And then you've constructed a Horizontal Dial. > > There's a widely-distributed graphical construction instruction that > models that construction. > > The formula: > > tan A = sin lat tan h > > ...comes directly from that construction. > > And yes, as I said in one of my recent posts here, any Vertical or > Reclining (but not Declining) flat dial can easily be shown to be a > horizontal dial for a different latitude....demonstrable with a globe. > > Of course the broad category that I described in the paragraph before this > one includes the Horizontal Dial, Disk Equatorial, and the Polar Dial as > special cases. > > So yes, all of what you said is true, but it's all surely been out there > for a long time. Dialing or dyalling has been studied and described for a > long time. > > I outlined a 5-day set of discussions to explain the construction of the > Reclining-Declining Dial. > > I'll just add that of course it's obvious that there are ways in which a > 3D working model of the 3 relevant co-ordinate-systems (Horizontal, > Equatorial, and Dial-Plate) could be made and used to construct a > Reclining-Declining Dial. I mention that to show that it's possible to > truly tell someone that they know of a way that such a dial could be made, > even if they haven't heard the 5-day explanation that I suggested. > > Or, as someone (but probably more than one person) else has suggested one > could also start with a Relining-Declining surface, and experimentally, > with a plumb-line, and a compass, north-star or pre-made landmark, align a > stick (in contact with the surface) so that it's 1) pointing northward, and > 2) elevated above the horizontal by an angle equal to your latitude. > ...and, from that, build the gnomon. > > ...and then, using, as reference-dial, any one of the Horizontal, > Reclining or Vertical (or Equatorial or Polar) dials described above, hour > lines could be drawn on the reclining declining surface where the > style-shadow is, when the reference dial says that it's a certain time. > > That might sound like cheating, but it's a legitimate way that such a dial > could be constructed, and for anyone who doesn't want to hear the 5-day > explanation, it's way that you could remind someone that they could make > such a dial. > > Michael Ossipoff > > > > > > On Tue, Jul 17, 2018 at 7:13 PM, rodwall1...@gmail.com < > rodwall1...@gmail.com> wrote: > >> Hi all, >> >> The following is what I think is the best way to describe how sundials >> work to kids or anyone. >> >> >> 1st start with the largest sundial in the world. Planet Earth the Master >> sundial clock. Stick a vertical stick into the ground at the South pole or >> North pole. And describe how the shadow shows the time throughout the day. >> Draw the 24 hour lines every 15 deg and that 15 deg x 24 hours = 360 deg >> one day. >> >> Then show how the Equatorial sundial relates to our stick sundial at the >> poles. Place it at the South or North pole. And show that the Equatorial >> sundial style edge is parallel with the stick and the axis of the Earth. >> And that the hour lines are the same every 15 deg. And that it will keep >> the same time. >> >> Then place the Equatorial sundial anywhere on earth. And show that the >> sundial is geared to Earth the largest sundial in the world (the Master >> sundial clock). Therefore it will keep the same time. Show how the sundial >> time markings relate to your local time. And that the style edge of the >> sundial must be parallel with the axis of the earth and parallel with the >> vertical stick at the poles. And that at night time the sundial is in the >> shadow of Earth. >> >> Then place a horizontal sundial at the same location. And describe that >> the style edge is also placed parallel with the stick and the axis of the >> Earth. And how the hour lines are projected every 15 deg from the >> horizontal sundial style. That is to place the style edge of the equatorial >> sundial onto the horizontal sundial style edge and use it to project the >> hour points onto the horizontal base of the horizontal sundial. Then draw >> the hour lines on the horizontal sundial. Then show the direction the >> horizontal sundial faces if in the Southern hemisphere or Northern >> hemisphere. To ensure that the Style edge is parallel with the axis of the >> Earth and stick. >> >> Then go through the same process as you did for the horizontal sundial >> but with a vertical sundial on a wall. And show that it must face North >> towards the sun if in the southern hemisphere and face South for the >> northern hemisphere. Yes I live in sunny Australia in the southern >> hemisphere. >> >> Then show how the angle of the style edge relates to the latitude of the >> location of the sundial. To make the style edge parallel with the stick and >> axis of the Earth. >> >> Depending on how far you want to go. Describe how a horizontal sundial >> that is not designed for the latitude of the location you are at. Can be >> corrected if a block is placed under the dial to make the style edge >> parallel with the axis of the Earth and the stick. >> >> Describe how longitude relates to your location. And that the longitude >> time zones are every 15 deg. 15 deg x 24 hours = 360 deg one day. >> >> One way to describe the above is to use a globe of the world or a large >> beach ball. With cardboard cutouts for sundials. And a lamp (sun) to make >> the shadows. Young kids and some adults learn better when learning in the >> sand pit (concrete learning) rather that just using symbolic words >> (symbolic learning). >> >> That is how I explain it. >> >> If anyone wants to publish the above. Please do and let them know where >> it came from. >> >> Have fun, >> >> Roderick Wall >> Sunny Australia. >> >> >> > On Wed, Jul 18, 2018 at 2:10 PM, Michael Ossipoff <email9648...@gmail.com> wrote: > Yes, the central-gnomon Equatorial dials, with their gnomon parallel to > the Earth's axis, and their circular measuring-scale parallel to plane of > the equator is educational, because its measurement of Solar Time is > completely direct. > > And yes, you're quite right: The construction of the Horizontal Dial is > easily described in terms of a central-gnomon equatorial (...say, a > Disk-Equatorial). Stand a disk-equatorial with its disk resting on the > ground and its gnomon (by choosing the right length for it) parallel to the > Earth's axis. Draw an east-west line through the point where the disk > contacts the ground. > > Extend the Disk-Equatorial's radial hour-lines (as "rays") to where they > meet that east-west line on the ground. > > Obviously, when the gnomon's shadow is along a radial hour-line of the > Disk-Equatorial, it will also go to the point where that ray meets the > ground. So draw a line from the ground-contacting end of the gnomon to that > point on the east-west line. > > And then you've constructed a Horizontal Dial. > > There's a widely-distributed graphical construction instruction that > models that construction. > > The formula: > > tan A = sin lat tan h > > ...comes directly from that construction. > > And yes, as I said in one of my recent posts here, any Vertical or > Reclining (but not Declining) flat dial can easily be shown to be a > horizontal dial for a different latitude....demonstrable with a globe. > > Of course the broad category that I described in the paragraph before this > one includes the Horizontal Dial, Disk Equatorial, and the Polar Dial as > special cases. > > So yes, all of what you said is true, but it's all surely been out there > for a long time. Dialing or dyalling has been studied and described for a > long time. > > I outlined a 5-day set of discussions to explain the construction of the > Reclining-Declining Dial. > > I'll just add that of course it's obvious that there are ways in which a > 3D working model of the 3 relevant co-ordinate-systems (Horizontal, > Equatorial, and Dial-Plate) could be made and used to construct a > Reclining-Declining Dial. I mention that to show that it's possible to > truly tell someone that they know of a way that such a dial could be made, > even if they haven't heard the 5-day explanation that I suggested. > > Or, as someone (but probably more than one person) else has suggested one > could also start with a Relining-Declining surface, and experimentally, > with a plumb-line, and a compass, north-star or pre-made landmark, align a > stick (in contact with the surface) so that it's 1) pointing northward, and > 2) elevated above the horizontal by an angle equal to your latitude. > ...and, from that, build the gnomon. > > ...and then, using, as reference-dial, any one of the Horizontal, > Reclining or Vertical (or Equatorial or Polar) dials described above, hour > lines could be drawn on the reclining declining surface where the > style-shadow is, when the reference dial says that it's a certain time. > > That might sound like cheating, but it's a legitimate way that such a dial > could be constructed, and for anyone who doesn't want to hear the 5-day > explanation, it's way that you could remind someone that they could make > such a dial. > > Michael Ossipoff > > > > > > On Tue, Jul 17, 2018 at 7:13 PM, rodwall1...@gmail.com < > rodwall1...@gmail.com> wrote: > >> Hi all, >> >> The following is what I think is the best way to describe how sundials >> work to kids or anyone. >> >> >> 1st start with the largest sundial in the world. Planet Earth the Master >> sundial clock. Stick a vertical stick into the ground at the South pole or >> North pole. And describe how the shadow shows the time throughout the day. >> Draw the 24 hour lines every 15 deg and that 15 deg x 24 hours = 360 deg >> one day. >> >> Then show how the Equatorial sundial relates to our stick sundial at the >> poles. Place it at the South or North pole. And show that the Equatorial >> sundial style edge is parallel with the stick and the axis of the Earth. >> And that the hour lines are the same every 15 deg. And that it will keep >> the same time. >> >> Then place the Equatorial sundial anywhere on earth. And show that the >> sundial is geared to Earth the largest sundial in the world (the Master >> sundial clock). Therefore it will keep the same time. Show how the sundial >> time markings relate to your local time. And that the style edge of the >> sundial must be parallel with the axis of the earth and parallel with the >> vertical stick at the poles. And that at night time the sundial is in the >> shadow of Earth. >> >> Then place a horizontal sundial at the same location. And describe that >> the style edge is also placed parallel with the stick and the axis of the >> Earth. And how the hour lines are projected every 15 deg from the >> horizontal sundial style. That is to place the style edge of the equatorial >> sundial onto the horizontal sundial style edge and use it to project the >> hour points onto the horizontal base of the horizontal sundial. Then draw >> the hour lines on the horizontal sundial. Then show the direction the >> horizontal sundial faces if in the Southern hemisphere or Northern >> hemisphere. To ensure that the Style edge is parallel with the axis of the >> Earth and stick. >> >> Then go through the same process as you did for the horizontal sundial >> but with a vertical sundial on a wall. And show that it must face North >> towards the sun if in the southern hemisphere and face South for the >> northern hemisphere. Yes I live in sunny Australia in the southern >> hemisphere. >> >> Then show how the angle of the style edge relates to the latitude of the >> location of the sundial. To make the style edge parallel with the stick and >> axis of the Earth. >> >> Depending on how far you want to go. Describe how a horizontal sundial >> that is not designed for the latitude of the location you are at. Can be >> corrected if a block is placed under the dial to make the style edge >> parallel with the axis of the Earth and the stick. >> >> Describe how longitude relates to your location. And that the longitude >> time zones are every 15 deg. 15 deg x 24 hours = 360 deg one day. >> >> One way to describe the above is to use a globe of the world or a large >> beach ball. With cardboard cutouts for sundials. And a lamp (sun) to make >> the shadows. Young kids and some adults learn better when learning in the >> sand pit (concrete learning) rather that just using symbolic words >> (symbolic learning). >> >> That is how I explain it. >> >> If anyone wants to publish the above. Please do and let them know where >> it came from. >> >> Have fun, >> >> Roderick Wall >> Sunny Australia. >> >> >> >
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