Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

On Sun, May 14, 2017 at 5:18 PM, Fred Sawyer  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.
>

What were some of the alternatives, and some of their relative advantages?

Michael Ossipoff

>
>
> Fred Sawyer
>
>
> On Sun, May 14, 2017 at 4:40 PM, Michael Ossipoff 
> 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

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

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 
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 
> 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  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

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

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 
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  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:
>>
>> 

Re: Capuchin and Regiomontanus dials

[Geoff Thurston]

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  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  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 > > 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 

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

I asked:

"Or, I don't know, is that a trigonometric fact that would be already known
to someone who is really experienced in trig?"

Well, alternative expressions for the product of two cosines is something
that might be basic and frequently-occurring enough to be written down
somewhere, where someone could look it up.

Maybe someone who'd thoroughly studied trig, and done a lot of it, would
know it without looking it up.

It might be especially notable that one of the alternative expressions for
sec x sec y is the square-root of the sum of two squares, a Pythagorean
distance.

Michael Ossipoff




On Mon, May 15, 2017 at 11:32 AM, Michael Ossipoff 
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  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 > > 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, 

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

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  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 
> 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 

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

Thanks, I'll check it out.

I used to be put off from the altitude dials by the noon inaccuracy. I was
concerned that Romans must have sometimes been late to noon appointments
and lunch-dates.

But I'd expect that, where lots of people are using altitude dials,
punctuality-critical events and appointments wouldn't be scheduled for
times near noon.

Altitude dials have the advantage that they combine a sundial with a
built-in indication of what time the sun will set.

Also, the polar-style dials, which need to be oriented by compass, or two
different kinds of dial (like a polar-style dial and an analemmatic dial)
on the same instrument, or a declination-measurement being correct--They
might be more demanding on the person using the dial, because it's
necessary to watch the compass, the declination-reading, or the time-match
of the two dials.

...So it seems as if an altitude dial might be easier to use. And the
pre-adjustable ones seem likely to be easier to use than a Shepard's dial
or Roman Flat altitude dial, with which you have to be concerned that the
gnomon-point-shadow is on right date's point on the lines.

But I've used compass tablet-dials, and mine were all accurate within 5
minutes, and one was usually accurate within 3 minutes.

I like the Marke dial too. I think you wrote an article about it in
Compendium.

But if I were to make a universal portable altitude dial for someone, I
think they'd prefer the easier declination and latitude adjustment of the
Regiomontanus dial.

The only sundials I've made have been those compass tablet-dials, but if I
have reason to make another portable dial, it will be an altitude dial.

Well, I might, at some time, make the Regiomontanus, Marke and
Ring-Equinoctical dials, just to find out what it's like to make and use
them, and to show them to my girlfriend.

I have a clever folded-cardboard version of the ring-equinoctical, from a
cut-out pattern in a book. It's on a 2X1 shaped cardboard, folded at the
middle to a right-angle, with a thread to limit it to that angle. A
quarter-circle of an equatorial sundial-scale is printed on the surfaces,
like a quarter of a Disk-Equatorial.

An edge serves as the polar style, A string-plumbline, used with one of the
degree-scales, facilitates the correct vertical tip, for the polar-gnomon.
Or one of the printed degree-scales can be used for vertical orientation by
sighting on the horizon, if the horizon is visible.

A small paper tab, sliding on one of the edges, and casting an edge-shadow
on a date-scale, gives a declination-reading that can be used to
azimuth-orient the dial.


Michael Ossipoff


On Sun, May 14, 2017 at 5:18 PM, Fred Sawyer  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 
> 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 

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

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. 

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

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 
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 

Re: Capuchin and Regiomontanus dials

[Michael Ossipoff]

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  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.)
>