Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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,

Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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.

Re: Capuchin and Regiomontanus dials

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

Re: Capuchin and Regiomontanus dials

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

Capuchin and Regiomontanus dials

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