Although I only briefly mentioned it in this post, this past summer I’d taken the quadrant to Queen’s Prize which is Calontir’s novice level A&S competition held once during each reign. One of the challenges I faced was that the quadrant didn’t entirely fit in any of the categories. While it was designed to function like a period instrument, it was built using decidedly modern methods, and the goal of the quadrant wasn’t the instrument itself, but the measurements it could take. So the overall reaction from the judges was “super cool, but it’s hard to judge on its own merits.”
Thus, it was suggested that a more appropriate format for me might be a research paper. As the measurements from the quadrant1 is still a project in process, I knew I would need to do a paper on something else. While I was sorely tempted to simply print out all of my Almagest posts as one massive paper, I was inspired when writing this post on applications of the rising time tables. In particular, a member of my Barony, Padraigin, is interested in medieval astrology which makes use of rising signs; something that Ptolemy describes how to calculate. However, based on the work done thus far, it was only simple to do for latitudes given in the rising time tables. My goal was to try to condense the methodology for computing a rising time table at any latitude into a relatively short paper.
Ultimately this proved more difficult than I anticipated. My first thought was that instead of calculating a full rising time table, we could be very judicious about what was calculated, using the Menlaus configuration from this post, since it seemed it should work for almost any arc.
However, what quickly became apparent is that there were lengths of the ecliptic for which it simply did not work. For my test case, I’d selected a point where the sun was in Libra. Plugging that directly into the equation made no sense and it quickly became apparent that the formula did not work beyond 90º. However, that should still allow us to quickly calculate 0º-90º. And from there we could use symmetries to easily fill out the rest of the table.
Unfortunately, it became apparent that the equation also did not work for 90º. While I still don’t have a full explanation on that, I suspect it’s due to the two legs of the Menelaus configuration becoming one, and thus the configuration falling apart. Thus, I had to follow Ptolemy’s method a bit more closely than I’d have liked and to get that final 10º arc at 90º, it involved using the length of the longest day for the given latitude.
The trouble was, Ptolemy didn’t state how he got this. While it was something I could look up easily enough, there is a trouble that Ptolemy’s values aren’t always “true”2. Rather, they’re idealized such that the sun is a point. There’s also probably some other hidden assumptions in there as well which I’m sure we’ll learn more about in the next book which is all about the sun’s motion. Regardless, since the values Ptolemy uses don’t match with what would be reported, I needed to make sure I stayed within the context of Ptolemy’s methodology. Thus, I determined that a series of linear approximations between the points listed early in book II would be the best approach.
Ultimately the paper ended up being 13 pages, with an additional 17 pages of reference material which included a brief section on the mathematical techniques, as well as extensive reference tables.
However, instead of doing all the calculations for the example in the paper on a calculator, I reduced my work load by performing many of them in Excel. Which got me thinking whether or not it would be possible to leave the latitude as a variable, pointing to a reference cell which would then calculate the entire new rising time table automatically.
For the most part, this was relatively simple. It just made use of the equation from the Menelaus configuration. But as noted above, this fails when the arc of the ecliptic is 90º and the trick with the length of the longest day must be applied. But the method by which this was to be determined (the series of linear approximations from what Ptolemy had early in book II), would be a series of 31 nested IF statements. But with the help of an online statement generator, I got it working.
So now that Queen’s Prize is over, I’m sharing both the paper and the Excel calculator.
Paper: Computing Rising Sign – Queen’s Prize Spring 2019
Calculator: Rising Time Table Calculator
Overall, the event itself was quite enjoyable although it took some time to get into. The space wasn’t big enough, with rows of tables crammed uncomfortably close together and the volume level nearly giving me an anxiety attack. Fortunately, I was in the first judging session so I quickly had my attention diverted into being able to discuss the topic.
The three judges I had were John Boyer, Hugo von Harlo, and Viktor von Wedel. Of the three, Viktor was the only one that said he was able to follow the paper all the way through. John made it through, but didn’t follow the math and only did so to comment on formatting and things from an editorial standpoint. Hugo made a good effort, but the math is pretty challenging.
The surprise came when Viktor said he was familiar with the concept of rising time and challenged me to, using his information, calculate his rising time with the method provided in the paper which I was able to do correctly.
Afterwards, I spent the majority of the rest of the day working on the next section of the Almagest, stopping to chat with anyone else dropping by to ask questions. I had visitors that stopped to discuss their related topics, such as sun dials and how time was measured. Many people remembered the quadrant from last time and came by for an update on that side of this project, having no warning that there was this deep theoretical side.
And ultimately that’s what I was there for. For many, Queen’s Prize is an event to show off your first steps into a new endeavor and get feedback from subject matter experts. But given I don’t believe there’s anyone in the SCA that has attempted to do anything like my project, Queen’s Prize is a chance for me to demonstrate to others where I’ve gotten which is always fun since the project is rather personal otherwise.
All in all, a good day.