Tracing the history of medieval astronomy
In this next chapter, we’ll be looking at the angle between the ecliptic and horizon. Obviously, this one will depend on the latitude so there will be more complexity than the previous chapter. But as with before, we’ll simplify the process by starting off with some symmetries.
Continue reading “Almagest Book II: Angle Between Ecliptic And Horizon – Symmetries”
Now that we’ve gotten a few symmetry rules developed, we can return to the main objective of calculating the angle between the ecliptic and meridian at different points along the ecliptic. Specifically, Ptolemy sets out to do this at the first point in every sign. But thanks to the previously derived symmetries, we’ll save ourselves a bit of work.
First Ptolemy does some very short proofs for these angles at the meridian and solstice, and then a slightly more complex one for the signs between them.
Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Angle Calculations”
In my last post, I mentioned that entered a paper based on the rising sign calculations presented in this post into an A&S competition. This was a very interesting piece to do because it showed how well woven the roots are, as doing so made use of almost every section we’ve gone through previously. As such, it felt like a good capstone for book II. But it doesn’t end there.
Rather, Ptolemy decides to go on for several more chapters as this book is focused on the great circles on the celestial sphere. While we’ve covered the ecliptic and celestial equator pretty extensively, we have done less with the horizon and meridian which is where Ptolemy seeks to go for the last few chapters in this book. Specifically, we’ll be covering:
All followed by another summary chapter at various latitudes. As the title of this post may have indicated, we’ll be covering the first of these in this post1. Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries”
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 quadrant2 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.
At this point we’ve spent some considerable time doing the work to develop our rising time tables. Now Ptolemy answers the question: What can we do with them?
Ptolemy provides several algorithms:
Seasonal Hours (Alternative Method)
Seasonal Hours to Equinoctial Hours
Upper Culmination (Alternative Method)
Continue reading “Almagest Book II: Applications of Rising-Time Tables”
Over the past several posts we have worked towards an understanding of the rising times of arcs of the ecliptic at various latitudes. With this work complete, Ptolemy presents a table of these rising times in 10º intervals for select latitudes. Continue reading “Almagest Book II: Rising Time of the Ecliptic Tables”
With the previous theorem about the ascensional differences complete, it’s time to move on to determine how to figure out the rising time of arcs of the ecliptic for 10º segments at various latitudes using what Ptolemy promises to be a shortcut in the math. In the modern sense it really doesn’t seem to be much of a shortcut, but that’s because with the assistance of calculator’s, the equations we were using previously seem much less daunting. If it had to be done by hand, I’m sure it would be far more tedious.
Instead, Ptolemy reduces the number of calculations by going through the proof regarding ascensional differences as well as making use of some previously calculated values to avoid having to do other calculations.
To get started, Ptolemy revises the previous drawing, making it a bit simpler by removing the ecliptic and renaming a few of the points, as well as changing a few definitions.
Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs”
Not content to simply figure out how long it would take a zodiacal constellation to rise at latitudes other than the equator, Ptolemy sets out to further divide the ecliptic into 10º arcs and he’s promised an easier method than what we’ve done previously. But before we can get there, Ptolemy gives a brief proof which he’ll make use of later.
To start, we begin with the vernal equinox on the horizon:
In the last post, we explored the rising time for one zodiacal sign which would comprise 30º of the ecliptic. Since the preliminary math is now out of the way, we can quickly do 60º of the ecliptic which constitutes Aries and Taurus. From there, we’ll be able to more quickly compute the remaining constellations as well. Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for Remaining Arcs”
Now that we’ve done a bit of preliminary theory, we’re ready to jump into an example problem. As with before, Ptolemy selects Rhodes at a latitude of 36º. Since we’ve been seeing several of the same diagrams here, I’m going to forego the buildup and give a mostly completed diagram:
Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua”