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Almagest Book II: Angle Between Ecliptic and Meridian – Angle Calculations

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.

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Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries

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:

  • The angles between the ecliptic and meridian
  • The ecliptic and horizon
  • The ecliptic and an arc from horizon to the zenith (an altitude circle)

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”

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Queen’s Prize Winter 2019 – Computing Astronomical Rising Signs for Any Latitude & Excel Calculator

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.

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Data: Stellar Quadrant Observations – 1/4/19

Last night was a nearly new moon. The temperature was uncommonly warm for this time of year1 but that didn’t end up disturbing the seeing too much as we had an exceptionally good night for observing. Combined with a rich field of stars in Taurus and Orion passing during the evening and Padraig and I had the most productive evening yet cataloging 59 stars, 16 of which were not in my original list of targets, but due to being able to see down to 5th magnitude, we were able to grab them.

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Almagest Book II: Applications of Rising-Time Tables

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:

Length of a Day

Seasonal Hours (Alternative Method)

Seasonal Hours to Equinoctial Hours

Horoscopes

Upper Culmination (Alternative Method)

Rising Point

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Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs

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.

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