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Almagest Book V: Lunar Parallax

Chapter 11 of Book 5 is one of those rare chapters that’s blessedly free of any actual math. Instead, Ptolemy gives an overview of the problem of lunar parallax, stating that it will need to be considered because “the earth does not bear the ratio of a point to the distance of the moon’s sphere.” In other words, the ratio of the diameter of the earth to the distance of the moon isn’t zero.

However, this does pose an interesting question. We’ve previously given the radius of the eccentre as $49;41^p$, but we haven’t given the radius of the Earth in the same units. Thus, how can this ratio even be taken to know this? Continue reading “Almagest Book V: Lunar Parallax”

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Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee

In the last post, we looked at how much the total equation of anomaly would change during syzygy due to the eccentre we added to the lunar model in this book, when the moon was at its greatest base equation of anomaly. As Ptolemy told us, it wasn’t much. However, there was a second effect that can also change the equation of anomaly, which was based on where we measure the movement around the epicycle from. Namely, the mean apogee instead of the true apogee. This has its maximum effect when the moon is near apogee or perigee so in this post, we’ll again quantify how much.

Let’s start off by building our diagram:

Continue reading “Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee”

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Almagest Book V: The Difference at Syzygies – Maximum Lunar Anomaly

Syzygy is one of those words that has popped up very little in the Almagest so every time it does, I’m always thrown off a bit1. Especially when Ptolemy is going to spend an entire chapter discussing a topic that has scarcely even come up. But here we have Ptolemy spending the entirety of chapter $10$, to demonstrate that these modifications we’ve made to the lunar model have a negligible effect because he fears readers might think it does since

the centre of the epicycle does not always … stand exactly at the apogee at those times, but can be removed from the apogee by an arc [of the eccentre] of considerable size, because location precisely at the apogee occurs at the mean syzygies, whereas the determination of true conjunction and opposition requires taking the anomalies of both luminaries into account.

Continue reading “Almagest Book V: The Difference at Syzygies – Maximum Lunar Anomaly”

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Data: Stellar Quadrant Observations – 12/9/2020

Yesterday, the highs here in Three Rivers climbed into the 60’s. It’s December. This isn’t normal. But the weather was absolutely stunning and not even the sort that whips around to bone-chilling cold the next day as it was again in the 60’s today. This brief stability in the weather brought clear, calm skies, so despite it being a work night, I knew I needed to get out to observe.

Fortunately, with it getting dark so early, I could observe for several hours and still be home by a reasonable hour, even if I headed out to the further site I like observing at, Danville Conservation Area. And boy what a night it was! Continue reading “Data: Stellar Quadrant Observations – 12/9/2020”

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Almagest Book V: Calculation of the Lunar Position

It’s been awhile since I’ve done an Almagest post so quick recap. In Book IV, we worked out a first model for the moon, which was a simple epicycle model inclined to the plane of the sun’s sphere. In this book, Ptolemy showed us that this model was insufficient as the moon’s speed varies more than should predict and so we added an eccentric as well as having the center of the eccentre rotate around the observer. Finally, we introduced the concept of a “mean apogee” which is the position on the epicycle we’ll need to measure from in order to do calculations.

That’s been a lot, but with all of this completed, we should now be able to use it to calculate lunar positions which Ptolemy walks us through in this chapter. Unfortunately, he does this in the form of a generic prescription of steps instead of a concrete example. Fortunately, Toomer provides an example2 that I’ll use to supplement Ptolemy’s narration. Continue reading “Almagest Book V: Calculation of the Lunar Position”

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Data: Stellar Quadrant Observations – 11/6/2020

This is normally the part of the year where it gets difficult to go observing due to extremely cold temperatures at night. However, this week has seen daily highs in the mid 70’s and overnight lows in the mid 50’s. So with clear skies, I knew I needed to get out and observe.

Unfortunately, we’re at roughly a third quarter moon so it rose around 9:30 and was high enough to start being problematic by a little after 10:00. As such, I didn’t feel like making the long drive to Danville and just headed out to Broemmelsiek. The skies there continue to grow more and more light polluted so I was limited to pretty bright stars as it was. Continue reading “Data: Stellar Quadrant Observations – 11/6/2020”

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Almagest Book V: Constructing the Lunar Anomaly Table

In our last post, we showed how it is possible to determine the equation of anomaly by knowing the motion around the epicycle and the double elongation. This, combined with the position of the mean moon3 gives the true position of the moon. As usual, Ptolemy is going to give us a new table to make this relatively easy to look up. But before doing so, Ptolemy wants to explain what this table is going to look like. Continue reading “Almagest Book V: Constructing the Lunar Anomaly Table”

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Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions

Now that we’ve revised our lunar model to include the position of the “mean apogee” from which we’ll measure motion around the epicycle, we need to discuss how we can use this to determine the true position of the moon.

As a general statement, we know how to do this: Take the position of the mean moon, determined by adding the motion since the beginning of the epoch, and add or subtract the equation of anomaly. The problem is that our revisions in this book mean the table for the lunar equation of anomaly we built in Book IV is no longer correct.

Instead, to determine the equation of anomaly, we’ll start with the motion around the epicycle4 and need to factor in the double elongation of the moon from the sun.

To see how to do so, let’s get started on a new diagram:

Continue reading “Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions”