Tag: Almagest

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Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax

Now that we’ve determined how far away from the nodes a lunar eclipse can occur, we’ll work on doing the same for a solar eclipse1. But before diving in, I want to say that this has been one of the most, if not the most challenging section of the the Almagest so far. One of the primary reasons is that Ptolemy shows no work and gives almost no explanation on how he did this. When such things happen, I often turn to Neugebauer’s History of Ancient Mathematical Astronomy which I did in this case. There, Neugebauer refers to Pappus of Alexandria, a fourth century mathematician who did commentary on the Almagest and walks through a process that arrives at the same values as Ptolemy.

However, there was a very large amount to unpack in just a few pages there and, unlike most cases where I can simply work along with it and see where things are going, this time I had to really understand the whole process before the first steps made any sense. This led me to agonize over what was going on with those first steps, amounting to several days of effort and rewriting this post from scratch several times. The result is twofold. First because I feel this section can only be approached by understanding the methodology before diving into the math, there’s going to be far more exposition than normal and, as a result, this is likely to be one of my longer posts. Second, the struggles I had with trying to understand the method and rewriting this post so many times has left me with a lot of fragments of thoughts in my brain and in the blog editor. I’ve done my best to clean it up, and maybe it’s just those thoughts swirling around in my brain, but this post just doesn’t feel as coherent as I like. Apologies in advance if you struggle to follow. Know I did as well.

Anyway, moving on to the topic at hand.

Normally, I like to start with a quote from Ptolemy to give us some direction, but I think Ptolemy did such a poor job of laying this section out, I’m going to avoid doing so for the majority of the post. Instead, let’s try to understand the process by recalling what we did with the moon and discussing how things will change. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax”

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Almagest Book VI: Lunar Diameter and Earth’s Shadow at Perigee During Syzygy

In the last post, we explored how to make use of the table of mean syzygies to calculate the true syzygies. However, that chapter focused mostly on finding the time when the moon and sun would have either the same or exactly opposite ecliptic latitude. But what got left by the wayside was the lunar ecliptic latitude. We did a bit of work on calculating the argument of it but, aside from my mention of it in the afterword of the post, we never really completed that calculation. And eclipses of either type cannot truly occur unless the lunar ecliptic latitude is reasonably close to zero.

So we could calculate the ecliptic latitude of the moon for every conjunction and opposition but instead, Ptolemy decides we should first do a bit of a sanity check before getting any more involved. To do so, Ptolemy wants to examine how far from a node is it even possible for the ecliptic longitude of the syzygy to occur and still have an eclipse. If it’s outside of these limits, then no further calculation is necessary. To do this, Ptolemy is going to need to know some additional values. In this post, we’ll explore the angular diameter when the moon is at the perigee of its epicycle at syzygy2 as well as determining the width of Earth’s shadow at that distance. Continue reading “Almagest Book VI: Lunar Diameter and Earth’s Shadow at Perigee During Syzygy”

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Almagest Book VI: How to Determine the Mean and True Syzygies

Now that we’ve created our table of conjunctions and oppositions, how do we go about using it? As usual, Ptolemy walks through the process in a vacuum, so to help, I’ll follow along with the example Neugebauer does in History of Ancient Mathematical Astronomy on pages $123-124$, although somewhat slimmed down. In particular, I’ll walk through finding the true opposition from the year $718$ in the epoch for the first opposition in the year. Continue reading “Almagest Book VI: How to Determine the Mean and True Syzygies”

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Almagest Book VI: Construction of the Table of Mean Syzygies

As promised in the last chapter, Ptolemy’s first task in eclipse prediction is going to be laying out a table of mean syzygies around which eclipses might be possible, so we can check those to see if an eclipse might occur instead of performing useless calculations where the sun and moon are nowhere near a syzygy. In this post, we’ll go over the construction of that table! Continue reading “Almagest Book VI: Construction of the Table of Mean Syzygies”

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Almagest Book VI: On Conjunctions and Oppositions of Sun and Moon

Finally we’re on Book VI. So far in the Almagest, we’ve had a few books which laid out some preliminary tables and concepts, a book on the sun, and two on the moon3. Now it’s time to put the sun and the moon together to start looking at some of the most dramatic astronomical phenomena: eclipses. To introduce this topic, Ptolemy begins with an uncharacteristically short chapter which is a single paragraph. Continue reading “Almagest Book VI: On Conjunctions and Oppositions of Sun and Moon”

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Almagest Book V: Components of Parallax – Correction Example

Now that Ptolemy has straightened us out regarding which angle we need to be using, Ptolemy sets out

a convention method for making the above kind of correction of the angles and arcs, if anyone wants to make it when the [differences] involved are so small.

Fortunately, we can use the same diagram we ended with last time which was the generic case:

Ptolemy lays his method on us in his familiar brick of text: Continue reading “Almagest Book V: Components of Parallax – Correction Example”

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Almagest Book V: Components of Parallax – Corrections

At the end of the last post, we noted that Ptolemy wasn’t quite satisfied with what we did previously because we used some rather faulty assumptions.

As Ptolemy states it:

For lunar parallaxes, we considered it sufficient to use the arcs and angles formed by the great circle through the poles of the horizon [i.e., an altitude circle] at the ecliptic, instead of those at the moon’s inclined circle. For we saw that the difference which would result at syzygies in which eclipses occur is imperceptible, and to set out the latter would have been complicated to demonstrate and laborious to calculate; for the distance of the moon from the node is not fixed for a given position of the moon on the ecliptic, but undergoes multiple changes in both the amount and relative position.

The key phrase here is the “at the ecliptic, instead of the moon’s inclined circle.” This got swept under the rug in that post because Ptolemy didn’t really explain why the algorithm he gave us should work. So to understand, let’s start by taking a harder look at what’s actually going on.

Continue reading “Almagest Book V: Components of Parallax – Corrections”

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Almagest Book V: Components of Parallax – A First Approximation

So far in this chapter, we’ve explored how to use the table of parallax to calculate the parallax of the moon and sun by knowing its distance from the zenith. But this in and of itself isn’t particularly useful. For example, when we did a sample lunar parallax calculation, we determined in that situation, the moon was about $1;10º$ off from its true position. But $1;10º$ in which direction?

Thus, the next step will be to break that down into its components, determining how far off in both ecliptic latitude and longitude the parallax makes the moon appear. Ptolemy again gives steps, but no example, so we’ll continue the previous example we did for the moon, following Neugebauer4. Continue reading “Almagest Book V: Components of Parallax – A First Approximation”

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Almagest Book V: Parallax Table

In the previous chapter, we worked out the parallax for the sun and the moon at 4 different positions. Next, we determined how we might make some corrections for the moon if it were between the positions we described, breaking the corrections into ones for the epicycle and for the eccentre. From that work, Chapter $18$ presents the results in tabular format. Since this one is pretty wide, I’ve again made the decision to put it into a Google Sheet.

Continue reading “Almagest Book V: Parallax Table”