Tag: sun

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Scholarly History of Commentary on Ptolemy’s Star Catalog: 137 CE – 1918 CE

I’m going to take a bit of a break from direct progress on the Almagest as we get to the star catalog. This is because there is, what I feel to be a fascinating and important discussion surrounding its legitimacy and I want to explore the history of this discussion, even though almost all of it is outside the range of the SCA period1. Namely, the discussion is whether or not Ptolemy’s star catalog is legitimate, one which he took the measurements himself, or if Ptolemy stole the data from an astronomer that came before him and tried to update it, but failed due to an incorrect value for the rate of precession.

Continue reading “Scholarly History of Commentary on Ptolemy’s Star Catalog: 137 CE – 1918 CE”

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Almagest Book IV: Adjustments to Intervals for Parallax

I stopped my previous post where I did because the material it covered is the end of the example problem Toomer provided. However, Ptolemy still has a few more paragraphs to go because

there is, in fact, a noticeable inequality in these intervals [of immersion/emersion] due, not to the anomalistic motion of the luminaries2, but to the moon’s parallax. The effect of this is to make each of the two intervals, separately, always greater than the amount derived by the above method, and, generally, unequal to each other.

In short, because the parallax changes over the course of the eclipse, it will cause the immersion and emersion durations to be longer than they would otherwise be.

We shall not neglect to take this into account, even if it is small.

Then let’s get to it. Continue reading “Almagest Book IV: Adjustments to Intervals for Parallax”

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Almagest Book VI: Predicting Solar Eclipses

Now that we understand how to predict lunar eclipses, we’ll turn our attention towards solar eclipses. However, Ptolemy warns us that these will be

more complicated to predict because of lunar parallax3.

Toomer again provides an example that we can follow along with4. This will be Example $12$ from Appendix A. Surprisingly, nowhere in the Almagest does Ptolemy describe the details of a solar eclipse. As such, Toomer has selected his own example. In this case, we are to determine the details of the solar eclipse of June $16$, $364$ CE (Nabonassar $1112$ in the month of Thoth), which was observed by Theon of Alexandria5. Upon observing the eclipse, Theon then followed Ptolemy’s methods in the Almagest and Handy Tables to compare the predictions against observations and his calculation are what Toomer follows as an example using Ptolemy’s methods6. Continue reading “Almagest Book VI: Predicting Solar Eclipses”

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Almagest Book VI: Solar Eclipses Separated by One Month

We have finally reached the final in this run of eclipse timing feasibility checks. In it Ptolemy wants to demonstrate that it is impossible to have two eclipses separated by one month

even if one assumes a combination of conditions which could not in fact all hold true at the same time, but which may be lumped together in a vain attempt to provide a possibility of the event in question happening.

In short, we’re going to assume an overly ambitious “best case” scenario which can’t actually happen because some of these best case conditions contradict one another. Continue reading “Almagest Book VI: Solar Eclipses Separated by One Month”

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Almagest Book VI: Solar Eclipses Separated by Five Months

Ptolemy next looks at whether or not it is possible for a solar eclipse to occur five months after a previous one. We’ve already done a fair bit of the heavy lifting for this topic as some of the math we did when considering lunar eclipses separated by five months will still apply. In that post, we determined that the moon would have moved on its inclined circle by $159;05º$ between true conjunctions. This does require we adopt the same assumptions of the sun moving its greatest distance and the moon moving its least.

What we’ll need to focus on for this post is redoing the eclipse limits for the situation in question. Continue reading “Almagest Book VI: Solar Eclipses Separated by Five Months”

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Almagest Book VI: Lunar Eclipses Separated by Seven Months

So far in this chapter, we’ve demonstrated that it’s quite possible that, if either a solar or lunar eclipse occurs, there will be another one six months later. Then, in the last post we showed that, if a lunar eclipse happens, it’s possible (although unlikely) that there can be another one five months later. In this post, we’ll explore whether or not, for lunar eclipses, you can have two separated by seven months.

The procedure will be the same as in the last post. First we’ll determine the motion of the luminaries, taking into account the discrepancies due to the anomalies, as well as the differences between mean and true syzygy, and then compare those to the eclipse windows which we’ll again recalculate for this situation. Continue reading “Almagest Book VI: Lunar Eclipses Separated by Seven Months”

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Almagest Book VI: Lunar Eclipses Separated by Five Months

In the previous post, we showed that , if a solar or lunar eclipse occurs, it is possible that another may occur six months later. Now, we’ll turn to ask whether or not another lunar eclipse can happen five months after a previous one. To answer this question, we’ll first work out how much the moon would have moved in that time period and then compare that to the eclipse window.. Continue reading “Almagest Book VI: Lunar Eclipses Separated by Five Months”

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

Now that we’ve determined how much further from the nodes parallax can cause solar eclipses to occur due to the latitudinal parallax, we need to consider the longitudinal effect. As with the last post, Ptolemy is absolutely no help in this. He simply tosses out some values with no explanation or work stating

When [the latitudinal] parallax is $0;08º$ northwards7, [the moon] has a maximum longitudinal parallax of about $0;30º$ … and when its [latitudinal] parallax is $0;58º$ southwards8, it has a maximum longitudinal parallax of about $0;15º$…

Seeking some assistance, I again refer to Neugebauer and Pappus, but immediately run into an issue. Neugebauer minces no words and states

Ptolemy is wrong in stating that $p_\lambda = 0;30º$ and $p_\lambda = 0;15º$ are the greatest longitudinal components of the parallax for locations between Meroe and the Borysthenes. It is difficult to explain how he arrived at this result.

Well… this will be interesting to try to untangle then. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Longitudinal Parallax”

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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 eclipse9. 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”