Tag: Ptolemy

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Almagest Book VI: Table for Magnitudes of Solar and Lunar Eclipses – Solar Eclipse Example

Previously, when discussing eclipses, we’ve discussed the amount that is obscured in terms of “digits” where each digit is $\frac{1}{12}$ of the diameter of the object. However, Ptolmey indicates that not everyone necessarily estimates the magnitude of eclipses in this way, stating,

most of those who observe [eclipses]… measure the size of the obscuration, not by the diameters of the disks, but, on the whole, by the total surface area of the disks, since, when one approaches the problem naively, the eye compares the whole part of the surface which is visible with the whole of that which is invisible1.

To deal with this Ptolemy provides “another little table” which will allow us to convert between the linear diameter obscured and the area of either the sun, or moon. Continue reading “Almagest Book VI: Table for Magnitudes of Solar and Lunar Eclipses – Solar Eclipse Example”

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Almagest Book VI: Table of Eclipse Correction

Having computed tables that give information about eclipses for the sun and moon at both greatest and least distance, Ptolemy now turns to creating a table to estimate the impact of the moon being at other positions about the epicycle besides apogee and perigee.

This is to be done by a table that uses the distance from apogee about the epicycle as the input and returns what proportion of the difference between apogee and perigee one should use, expressed in sixtieths. The good news is that we’re not really going to have to do any calculations here because we’ve actually already done them when we were putting together our Parallax Table. Continue reading “Almagest Book VI: Table of Eclipse Correction”

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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: Solar and Lunar Eclipses Separated by Six Months

Continuing in the theme of checking as few as possible syzygies for eclipses, Ptolemy now turns his attention towards

the problem of intervals at which, in general, it is possible for ecliptic syzygies to occur, so that, once we have determined a single example of of an ecliptic syzygy, we need not apply our examination to the [ecliptic] limits to every succeeding syzygy in turn, but only to those which are separated [from the first] by an interval of months at which it is possible for an eclipse to recur.

Continue reading “Almagest Book VI: Solar and Lunar Eclipses Separated by Six Months”

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Almagest Book VI: Eclipse Limits for Solar Eclipses – Solar & Lunar Anomalies

So far, when considering the distance the sun/moon can be from one of the nodes, we’ve worked out how much the longitudinal and latitudinal parallax impact things and all that’s left now is the fact that the sun and moon aren’t always at their mean position. They both have anomalies which we’ll need to consider. This is because the big goal of this book, so far, is to reduce the amount of math we have to do when checking for an eclipse. While we could go through all the effort of calculating the true position, that’s extra steps. Wouldn’t it be nicer if we could just stop at the mean position if it’s not in the window in which an eclipse can occur?

To that end, our final step in this series of posts exploring the limits for solar eclipses is to translate the true positions to the mean positions.

Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Solar & Lunar Anomalies”

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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 $0;08º$ northwards2, [the moon] has a maximum longitudinal parallax of about $0;30º$ … and when its [latitudinal] parallax is $0;58º$ southwards3, 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 eclipse4. 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: 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”