Tag: solar eclipse

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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 luminaries1, 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 parallax1.

Toomer again provides an example that we can follow along with2. 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 Alexandria3. 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 methods4. Continue reading “Almagest Book VI: Predicting Solar Eclipses”

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Almagest Book VI: Eclipse Tables

Having spent five posts building eclipse tables, here’s the full tables. As usual, I’ve placed them in a Google Doc for easy access.

Do note that this table is broken up into four tabs.

Toomer notes that there are a number of errors in the table, but it’s not clear whether they originate with Ptolemy’s calculations or are a result of later transcription errors. Ones that can be confirmed as scribal errors he notes were corrected in the translation.

 

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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: Construction of the Eclipse Tables – Solar Eclipse Tables

In the first post on Book VI, I stated that, while we could calculate the position of the sun and moon every day to determine whether an eclipse was happening, we wanted to rule out as much as possible. To that end, we’ve spent most of our time trying to figure out when we do or do not need to worry about there being an eclipse. First we looked at determining mean conjunctions, then showed how to get from mean to true syzygy, then looked at how far away from a mean syzygy an eclipse could occur, and finally, in the last chapter, we looked at numerous periods to see whether or not they would be possible.

However, we’ve now run out of things that Ptolemy wants to rule out. As such, in what’s left, we’ll need to actually go through at least some of the calculations. Specifically, in this chapter we’re going to work on some tables that, if we input the argument of

the moon’s position in latitude [for a given syzygy, we will know] which of those syzygies will definitely produce an eclipse, as well as the magnitudes and times of obscuration for these eclipses.

Continue reading “Almagest Book VI: Construction of the Eclipse Tables – Solar Eclipse Tables”

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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 Seven Months

Having established that two solar eclipses separated by the five months from the same location are just barely possible, Ptolemy then works on whether it will be possible for the same to occur over a period of seven months concluding that it is possible, provided it happen in the “shortest $7$-month interval”1. Continue reading “Almagest Book VI: Solar Eclipses Separated by Seven Months”

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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: 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”