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Commentaries of Pappus and Theon of Alexandria on the Almagest – A. Rome (1931)

Towards the end of $2021$, I got stuck on a particular calculation and it took me over a month and a half to resolve. Frustratingly, Ptolemy showed no work, Neugebauer made no comment (which generally indicates he found no fault in the calculation), and Pedersen skipped the chapter entirely. It seemed there was little to no help available save a footnote in the Toomer translation I’m using which stated

a somewhat unsatisfactory numerical verification of [the calculation] (using the Handy Tables) is in Pappus’ commentary (Rome[$1$] $232-4$).

The citation here is to a text entitled Commentaires de Pappus et de Theon d’Alexandrie sur l’Almageste(Commentaries of Pappus and Theon of Alexandria) by Adolphe Rome.

Pappus was a $4^{th}$ century astronomer/mathematician so my hope was that the source by Rome that Toomer was citing was a translation. As was obvious from the title, the work was written in French and I’d taken enough French that I hoped I would be able to muddle through a translation and so I requested the text through an interlibrary loan.

Continue reading “Commentaries of Pappus and Theon of Alexandria on the Almagest – A. Rome (1931)”

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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: 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”