Tag: Almagest

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Almagest Book VI: Angles of Inclination at Eclipses

The final few chapters of Book VI are rather odd. Now that we’ve completed the discussion of eclipse prediction, Ptolemy wants to do an “examination of the inclination which are formed at eclipses.” However, he doesn’t appear to provide any motivation for doing so. Toomer and Neugebauer both indicate that the actual reason was likely weather prediction1, but the Almagest doesn’t contain any information on how this is to be used. Neugebauer indicates that,

[T]he technical term connected with this problem is “prosneusis”…developed from the original meaning of the verb νευειν (to nod, to incline the head, etc…). According to the terminology of hellenistic astrology, the planets or moon can, e.g., give their consent by “inclining” toward a certain position, i.e., by being found in a favorable configuration.

However, aside from these astrological purposes, these last few chapters are essentially left as a free-floating bit of material. Continue reading “Almagest Book VI: Angles of Inclination at Eclipses”

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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: Predicting Lunar Eclipses

Having set out the above as a preliminary, we can predict lunar eclipses in the following manner.

As Ptolemy states in opening this chapter, we’re finally done with the preliminary work and we’re ready to start diving into how to actually use everything we’ve done to predict eclipses. As usual, Ptolemy walks us through the steps, but does not provide an example, so I will follow my usual procedure of using example $11$ in Appendix A of Toomer’s translation7.In that example, Toomer invites us to examine lunar eclipses around Nabonassar 28, in the month of Thoth (the first month of the Egyptian year). Continue reading “Almagest Book VI: Predicting Lunar 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 – Lunar Eclipse Example

Having  completed an example calculation for converting linear digits to area digits in the previous post, we’ll now do the same calculation for a lunar eclipse. The good news is the setup is the same. While I don’t strictly need to redraw the diagram, I’m going to anyway because the earth’s shadow is so much larger than the moon and drawing it as such helps me visualize things mentally although the respective position of the points doesn’t change at all.

Continue reading “Almagest Book VI: Table for Magnitudes of Solar and Lunar Eclipses – Lunar Eclipse Example”

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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 invisible8.

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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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)”