Tag: Ptolemy

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Almagest Book VII: On the Relative Positions of the Stars

Now, to deal with the next part of the theory, we shall begin discussing the stars, and first, in accordance with the logical order, the so-called fixed stars.

To introduce the next book of the Almagest, Ptolemy informs the reader that we will be taking some time to understand the fixed stars. There are two important things to note about this sentence. The first is that Ptolemy is taking extra care to make sure the reader knows we are discussing the fixed stars. This is because the Greek concept of “stars” could include the planets. Continue reading “Almagest Book VII: On the Relative Positions of the Stars”

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Almagest Book VI: Determining the Inclinations

And here we are. Final chapter of Book VI. Here, we’ll finish out Ptolemy’s discussion of the inclinations of eclipses. In the previous post, we discussed a few things we’ll need to determine that1, but we really haven’t discussed is what the “inclination” means. So I’ll begin by doing that as Ptolemy doesn’t really do a clear job of doing so. Continue reading “Almagest Book VI: Determining the Inclinations”

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Almagest Book VI: Horizon Diagram

In the previous chapter, we mentioned that one of the things we’ll need for Ptolemy’s weather prediction is the angle between the ecliptic and celestial equator as measured along the horizon. This is something we calculated back in II.2 and only now are we getting around to displaying a diagram to summarize it. So I’ll display the diagram here2 and discussion about it will be beneath the fold.

Continue reading “Almagest Book VI: Horizon Diagram”

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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 prediction3, 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 luminaries4, 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 parallax5.

Toomer again provides an example that we can follow along with6. 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 Alexandria7. 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 methods8. 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 translation9.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”